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Dr4noel
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  1. \documentclass[11pt, twoside, letterpaper, leqno]{article}
  2.  
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  4. \usepackage[top=3.30cm, bottom=3.87cm, left=30mm, right=30mm]{geometry}
  5. \usepackage[utf8]{inputenc}
  6. \usepackage{graphicx}
  7. \usepackage{titling}
  8. \usepackage{relsize}
  9. \usepackage{enumitem}
  10. \usepackage{textcomp}
  11. %mathematical packages
  12. \usepackage{amscd}
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  14. \usepackage{amsthm}
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  17. \usepackage{mathrsfs}
  18. %---------------------------
  19. \usepackage{blindtext}
  20. \usepackage[indentafter]{titlesec}
  21. \usepackage{abstract}
  22. %-----------HEADERS AND FOOTERS-------------
  23. \usepackage{fancyhdr}
  24. \renewcommand{\headrulewidth}{0pt}
  25. \pagestyle{fancy}
  26. \fancyhf{}
  27. \fancyhead[LE,RO]{\footnotesize{\thepage}}
  28. \fancyhead[CE]{\scriptsize{NIKOLAOS S. PAPAGEORGIOU, VICEN\c{T}IU D. R\u{A}DULESCU, AND DUSAN REPOVS}}
  29. \fancyhead[CO]{\scriptsize{POSITIVE SOLUTIONS FOR THE ROBIN P-LAPLACIAN PLUS AN INDEFINITE POTENTIAL}}
  30. \fancyfoot[CE,CO]{\thepage}
  31.  
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  40. %section like the ones in amsart
  41. \titleformat{name=\section}{}{\thetitle.}{0.8em}{\centering\scshape}
  42. \titleformat{name=\subsection}[runin]{}{\thetitle.}{0.5em}{\bfseries}[.]
  43. \titleformat{name=\subsubsection}[runin]{}{\thetitle.}{0.5em}{\itshape}[.]
  44. \titleformat{name=\paragraph,numberless}[runin]{}{}{0em}{}[.]
  45. \titlespacing{\paragraph}{0em}{0em}{0.5em}
  46. \titleformat{name=\subparagraph,numberless}[runin]{}{}{0em}{}[.]
  47. \titlespacing{\subparagraph}{0em}{0em}{0.5em}
  48. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  49.  
  50. %New commands
  51. \newtheorem{theorem}{Theorem}
  52. \newtheorem{remark}{Remark}
  53. \newtheorem{prop}{Proposition}
  54. \newtheorem{lemma}[theorem]{Lemma}
  55. \newtheorem{corollary}{Corollary}
  56. %%%%%%%%%%%%%%%%%%%%%%555
  57.  
  58. %text values change
  59. \textwidth=6in \textheight=664pt
  60. %\headheight=12pt \marginparsep=10pt
  61. %\footskip=30pt \hoffset=0pt \headsep=25pt \textwidth=426pt
  62. %\marginparwidth=50pt \voffset=9pt
  63. %\oddsidemargin=13pt
  64. %\oddsidemargin=0.5cm \evensidemargin=0.5cm
  65.  
  66. %start of the documents
  67. \begin{document}
  68. %equation space above and below
  69. \setlength{\abovedisplayskip}{5pt}
  70. \setlength{\belowdisplayskip}{5pt}
  71. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  72.  
  73. \input{titlepage}
  74. \vspace{.3cm}
  75. \section{\normalsize{Introduction}}
  76. Let $\Omega\subseteq\mathbb{R}^{\mathbb{N}}$ be a bounded domain with with a $C^2$-boundary $\partial\Omega$. In this paper we study the following nonlinear parametric Robin problem
  77. \begin{equation}
  78. \left\{
  79. \begin{array}{ll}
  80. \mbox{-}\Delta_p u(z) + \xi(z)u(z)^{p-1} - f(z,u(z),\lambda) + g(z,u(z)) \ \mbox{in} \ \Omega, \\
  81. \mathlarger{\frac{\partial u}{\partial n_p}} + \beta(z)u(z)^{p-1} = 0 \ \mbox{on} \ \partial\Omega, u > 0, \lambda > 0.
  82. \end{array}
  83. \right\}
  84. \tag{$P_{\lambda}$}
  85. \label{eqp}
  86. \end{equation}
  87.  
  88. In this problem $\Delta_p$ denotes the p-Laplacia \let\thefootnote\relax\footnote{{\textit{Key words and phrases}. p-Laplacian, local minimizers, strong comparison, positive solutions, nonlinear regularity, minimal solution, indefinite potential\\
  89. \indent\indent 2010 AMS Subject Classification 35J20, 35J60}}
  90. \addtocounter{footnote}{-1}\let\thefootnote\svthefootnote differential operator defined by
  91. \begin{equation*}
  92. \Delta_p u = \ \mbox{div} (|Du|^{p-2}Du) \ \mbox{for all} \ u \in W^{1,p}(\Omega), 1 < p < \infty.
  93. \label{eqld}
  94. \end{equation*}
  95.  
  96. The potential function $\xi \in L^\infty(\Omega)$ is in general indefinite (that is, sign changing). Therefore the differential operator (left hand side of $(P_\lambda)$) need not be coercive. In the reaction (right hand side of $(P_\lambda)$), we have the competing effects of two terms. The first is a parametric function which is strictly $(p - 1)- \ \mbox{sublinear near} \ + \infty$. The second function (the perturbation of the parametric term), is $(p - 1) - linear \ \mbox{near} \ +\infty$. Both functions are Caratheodory ($ \mbox{that is, for all} \ x \in \mathbb{R} \ z \rightarrow f(z, x, \lambda), g(z, x)$) are measurable and for all $z \in \Omega x \rightarrow f(z, x, \lambda), g(z, x)$ are continuous. In the boundary condition $\mathlarger{\frac{\partial u}{\partial n_p}}$ denotes the conormal derivates of u, defined by extesion on the map
  97.  
  98. \vspace{-0.3cm}
  99.  
  100. \begin{equation*}
  101. C' (\overline\Omega) \ni u \rightarrow |Du|^{p-2}(Du,n)_\mathbb{R^\mathbb{N}} = |Du|^{p-2}\mathlarger{\frac{\partial u}{\partial n}},
  102. \label{eqcdu}
  103. \end{equation*}
  104.  
  105. \vspace{-0.25cm}
  106.  
  107. \noindent with $n(\cdot)$ being the outward unit normal on $\partial\Omega$.
  108.  
  109. Our aim here is to study the nonexistance, existence and multiplicity of positive solutions for problem $(P_\lambda)$ as the parameter $\lambda$ moves on the positive semiaxis $(0, +\infty)$. We prove a bifurcation-type result for large values of the parameter. More precisely, we show that there is a critical parameter value $\lambda^* > 0$ such that
  110.  
  111. \begin{itemize}[noitemsep, topsep=0pt]
  112. \setlength{\itemindent}{.2in}
  113. \item\indent for all $\lambda > \lambda^*$ problem $P_\lambda$ has at least two positive solutions;
  114. \item\indent for all $\lambda = \lambda^*$ problem $P_\lambda$ has at least one positive solution;
  115. \item\indent for all $0 < \lambda < \lambda^*$ problem $P_\lambda$ has no positive solutions;
  116. \end{itemize}
  117.  
  118. Moreover, we show that for every admissible parameter $\lambda \in [\lambda^*, +\infty)$, problem $(P_\lambda)$ has a smallest positive solution $\overline{u}_\lambda$ and we examine the continuity and monotocicity properties of the map $\lambda \rightarrow \overline{u}_\lambda$.
  119.  
  120. The first such bifurcation-type result for parametric elliptic equations with competing nonlinearities, was prove by Ambrosetti-Brezis-Cerami \cite{2} (semilinear Dirichlet problems with concave-context reaction). Their work was extended to Dirichlet p-Laplace equations by Garcia Azorero-Manfredi-Peral Alonso \cite{6}, Guo-Zhang \cite{9}, Hi-Papageorgiou \cite{10}. For equations of logistic type we have the works of Radulescu-Repovs \cite{19} (semilinear Dirichlet problems) and Cardinali-Papageorgiou-Rubbioni \cite{4} (nonlinear Neumann problems). For Robin problems, we mention the work of Papageorgiou-Radulescu \cite{15}. In all aforementioned works the differential operator is coercive and the reaction has a different pair of competing nonlinearities. Here we have a new class of competition phenomena, which lead to bifurcation type results. In fact the behaviour of the set of positive solutions as the parameter $\lambda > 0$ varies, is similar to that of superdiffusive logistic equations, since the "bifurcation" occurs for large values of $\lambda > 0$.
  121.  
  122. Our method of proof uses variational tools from critical point theory together with suitable truncation, perturbation and comparison arguments.
  123.  
  124. $$\sum_{i = 0}^n$$
  125.  
  126. Here will be just a paracgraph to prove that I used $\sum_{i = 0}^n\limits$ in a paragraph
  127.  
  128. \section{\normalsize{Mathematical Background-Hypotheses}}
  129.  
  130. Suppose that $X$ is a Banach space. By $X^*$ we denote the topological dual of $X$ and by $\left\langle \cdot, \cdot \right\rangle$ the duality brackets for the pair $(X^*, X)$. Given $\varphi \in C'(X, \mathbb{R})$ we say that $\varphi$ satisfies the "Palais-Smale condition" (the "PS-condition" for short) if the following property holds:
  131.  
  132. \vspace{-0.3cm}
  133.  
  134. \begin{center}
  135. "Every sequence $\left\{u_n\right\}_{n\geqslant1} \subseteq X$ such that \\ $\left\{\varphi(u_n)\right\}_{n\geqslant1} \subseteq \mathbb{R}$ is bounded and $\varphi'(u_n) \rightarrow 0 \ \mbox{in} \ X^* \ \mbox{as} \ n \rightarrow \infty$, \\ admits a strongly convergent subsequence".
  136. \end{center}
  137.  
  138. \vspace{-0.3cm}
  139.  
  140. This is a compactness-type condition on the functional $\varphi$. Using this condition, one can prove a deformation theorem from which follows the minimax theory for the critical values of $\varphi$. Prominent is that theory, us the so-called "mountain pass theorem" which we recall here because we will use it in the sequel.
  141.  
  142. \begin{theorem}
  143. $\textit{If} \ \varphi \in C' (X, \mathbb{R}) \ \textit{satisfies the PS-condition}, \ u_0, u_1 \in X, ||u_1 - u_2 || > \rho > 0$, \\
  144. \centerline{$\max \left\{\varphi(u_0),\varphi(u_1)\right\} < \ \inf \left[\varphi(u) : ||u - u_0|| = \rho\right] = \eta_\rho$}
  145. $and \ c = \inf_{\gamma\in\Gamma}\limits\max_{0\leqslant t \leqslant 1}\limits \varphi(\gamma(t)) \ with \ \Gamma = \left\{\gamma \in C(\left[0,1\right],X) : \gamma(0) = u_0, \gamma(1) = u_1 \right\}, \ then \ c\geqslant\eta_\rho \\ \mbox{and c is a critical value of} \ \varphi \ (\mbox{that is, we can find} \ \widehat{u} \in X \ \mbox{such that} \ \varphi'(\widehat{u}) = 0 \ and \ \varphi\widehat{u} = c)$.
  146. \end{theorem}
  147.  
  148. \begin{remark}
  149. We mention that if $\varphi' = A + K$ with $A : X \rightarrow X^*$ a continuous map of type $(S)_+$ (that is, if $u_n \xrightarrow{w} u$ in $X \ and \ \limsup_{n\rightarrow\infty}\limits\left\langle A(u_n), u_n - u \right\rangle \leqslant 0, \ then \ u_n \rightarrow u \ in \ X)$ and $K : X \rightarrow X^*$ is completely continuous (that is, if $u_n \xrightarrow{w} u \ in \ X, \ then \ K(u_n) \rightarrow K(u) \ in \ X^*$), then $\varphi$ satisfies the PS-condition (see Marano-Papageorgiou \cite{13}, Proposition 2.2). This is the case in our setting.
  150. \end{remark}
  151.  
  152. The analysis of problem $(P_\lambda)$ involves the Sobolev space $W^{1,p}(\Omega)$, the Banach space $C'(\overline{\Omega})$ and the "boundary" Lebesgue space $L^p(\partial\Omega)$.
  153.  
  154. By $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$ defined by
  155.  
  156. $$||u|| = \left[||u||_p^p + || Du||_p^p\right]^\frac{1}{p} \ \mbox{for all} \ u \in W^{1,p}(\Omega).$$
  157.  
  158. The space $C'\Omega$ is an ordered Banach space with positive (order) cone
  159.  
  160. $$C_+ =\left\{u \in C'(\overline{\Omega}) : u(z) \geqslant 0 \ \mbox{for all} \ z \in \overline{\Omega} \right\}.$$
  161.  
  162. This cone has a nonempty interior given by
  163.  
  164. $$ int C_+ = \left\{u \in C_+ : u(z) > 0 \ \mbox{for all} \ z \in \Omega, \frac{\partial u}{\partial n}|_{\partial\Omega\cap u^{-1}(0)} < 0 \right\} $$
  165.  
  166. We will also use the set $D_+ \subseteq C_+$ defined by
  167.  
  168. $$D_+ = \left\{u \in C_+ : u(z) > 0 \ \mbox{for all} \ z \in \overline{\Omega}\right\}.$$
  169.  
  170. Evidently $D_+$ is open in $C'(\overline\Omega)$ and $D_+ \subseteq intC_+$. In fact $D_+$ is the interior of $C_+$ when $C'(\overline\Omega)$ is furnished with the coarser relative $C(\overline\Omega)$-norm topology.
  171.  
  172. On $\partial\Omega$ we introduce the $(N - 1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. Using $\sigma(\cdot)$ we can define in the usual way the boundary Lebesgue spaces $L^q(\partial\Omega), 1 \leqslant q \leqslant \infty$. From the theory of Sobolev spaces, we know that there exists a unique continuous linear map $\gamma_0 : W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$, know as the "trace map", such that
  173.  
  174. $$\gamma_0(u) = u|_{\partial\Omega} \ \mbox{for all} \ u \in W^{1,p}(\Omega)\cap C(\overline\Omega).$$
  175.  
  176. So, the trace map gives meaning to the notice of "boundary values" for any Sobolev function. The trace map is not surjective (in fact $im\gamma_0 = W^{\frac{1}{p'},p}(\partial\Omega)$ with $\frac{1}{p} + \frac{1}{p'} = 1$) and $ker\gamma_0 = W_0^{1,p}(\Omega)$. Moreover, $\gamma_0$ is compact into $L^q(\partial\Omega) \ \mbox{for all} \ q \in [1, \mathlarger{\frac{(N - 1)p}{p}}]$ if $p < N$ and into $L^p(\partial\Omega) \ \mbox{for all} \ 1 \leqslant q < \infty$ if $N \leqslant p$. In the sequel, for the sake of notational simplicity, we drop the use of the trace map $\gamma_0$. All restrictions of Sobolev functions on $\partial\Omega$ are understood in the sense of traces.
  177.  
  178. Let $A : W^{1, p}(\Omega) \rightarrow W^{1, p}(\Omega)^*$ be the nonlinear map defined by
  179.  
  180. $$\left\langle A(u), h\right\rangle = \int_\Omega |Du|^{p-2}(Du, Dh)_{\mathbb{R}^\mathbb{N}}dz \ \mbox{for all} \ u, h \in W^{1, p}(\Omega).$$
  181.  
  182. In the next proposition, we have collected the main properties of this map (see Gasinski-Papageorgiou \cite{8}, p.279).
  183.  
  184. \stepcounter{prop}
  185. \begin{prop}
  186. The map $A(\cdot)$ is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (thus, maximal monotone too) and of type $(S)_+.$ Now we introduce our conditions on the potential function $\xi(\cdot)$ and on the boundary coefficient $\beta(\cdot)$.
  187. \end{prop}
  188.  
  189. $H(\xi) : \xi \in L^\infty(\Omega)$
  190.  
  191. $H(\beta) : \beta\in C^{0,\alpha}(\partial\Omega)$ for some $0 < \alpha < 1$ and $\beta(z)\geqslant0$ for all $z \in \partial\Omega$
  192.  
  193. \begin{remark}
  194. When $\beta\equiv0$ we have the Neumann problem.
  195. \end{remark}
  196.  
  197.  
  198. Let $\gamma_p : W^{1,p}(\Omega)\rightarrow\mathbb{R} \ \mbox{be the} \ C^1$-functional defined by
  199.  
  200. $$\gamma_p(u) = ||Du||_p^p+\int_\Omega\xi(z)|u|^pdz + \int_{\partial\Omega}\beta(z)|u|^pd\sigma \ \mbox{for all} \ u \in W^{1,p}(\Omega).$$
  201.  
  202. Also let $f_0 : \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ be a Caratheodory function which satisfies
  203.  
  204. $$|f_0(z,x)| \leqslant a(z)\left[1+|x|^{r-1}\right] \ \mbox{for almost all} \ z \in \Omega, \ \mbox{all} \ x \in \mathbb{R},$$
  205. with $a_0 \in L^\infty(\Omega),1 < r \leqslant p^* =
  206. \begin{cases}
  207. \mathlarger{\frac{N_p}{N - p}}& \mbox{if} \ p < N \\
  208. +\infty & \mbox{if} \ N \leqslant p
  209. \end{cases}$ (the critical Sobolev exponent). We set $F_0(z,x) = \mathlarger{\int_0^x} f_0(z,s)ds$ and consider the $C^1$-funtional $\varphi_0 : W^{1,p}(\Omega) \rightarrow \mathbb{R}$ defined by
  210.  
  211. $$\varphi_0(u) = \frac{1}{p}\gamma_p(u) - \int_\Omega F_0(z, u)dz \ \mbox{for all} \ u \in W^{1,p}(\Omega).$$
  212.  
  213. In the framework of variational methods, the local minimizers of $\varphi_0$ play an important role. As we will see in the sequel, solutions of the problem are often generated by minimizing $\varphi_0$ on a constraint set defined using the usual poitwise order on $W^{1, p}\Omega$ (this is done, via truncation of $f_0(z, \cdot)$). It is well known that the order cone $W_+ = \left\{u \in W^{1,p}(\Omega) : u(z) \geqslant 0 \ \mbox{for almost all} \ z \in \Omega \right\} \ \mbox{of} \ W^{1,p}(\Omega)$, has an empty interior. So, it is not clear if the constrained minimizer is in fact an unconstrained local minimizer of $\varphi_0$ over all of $W^{1,p}(\Omega)$.
  214.  
  215. The next result is helpful in this direction. It is a particular case of a more general result that can be found in Papageorgiou-Radulescu \cite{16}. The first to prove this relation between H\"{0}lder and Sobolev local minimizers, were Brenzis-Nirenberg \cite{3}.
  216.  
  217. \begin{prop}
  218. If $u_0 \in W^{1, p}(\Omega)$ is a local $C'(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0 > 0$ such that
  219.  
  220. $$\varphi_0(u_0) \leqslant \varphi_0(u_0 + h) \ \mbox{for all} \ h \in C'(\overline{\Omega}) \ with \ ||h||_{C'(\overline{\Omega})} \leqslant \rho_0,$$
  221. then $u_0 \in C^{',\vartheta}(\overline{\Omega}) \ with \ \vartheta \in (0, 1) \ and \ u_0 \ \mbox{is also a local} \ W^{1,p}(\Omega)$ - minimizer of $\varphi_0$, that is, there exists $\rho_1 > 0$ such that
  222.  
  223. $$\varphi_0(u_0) \leqslant \varphi_0(u_0 + h) \ \mbox{for all} \ h \in W^{1,p}(\Omega) \ with \ ||h|| \leqslant \rho_1.$$
  224. \end{prop}
  225.  
  226. As we already mentioned in the Introduction, our approach involves also comparison arguments. The next proposition will be helpful in this direction. Is a special case of a more general result of Papageorgiou-Radulescu-Repovs \cite{18}
  227.  
  228. \begin{prop}
  229. If $h_1, h_2, \vartheta \in L^\infty(\Omega), \vartheta(z) \geqslant 0 \ \mbox{for almost all} \ z \in \Omega$
  230.  
  231. $$0 < \eta \leqslant h_2(z) - h_1(z) \ \mbox{for almost all} \ z \in \Omega $$
  232. and $u_1, u_2 \in C^{1, \mu}(\overline{\Omega}) \ with \ 0 < \mu \leqslant 1 \ \mbox{are such that} \ u_1 \leqslant u_2 \ and $
  233.  
  234. \begin{align*}
  235. &-\Delta_p u_1 + \vartheta(z)|u_1|^{p-2}u_1 = h_1, \\
  236. &-\Delta_p u_2 + \vartheta(z)|u_2|^{p-2}u_2 = h_2 \ \mbox{for almost all} \ z \in \Omega,
  237. \end{align*}
  238. then $u_2 - u_2 \in \ intC_+.$
  239. \end{prop}
  240.  
  241. Next we consider the following nonlinear eigenvalue problem
  242.  
  243. \begin{equation}
  244. \left\{
  245. \begin{array}{ll}
  246. -\Delta_pu(z) + \xi(z)|u(z)|^{p - 2}u(z) = \hat{\lambda}|u(z)|^{p-2}u(z) in \Omega,\\
  247. \mathlarger{\frac{\partial u}{\partial n_p}} + \beta(z)|u|^{p-2}u = 0 \ \mbox{on} \ \partial\Omega.
  248. \end{array}
  249. \right\}\label{eq: NonlinearEigenvalue}
  250. \end{equation}
  251.  
  252. We say that $\hat{\lambda} \in \mathbb{R}$ is an "eigenvalue", if (\ref{eq: NonlinearEigenvalue}) admits a nontrivial solution $\hat{u}$ known as an "eigenfunction" corresponding to $\hat{\lambda}$. By $\hat{\sigma}(p)$ we denote the set of eigenvalues of problem (\ref{eq: NonlinearEigenvalue}). Is is easy to that $\hat{\sigma}(p) \subseteq \mathbb{R}$ is closed and it has a smallest element $\hat{\lambda}_1 = \hat{\lambda}_1(p, \xi, \beta) \in \mathbb{R}$ (first eigenvalue), which has the following properties (for details, we refer to Papageorgiou-Radulescu \cite{15} and Fragnelli-Mugnai-Papageorgiou \cite{5}).
  253.  
  254. \begin{prop}
  255. If hypotheses $H(\xi), H(\beta)$, then problem (\ref{eq: NonlinearEigenvalue}) has a smallest eigenvalue $\hat{\lambda}_1 \in \mathbb{R}$ such that
  256. \begin{enumerate}[label=\emph{(\alph*)}]
  257. \item $\hat{\lambda}_1 \ \mbox{is isolated in} \ \hat{\sigma}(p) (\mbox{that is, there is} \ \epsilon > 0 \ \mbox{such that} \ (\hat{\lambda}_1, \hat{\lambda}, +\xi) \cup \hat{\sigma}(p) = \varnothing)$
  258. \item $\hat{\lambda}_1 \ \mbox{is simple (that is, if} \ \hat{u}, \hat{v} \ \mbox{are eigenfunctions corresponding to} \ \hat{\lambda}_1 \ then \ \hat{u} = \eta\hat{v} \\ \mbox{for some} \ \eta \in \mathbb{R}\backslash\{0\})$;
  259. \item \hfill \makebox[0pt][r]{%
  260. \begin{minipage}[b]{15.45cm}
  261. \begin{equation}
  262. \hat{\lambda}_1 = in f\left[\frac{\gamma_0(u)}{||u||_p^p} : u \in W^{1, p}(\Omega), u \neq 0\right].
  263. \label{eq: 3rdElemOfNumList}
  264. \end{equation}
  265. \end{minipage}}
  266. \end{enumerate}
  267. \end{prop}
  268.  
  269. \begin{remark}
  270. The infimum in (\ref{eq: 3rdElemOfNumList}) is realized on the corresponding one dimensional eigenspace.
  271. \end{remark}
  272.  
  273. From (\ref{eq: 3rdElemOfNumList}) it follows that the elements of this eigenspace have fixed sign. By $\hat{u}_1$ we denote he positive, $L^p$-normalized (that is, $||\hat{u}_1||_p = 1$) eigenfunction corresponding to $\hat{\lambda}_1$. We know that $\hat{u}_1 \in D_+$ (see \cite{15}, \cite{5}). Also, every eigenvalue different from $\hat{\lambda}_1$ has eigenfunctions in $C'(\overline{\Omega})$ which are nodal (that is, sign changing). Finally, if $\xi \in L^\infty(\Omega), \xi(x) \geqslant 0$ for almost all $z \in \Omega$ and either $\xi \not\equiv 0$ or $\beta \not\equiv 0$, the $\hat{\lambda} > 0$.
  274.  
  275. An easy consequence of the above properties, is the following lemma (see Mugnai-Papageorgiou \cite{14}, Lemma 4.11).
  276.  
  277. \begin{lemma}
  278. If hypotheses $H(\xi), H(\beta) \ hold \, \eta \in L^\infty(\Omega), \eta(z) \leqslant \hat{\lambda}_1$ for almost all $z \in \Omega$ and the inequality is strict on a set of positive measure, then there exists $c_0 > 0$ such that
  279.  
  280. $$c_0||u||^p \geqslant \gamma_p(u) - \int_\Omega \eta(z)|u|^pdz \ \mbox{for all} \ u \in W^{1,p}(\Omega).$$
  281. \end{lemma}
  282.  
  283. Here is some preview because we love limits
  284.  
  285. \begin{align*}
  286. & \limsup_{i\rightarrow\infty}\limits \\
  287. & \liminf_{n\geqslant 0}x^{n^\frac{4}{a}}
  288. \end{align*}W
  289.  
  290. The hypotheses on the two terms of the reaction of $(P_\lambda)$ are the following. $H(f) f : \Omega \times \mathbb{R} \times (0, +\infty) \rightarrow \mathbb{R}$ is a Caratheodory function such that for all $\lambda > 0, f(z,x,\lambda) \geqslant 0$ for almost all $z \in \Omega$ all $x \geqslant 0, f(z, 0, \lambda) = 0$ for allmost all $z \in \Omega$ and
  291.  
  292. \begin{enumerate}[label=(\roman*)]
  293. \item for every $\rho > 0$ and every $\lambda_0 > 0,$ there exists $a_{\rho,\lambda_0} \in L^\infty(\Omega)$ such that $0 \leqslant f(z,x, \lambda) \leqslant a_{\rho,\lambda_0}(z)$ for almost all $z \in \Omega,$ all $0 \leqslant x \leqslant \rho,$ all $0 < \lambda \leqslant \lambda_0$;
  294. \item for every $\lambda_0 > 0$ we have \\
  295. $\lim_{x \rightarrow +\infty}\limits \mathlarger{\frac{f(z, x, \lambda)}{x^{p-1}}} = \lim_{x\rightarrow 0^+}\limits\mathlarger{\frac{f(z, x, \lambda)}{x^{p - 1}}} = 0$ uniformly for allmost all $z \in \Omega$;
  296. \item if $F(z, x, \lambda) = \mathlarger{\int_0^x} f(z, s, \lambda)ds,$ then there exist $\upsilon_0 \in L^p(\Omega)$ and $\widetilde{\lambda} > 0$ such that $\mathlarger{\int_\Omega} F(z, \upsilon_0(z),\lambda)dz > 0$ for all $\lambda > \widetilde{\lambda}$;
  297. \item $- f(z, x, \lambda) \rightarrow 0^+$ as $\lambda \rightarrow 0^+$ uniformly for almost all $z \in \Omega$ all $x \in c \subseteq \mathbb{R}_+$ bounded, $f(z, x, \lambda) \rightarrow +\infty$ as $\lambda \rightarrow +\infty$ for almost all $z \in \Omega$ all $x > 0$; \\ $- \ \mbox{for every} \ s > 0$, we can find $\widetilde{\eta}_s > 0$ such that \\
  298. $0 < \widetilde{\eta}_s \leqslant f(z, x, \mu) - f(z,x\lambda)$ for almost all $z \in \Omega,$ all $x \geqslant s,$ all $0 < \lambda < \mu$.
  299. \end{enumerate}
  300.  
  301. \begin{remark}
  302. Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis $\mathbb{R}_+ = [0, +\infty),$ without any loss of generality, we may assume that
  303.  
  304. \begin{equation}
  305. f(z, \cdot, \lambda)|_{(-\infty,0]} = 0 \ \mbox{for almost all} \ z \in \Omega, \ \mbox{all} \ \lambda > 0.
  306. \label{eq: renmarkfour}
  307. \end{equation}
  308. \end{remark}
  309.  
  310. Note that hypothesis $H(f)(ii)$ implies that $f(z, \cdot, \lambda)$ is strictly
  311. (p - 1)-sublinear near $+\infty$ and also near $0^+$. Hypothesis H(f)(iii) is satisfied if there is a $\widetilde{\lambda} > 0$ such that $L(z) = \left\{x \in \mathbb{R} : f(z, x, \lambda) > 0\right\}$ is nonempty for almost all $z \in \Omega$ all $\lambda > \widetilde{\lambda}$. Finally note that from hypothesis $H(f)(i\upsilon)$ implies that for almost all $z \in \Omega$, all $x > 0, \lambda \rightarrow f(z, x, \lambda)$ is stictly increasing.
  312.  
  313. $H(g) : g : \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Caratheodory function such that $g(z,0) = 0$ for almost all $z \in \Omega$ and
  314.  
  315. \begin{enumerate}[label=(\roman*)]
  316. \item there exist a $\in L^\infty(\Omega)$ and $p \leqslant r < p^*$ such that
  317. $$(g(z,x)) \leqslant a(1)[1 + x^{r - 1}] \ \mbox{for almost all} \ z \in \Omega, \ \mbox{all} \ x \geqslant 0;$$
  318. \item there exists a function $\eta_0 \in L^\infty(\Omega)$ such that\\
  319. $\eta_0(z) \leqslant \hat{\lambda}_1$ for almost all $z \in \Omega, \eta_0 \not\equiv \hat{\lambda}_1,$ \\
  320. $\limsup_{x\rightarrow +\infty}\limits\mathlarger{\frac{g(z,x)}{x^{p - 1}}} \leqslant \eta_0(z)$ and $\limsup_{x\rightarrow 0^+}\limits\mathlarger{\frac{g(z, x)}{x^{p - 1}}} \leqslant \eta_0(z)$ uniformly for almost all $z \in \Omega$
  321. \item for almost all $z \in \Omega \ x \rightarrow \frac{g(z, x)}{x^{p - 1}}$ is nondecreasing on $(0, +\infty)$
  322. \end{enumerate}
  323.  
  324. \begin{remark}
  325. As we did for $f(z, \cdot, \lambda),$ without any loss of generality, we may assume that
  326. \begin{equation}
  327. g(z, \cdot)|_{(-\infty,0]} = 0 \ \mbox{for almost all} \ z \in \Omega
  328. \label{eq: remark5}
  329. \end{equation}
  330. \end{remark}
  331.  
  332. Hypothesis $H(g)(ii)$ says that asymptotically at $+\infty$ and at $0^+$ we have nonuniform nonresonance with respect to $\hat{\lambda}_1$ from the left.
  333.  
  334. $H_0$ : for every $\rho > 0$ and every $\widetilde{\lambda} > 0,$ we can find $\hat{\xi}_0^{\hat{\lambda}} > 0$ such that for almost all $z \in \Omega$ and all $0 < \lambda \leqslant \lambda_0$, the function $x \rightarrow f(z, x, \lambda) + g(z, x) + \hat{\xi}_\rho^{\hat{\lambda}}x^{p - 1}$ is nondecreasing on $[0, \rho]$.
  335.  
  336. \begin{remark}
  337. This hypothesis is satisfied if, for example, for almost all $z \in \Omega$ and every $\lambda > 0,$ the functions $f(z, \cdot, \lambda)$ and $g(z, \cdot)$ are differentiable and for every $\rho > 0$ and $\hat{\lambda} > 0,$ there exists $\hat{\xi}_\rho^{\widetilde{\lambda}} > 0$ such that
  338.  
  339. $$(f'(z, x, \lambda) + g'_x(z, x)) x \geqslant - \hat{\xi}_\rho^{\widetilde{\lambda}}x^{p - 1} \ \mbox{for almost all} \ z \in \Omega, \ \mbox{all} \ 0 \leqslant x \leqslant \rho.$$
  340. \end{remark}
  341.  
  342. \textit{Examples:} The following pairs of functions $f$ and $g$ satisfy hypotheses $H(f), H(g), H(0),$ for the sale of simplicity we drop the z-dependence. Also recall (\ref{eq: renmarkfour}) and (\ref{eq: remark5}).
  343.  
  344. \begin{align*}
  345. &f_1(x, \lambda) = \lambda x^{q - 1} & &\mbox{for } x \geqslant 0,1 < q < p, \\
  346. &g_1(x) = \eta x^{r - 1} & &\mbox{for } x \geqslant 0, \eta < \hat{\lambda}_1,\\
  347. &f_2(x, \lambda) = \begin{cases} \lambda x^{r-1} & \mbox{if } 0 \leqslant x \leqslant 1 \\ \lambda x^{q - 1} & \mbox{if } 1 < x \end{cases} & &1< q < p < r, \\
  348. &g_2(x) = \begin{cases} cx^{\tau - 1} - x^{q - 1} & \mbox{if } 0 \leqslant x \leqslant 1 \\ \eta x^{p - 1} + (c - 1 - \eta) & \mbox{if } 1 < x \end{cases} & &1 < q < p \leqslant \tau, \eta < \hat{\lambda}_1 c > \max\{\eta + 1,0\}, \\
  349. &f_3(x, \lambda) = \begin{cases} \lambda(x^{\tau - 1} - x^{r - 1}) & \mbox{if } 0 \leqslant x \leqslant 1 \\ \lambda x^{q - 1}|_{nx} & \mbox{if } 1 < x \end{cases}, & &1 < q < p < \tau < r, \\
  350. &g_3(x) = \begin{cases} \eta[x^{p - 1} + x^{r - 1}] & \mbox{if } 0 \leqslant x \leqslant 1 \\ \eta[x^{p - 1} + x^{q - 1}] & \mbox{if } 1 < x \end{cases} & &1 < q < p < r, \eta < \hat{\lambda}_1, \\
  351. &f_4(x, \lambda) = \begin{cases} x^{\tau - 1} & \mbox{if } 0 \leqslant x \leqslant \rho(\lambda) \\ x^{q - 1} + \mu(\lambda) & \mbox{if } \rho(\lambda) < x \end{cases}, \\
  352. &g_4(x) = \eta x^{p - 1}
  353. \end{align*}
  354. with $\rho : (0, +\infty) \rightarrow (0, +\infty)$ strictly increasing, continuous, $\rho(\lambda) \rightarrow 0^+$ as $\lambda \rightarrow 0^+, \rho(\lambda) \rightarrow +\infty$ as $\lambda \rightarrow +\infty, \mu(\lambda) = [\rho(\lambda)^{\tau - 1} - 1]\rho(\lambda)^{q - 1}, 1 < q < p < \tau$ and $\eta < \hat{\lambda}_1$.
  355.  
  356. Finally we fix some basic notation that we will use throught this work. Let $x \in \mathbb{R}$ and set $x^\pm = \max\{\pm x,0\}$. Then for $u \in W^{1, p}(\Omega)$ we define $u^\pm(\cdot) = u(\cdot)^\pm$. We know that
  357.  
  358. $$u^\pm \in W^{1, p}(\Omega), u = u^+ - u^-, |u| = u^+ + u^-.$$
  359.  
  360. Also, if $u, \hat{u} \in W^{1, p}(\Omega)$ and $u \leqslant \hat{u},$ then
  361.  
  362. $$[u, \hat{u}] = \{\upsilon \in W^{1, p}(\Omega) : u(z) \leqslant \upsilon(z) \leqslant \hat{u}(z) \mbox{ for almost all } z \in \Omega \}$$
  363.  
  364. By $int_{C'(\overline{\Omega}})[u, \hat{u}]$ we denote the interior in $C'(\overline{\Omega})$ of $[u, \hat{u}] \cap C' (\overline{\Omega})$.
  365.  
  366. Finally we fix some basic notation that we will use throught this work. Let $x \in \mathbb{R}$ and set $x^\pm = \max\{\pm x, 0\}$. Then for $u \in W^{1, p}(\Omega)$ we define $u^\pm(\cdot) = u(\cdot)^\pm$. We know that
  367.  
  368. $$u^\pm \in W^{1, p}(\Omega), u = u^+ - u^-, |u| = u^+ + u^-.$$
  369.  
  370. Also, if $u, \hat{u} \in W^{1, p}(\Omega)$ and $u \leqslant \hat{u},$ then
  371.  
  372. $$[u,\hat{u}] = \{\upsilon \in W^{1, p}(\Omega) : u(z) \leqslant \upsilon(z) \mbox{ for almost all } z \in \Omega.\}$$
  373.  
  374. By $int_{C'(\overline{\Omega})}[u,\hat{u}]$ we denote the interior in $C'(\overline{\Omega})$ of $[u, \hat{u}] \cup C'(\overline{\Omega})$.
  375.  
  376. Finally, if $\varphi \in C'(X,\mathbb{R})$, then by $K_\varphi$ we denote the critical set of $\varphi$, that is,
  377.  
  378. $$K_\varphi = \{u \in X : \varphi'(u) = 0\}.$$
  379.  
  380. \section{Positive Solutions}
  381.  
  382. We introduce the following two sets
  383.  
  384. \begin{align*}
  385. & \mathcal{L} = \{\lambda > 0 : \mbox{ problem } (P_\lambda) \mbox{ admits a positive solution }\}, \\
  386. & S(\lambda) = \mbox{ set pf positive solutions for problem } (P_\lambda).
  387. \end{align*}
  388.  
  389. We set $\lambda^* = \sup\mathcal{L}$ with the usual convention that $\inf \varnothing = +\infty$.
  390.  
  391. \stepcounter{prop}
  392. \begin{prop}
  393. If hypotheses $H(\xi), H(\beta), H(f), H(g), H_0$ hold, then for every $\lambda \in \mathcal{L} S(\lambda)\subseteq D_+$ and $\lambda^* > 0$.
  394. \end{prop}
  395.  
  396. \begin{proof}
  397. Let $\lambda \in \mathcal{L}$ and let $u \in S(\lambda)$. From Papageorgiou-Radulescu \cite{15}, we know that
  398.  
  399. \begin{equation}
  400. \left\{
  401. \begin{array}{ll}
  402. \mbox{-}\Delta_pu(z) + \xi(z)u(z)^{p -1} = f(z, u(z), \lambda) + g(z, u(z))\mbox{ for almost all }z \in \Omega, \\
  403. \mathlarger{\frac{\partial u}{\partial n_p}} + \beta(z)u^{p - 1} = 0 \mbox{ on } \partial\Omega.
  404. \end{array}
  405. \right\}
  406. \label{eq: propsevenproof}
  407. \end{equation}
  408. \end{proof}
  409.  
  410. From (\ref{eq: propsevenproof}) and Papageorgiou-Radulescu \cite{16}, we have
  411.  
  412. $$u \in L^\infty(\Omega).$$
  413.  
  414. Invoking Theorem 2 of Lieberman \cite{12}, we infer that
  415.  
  416. $$u \in C_+\backslash\{0\}.$$
  417.  
  418. Let $\rho = ||u||_\infty$ and let $\hat{\xi}_\rho^\lambda > 0$ be as postured by hypothesis $H_0$. Then
  419.  
  420. \begin{equation}
  421. \Delta_pu(z) \leqslant \left[||\xi||_\infty + \xi_\rho^\lambda\right] u(z)^{p - 1} \mbox{ for almost all } z \in \Omega.
  422. \label{eq: PostulatedHypothesis}
  423. \end{equation}
  424.  
  425. From (\ref{eq: PostulatedHypothesis}) and from the nonlinear maximum priciple (see, for example, Gasisnski - Papageorgiou \cite{7}, p.738), we have
  426.  
  427. \begin{align*}
  428. & u \in D_+,\\
  429. \Rightarrow & S(\lambda) \subseteq D_+ \mbox{ for all } \lambda > 0.
  430. \end{align*}
  431.  
  432. Next we show that $\lambda^* = \inf\mathcal{L} > 0$. Hypotheses $H(f)(i),(ii)(i\upsilon)$ imply that given $\epsilon > 0,$ we can find $\overline{\lambda} > 0$ such that
  433.  
  434. \begin{equation}
  435. 0 \leqslant f(z, x, \overline{\lambda}) \leqslant \epsilon x^{p - 1} \mbox{ for almost all } z \in \Omega, \mbox{ all } x \geqslant 0.
  436. \label{eq: equationseven}
  437. \end{equation}
  438.  
  439. Hypothesis $H(g)(ii)$ implies that we can find $M,\delta > 0$ such that
  440.  
  441. \begin{equation}
  442. g(z, x) \leqslant [\eta_0(z) + \epsilon]x^{p - 1} \mbox{ for almost all } z \in \Omega \mbox{ and all } \delta \leqslant x \leqslant \delta.
  443. \label{eq: equationeight}
  444. \end{equation}
  445.  
  446. On the other hand by hypothesis $H(g)(iii)$ for almost all $z \in \Omega$ and all $\delta \leqslant x \leqslant M$ we have
  447.  
  448. \begin{equation}
  449. \begin{split}
  450. \frac{g(z, x)}{x^{p - 1}} &\leqslant \frac{g(z, M)}{M^{p - 1}}, \\
  451. \Rightarrow g(z,x) &\leqslant \frac{g(z, M)}{M^{p - 1}}x^{p - 1} \\
  452. &\leqslant [\eta_0(z) + \epsilon]x^{p - 1}(\mbox{see}(\ref{eq: equationeight}))
  453. \end{split}
  454. \label{eq: equationnine}
  455. \end{equation}
  456.  
  457. So, from (\ref{eq: equationeight}) and (\ref{eq: equationnine}), we infer that
  458.  
  459. \begin{equation}
  460. g(z, x) \leqslant [\eta_0(z) + \epsilon]x^{p - 1} \mbox{ for almost all } z \in \Omega, \mbox{ all } x \geqslant 0.
  461. \label{eq: equationten}
  462. \end{equation}
  463.  
  464. Let $\lambda \in (0, \overline{\lambda})$ (see (\ref{eq: equationseven})) and asume that $\lambda \in \mathcal{L}$. Then from the first part of the proof, we know that we can find $u_\lambda \in S(\lambda) \subseteq D_+$. For every $h \in W^{p - 1}(\Omega), h \geqslant 0$ we have
  465.  
  466. \begin{equation}
  467. \begin{split}
  468. & \left\langle A(u_\lambda),h\right\rangle + \int_\Omega \xi(z)u_\lambda^{p - 1}hdz + \int_{\partial\Omega}\beta (z)u_\lambda^{p - 1}hd\sigma \\
  469. = & \int_\Omega [f(z, u_\lambda), \lambda] + g(z, u_\lambda)]hdz \\
  470. \leqslant & \int_\Omega[\eta_0(z) + 2\xi]u_\lambda^{p - 1}hdz (\mbox{ see }(\ref{eq: equationseven}), (\ref{eq: equationten}) \mbox{ and hypothesis } H(f)(i\upsilon)).
  471. \end{split}
  472. \label{eq: equationeleven}
  473. \end{equation}
  474.  
  475. In (\ref{eq: equationeleven}) we choose $h = u_\lambda \in W^{1, p}(\Omega), u_\lambda \geqslant 0.$ Then
  476.  
  477. \begin{align*}
  478. & \gamma_p(u_\lambda) - \int_\Omega\eta_0(z)u_\lambda^{p - 1}dz \leqslant 2\epsilon||u_\lambda||^p, \\
  479. \Rightarrow & c_0 \leqslant 2\epsilon (\mbox{ see Lemma 6 })
  480. \end{align*}
  481.  
  482. Choosing $\epsilon \in (0, \frac{c_0}{2})$, we have a contradiction. Therefore $\lambda \not\in \mathcal{L}$ and so
  483.  
  484. $$0 < \overline{\lambda} \leqslant \lambda^*$$.
  485.  
  486. \begin{prop}
  487. If hypothesis $H(\xi), H(\beta), H(f), H(g), H_0$ hold, then $\mathcal{L} \not\equiv \varnothing$ ans so $0 < \lambda^* < +\infty$
  488. \end{prop}
  489.  
  490. \begin{proof}
  491. From hypotheses $H(f)(i),(ii)$ we, set that given $\epsilon > 0$ and $\lambda > 0$, we can find $c_1 = c_1(\epsilon, \lambda) > 0$ such that
  492. \end{proof}
  493.  
  494. \begin{equation}
  495. F(z, x, \lambda) \leqslant \frac{\epsilon}{p}x^p + c_1 \mbox{ for almost all } z \in \Omega, \mbox{ all } x \geqslant 0
  496. \label{eq: equationtwelve}
  497. \end{equation}
  498.  
  499. Similarly hypothesses $H(g)(i),(ii)$ imply that we can find $c_2 = c_2(\epsilon) > 0$ such that
  500.  
  501. \begin{equation}
  502. F(z, x, \lambda) \leqslant [\eta_0(z) + \epsilon]x^p + c_2 \mbox{ for almost all } z \in \Omega, \mbox{ all } x \geqslant 0
  503. \label{eq: equationthreeteen}
  504. \end{equation}
  505.  
  506. Let $\mu > ||\xi||_\infty$ (see hypothesis $H(\xi)$) and consider the Caratheodory function $k_\lambda(z, x)$ defined by
  507.  
  508. $$k_\lambda(z, x) = f(z, x, \lambda) + g(z, x) \mbox{ for all } (z, x) \in \Omega \times \mathbb{R}, \mbox{ all } \lambda > 0 (\mbox{ see } (\ref{eq: renmarkfour}),(\ref{eq: remark5})).$$
  509.  
  510. We set $K_\lambda(z, x) = \mathlarger{\int_o^x} k_\lambda(z, s)ds$ and consider the $C^1$-functional $\Psi_\lambda : W^{1, p} (\Omega) \rightarrow \mathbb{R}$ defined by
  511.  
  512. $$\Psi_\lambda(u) = \frac{1}{p}\gamma_p(u) + \frac{\mu}{p} ||u^-||_p^p - \int_\Omega K_\lambda(z, u)dz \mbox{ for all } u \in W^{1, p}(\Omega).$$
  513.  
  514. Using (\ref{eq: equationtwelve}) and (\ref{eq: equationthreeteen}), we have for all $u \in W^{1, p}(\Omega).$
  515.  
  516. \begin{align*}
  517. \Psi_\lambda(u) & \geqslant c_3 ||u^-||^p + \frac{1}{p}\gamma_p(u^+) - \frac{1}{p}\int_\Omega[\eta_0(z) + 2\epsilon](u^+)^pdz - c_4 \\
  518. & \mbox{ for some } c_3, c_4 > 0 (\mbox{ recall } \mu > ||\xi||_\infty) \\
  519. & \geqslant c_3||u^-||^p + [c_0 - 2\xi]||u^+||^p - c_4
  520. \end{align*}
  521.  
  522. Choosing $\xi \in (0, \frac{c_0}{2}),$ we obtain
  523.  
  524. \begin{align*}
  525. & \Psi_\lambda(u) \geqslant c_5||u||^p - c_4 \mbox{ for some } c_5 > 0, \mbox{ all } u \in W^{1, p}(\Omega),\\
  526. \Rightarrow \ \ \ \ \ & \Psi_\lambda(\cdot) \mbox{ is coercive }.
  527. \end{align*}
  528.  
  529. Also, using the Sobolev embedding theorem and the compactness of the trace map, we see that
  530.  
  531. $$\Psi_\lambda(\cdot) \mbox{ is sequentially weakly lower semicontinuous }.$$
  532.  
  533. By the Weierstrass-Tonelli theorem, we can find $u_\lambda \in W^{1, p}(\Omega)$ such that
  534.  
  535. \begin{equation}
  536. \Psi_\lambda = inf[\Psi_\lambda(u) : u \in W^{1, p}(\Omega)]
  537. \label{eq: equationfourteen}
  538. \end{equation}
  539.  
  540. Hypotheses $H(f)(i); (ii)$ imply that for every $\lambda > 0$, we can find $c_6 = c_6(\lambda) > 0$ such that
  541.  
  542. $$0 \leqslant F(z, x, \lambda) \leqslant c_6x^p \mbox{ for almost all } z \in \Omega, \mbox{ all } x \geqslant 0.$$
  543.  
  544. Evidently in hypothesis $H(f)(iii)$ we can have $\upsilon_0\geqslant 0 (\mbox{see }(\ref{eq: renmarkfour}))$. Consider the continuous integral functional $i_\lambda : L^p(\Omega) \rightarrow \mathbb{R}$ defined by
  545.  
  546. \begin{align*}
  547. & i_\lambda(\upsilon) = \int_\Omega F(z, \upsilon(z), \lambda)dz \mbox{ for all } \upsilon \in L^p(\Omega), \\
  548. \Rightarrow \ \ \ \ \ & i_\lambda(\upsilon_0) > 0 \mbox{ for all } \lambda > \tilde{\lambda} > 0 (\mbox{see hypothesis } H(f)(iii)).
  549. \end{align*}
  550.  
  551. Exploating the density of $W^{1, p}(\Omega)$ in $L^p(\Omega)$, we can fin $\tilde{\upsilon}_0 \in W^{1, p}(\Omega) \tilde{\upsilon}_0 \geqslant 0, \tilde{\upsilon}_0 \not\equiv 0$ such that
  552.  
  553. $$i_\lambda(\tilde{\upsilon}_0) > 0 \mbox{ for all } \lambda > \tilde{\lambda}.$$
  554.  
  555. The using hypothesis $H(f)(i\upsilon)$ and Fatou's lemma, we infer that
  556.  
  557. \begin{equation}
  558. \lim_{\lambda\rightarrow +\infty}\limits\int_\Omega F(z, \tilde{\upsilon}_0, \lambda)dz = +\infty
  559. \label{eq: equationfifthteen}
  560. \end{equation}
  561.  
  562. On the other hand hypothesis $H(g)(i)$ implies that, if $G(z, x) = \mathlarger{\int_0^x} g(z, s)ds$, then
  563.  
  564. \begin{equation}
  565. \left|\int_\Omega G(z,\tilde{\upsilon}_0)dz\right| \leqslant c_7 \mbox{ for some } c_7 > 0.
  566. \label{eq: equationsixteen}
  567. \end{equation}
  568.  
  569. Then from (\ref{eq: equationfifthteen}) and (\ref{eq: equationsixteen}) we see that for $\lambda > \tilde{\lambda}$ big, we will have
  570.  
  571. \begin{align*}
  572. & \Psi_\lambda(\tilde{\upsilon}_0) < 0, \\
  573. \Rightarrow \ \ \ \ \ & \Psi_\lambda(u_\lambda) < 0 = \Psi_\lambda(0) (\mbox{see }(\ref{eq: equationfourteen})) \\
  574. \Rightarrow \ \ \ \ \ & u_\lambda \not\equiv 0.
  575. \end{align*}
  576.  
  577. From (\ref{eq: equationfourteen}) we have
  578.  
  579. \begin{equation}
  580. \begin{split}
  581. & \Psi_\lambda' (u_\lambda) = 0, \\
  582. \Rightarrow \ \ \ \ \ & \langle A(u_\lambda), h\rangle + \int_\Omega \xi(z)|u_\lambda|^{p - 2}u_\lambda hd\sigma \int_{\partial\Omega} \beta(z)|u_\lambda|^{p - 2}u_\lambda hd\sigma - \int_\Omega \mu(u_\lambda^-)^{p - 1}hd\sigma \\
  583. = \ \ \ \ \ & \int_\Omega[f(z, u_\lambda, \lambda) + g(z, u_\lambda)]hdz \mbox{ for all } h \in W^{1, p}(\Omega).
  584. \end{split}
  585. \label{eq: equationseventeen}
  586. \end{equation}
  587.  
  588. In (\ref{eq: equationseventeen}) we choose $h = -u_\lambda^- \in W^{1, p}(\Omega)$. Then
  589.  
  590. \begin{align*}
  591. & \gamma_p(u_\lambda)^- + \mu ||u_\lambda^-||_p^p = 0 (\mbox{see }(\ref{eq: renmarkfour}),(\ref{eq: remark5})), \\
  592. \Rightarrow \ \ \ \ \ & c_8||u_\lambda^-|| \leqslant 0 \mbox{ for some } c_8 > 0 (\mbox{reacall that } \mu > ||\xi||_\infty),\\
  593. \Rightarrow \ \ \ \ \ & u_\lambda \geqslant 0, u_\lambda \not\equiv 0.
  594. \end{align*}
  595.  
  596. Then from (\ref{eq: equationseventeen}) it follows that $u_\lambda \in S_\lambda \subseteq D_+$ and so for big $\lambda > \tilde{\lambda}$, we have $\lambda \in \mathcal{L}$, hence $\mathcal{L} \not\equiv \varnothing$
  597.  
  598. Next we show that $\mathcal{L}$ is half-line.
  599.  
  600. \begin{prop}
  601. If hypotheses $H(\xi),H(\beta),H(f),H(g),H_0$ hold and $\lambda \in \mathcal{L}$, then $[\lambda, +\infty \subseteq \mathcal{L})$.
  602. \end{prop}
  603.  
  604. \begin{proof}
  605. Since $\lambda \in \mathcal{L}$, we can find $u_\lambda \in S(\lambda) \subseteq D_+$ (see Proposition 7). Let $\vartheta > \lambda$ and consider the following truncation-perturbation of the reaction in problem $(P_\vartheta)$
  606. \end{proof}
  607.  
  608. \begin{equation}
  609. \hat{k}_\vartheta(z, x) =
  610. \begin{cases}
  611. f(z, u_\lambda, \vartheta) + g(z, u_\lambda(z)) + \mu u_\lambda(z)^{p - 1} & \mbox{if } u_\lambda(z) \\
  612. f(z, x, \vartheta) + g(z, x) + \mu x^{p - 1} & \mbox{if } u_\lambda(z) < x.
  613. \end{cases}
  614. \label{eq: equationeightteen}
  615. \end{equation}
  616.  
  617. Recall that $\mu > ||\xi||_\infty$. We set $\hat{K}_\vartheta(z, x) = \mathlarger{\int_0^x} \hat{k}_\vartheta(z, s)ds$ and consider the $C^1$-functional $\hat{\psi}_\vartheta : W^{1, p}(\Omega)\rightarrow\mathbb{R}$ defined by
  618.  
  619. $$\hat{\psi}_\vartheta(u) = \frac{1}{p}\gamma_p(u) +\frac{\mu}{p}||u||_p^p - \int_\Omega \hat{K}_\vartheta(z, u)dz \mbox{ for all } u \in W^{1, p}(\Omega).$$
  620.  
  621. Reasoning as in the proof of Proposition 8, we cam show that
  622.  
  623. \begin{itemize}
  624. \item $\hat{\psi}_\vartheta(\cdot) \mbox{ is coercive };$
  625. \item $\hat{\psi}_\vartheta(\cdot) \mbox{ is sequentially weakly lower semicontinuous }.$
  626. \end{itemize}
  627.  
  628. So, we can find $u_\vartheta \in W^{1, p}(\Omega)$ such that
  629.  
  630. \begin{align*}
  631. &\hat{\psi}_\vartheta = inf[\hat{\psi}_\vartheta(u) : u \in W^{1, p}(\Omega)], \\
  632. \Rightarrow \ \ \ \ \ & \hat{\psi}_\vartheta'(u_\vartheta) = 0, \\
  633. \Rightarrow \ \ \ \ \ \langle A(u_\vartheta),h\rangle & + \int_\Omega[\xi(z) + \mu]|u_\vartheta|^{p - 2}u_\vartheta hdz + \int_{\partial\Omega}\beta(z)|u_\vartheta|^{p - 2}u_\vartheta hd\sigma =
  634. \end{align*}
  635. \begin{equation}
  636. \int_\Omega\hat{k}_\vartheta(z, u_\vartheta)hdz \mbox{ for all } W^{1, p}(\Omega)
  637. \label{eq: equationnineteen}
  638. \end{equation}
  639.  
  640. In (\ref{eq: equationnineteen}) we choose $h = (u_\lambda - u_\vartheta)^+ \in W^{1, p}(\Omega)$. Then we have
  641.  
  642. \begin{align*}
  643. & \langle A(u_\vartheta), (u_\lambda - u_\vartheta)^+ \rangle + \int_\Omega[\xi(z) + \mu]|u_\vartheta|^{p - 2}u_\vartheta(u_\lambda - u_\vartheta)^+ dz + \\
  644. & \ \ \ \ \ \ \int_{\partial\Omega}\beta(z)|u_\vartheta|^{p - 2}u_\vartheta(u_\lambda - u_\vartheta)^+ d\sigma \\
  645. = \ \ \ & \int_\Omega[f(z, u_\lambda), \vartheta) + g(z, u_\lambda) + \mu u_\lambda^{p - 1}](u_\lambda - u_\vartheta)^+ dz (\mbox{see }(\ref{eq: equationeightteen})) \\
  646. \geqslant \ \ \ & \int_\Omega[f(z, u_\lambda, \lambda) + g(z, u_\lambda) + \mu_\lambda^{p - 1}](u_\lambda - u_\vartheta)^+dz (\mbox{since } \lambda < \vartheta, \\
  647. & \mbox{ see hypothesis }H(f)(i\upsilon)) \\
  648. = \ \ \ & \langle A(u_\lambda), (u_\lambda - u_\vartheta)^+ \rangle + \int_\Omega[\xi(z) + \mu]u_\lambda^{p - 1}(u_\lambda - u_\vartheta)^+dz + \int_{\partial\Omega}\beta(z)u_\lambda^{p - 1}(u_\lambda - u_\vartheta)^+ d\sigma \\
  649. & (since u_\lambda \in S(\lambda)), \\
  650. \Rightarrow \ \ \ & u_\lambda \leqslant u_\vartheta \ (\mbox{see Proposition 2 and recall that } \mu > ||\xi||_\infty).
  651. \end{align*}
  652.  
  653. Then equation (\ref{eq: equationnineteen}) becomes
  654.  
  655. \begin{align*}
  656. & \langle A(u_\vartheta),h\rangle + \int_\Omega\xi(z)u_\vartheta^{p - 1}hdz + \int_{\partial\Omega}\beta(z)u_\vartheta^{p - 1}hd\sigma = \int_\Omega[f(z, u_\vartheta, \vartheta) + g(z, u_\vartheta)]hdz \\
  657. & \mbox{for all } h \in W^{1, p}(\Omega), \\
  658. \Rightarrow & u_\vartheta \in S(\vartheta) \subseteq D_+ \mbox{ and so } \vartheta \in \mathcal{L}
  659. \end{align*}
  660.  
  661. Therefore we conclude that
  662.  
  663. $$[\lambda, +\infty) \subseteq \mathcal{L}.$$
  664.  
  665. An interesting by product of the above proof is the foolowing corollary.
  666.  
  667. \setcounter{corollary}{9}
  668. \begin{corollary}
  669. If hypotheses $H(\xi), H(\beta), H(f), H(g), H_0$ hold, $\lambda \in \mathcal{L}, \vartheta > \lambda$ and $u_\lambda \in S(\lambda) \subseteq D_+$, then $\vartheta \in \mathcal{L}$ and we can find $u_\vartheta \in S(\vartheta) \subseteq D_+$ such that $u_\lambda \leqslant u_\vartheta, u_\vartheta \not\equiv u_\lambda.$
  670. \end{corollary}
  671.  
  672. In fact we can improve the conclusion of this corollary.
  673.  
  674. \stepcounter{prop}
  675. \begin{prop}
  676. If hypotheses $H(\xi), H(\beta), H(f), H(g), H_0$ hold, $\lambda \in \mathcal{L}, \vartheta > \lambda$ and $u_\lambda \in S(\lambda) \subseteq D_+$ then $\vartheta \in \mathcal{L}$ and we can find $u_\vartheta \in S(\vartheta) \subseteq D_+$ such that $u_\vartheta - u_\lambda \in intC_+$.
  677. \end{prop}
  678.  
  679. \begin{proof}
  680. From Corollary 10 we already know that $\vartheta \in \mathcal{L}$ and that there exists $u_\vartheta \in S(\vartheta) \subseteq D_+$ such that
  681.  
  682. $$u_\vartheta - u_\lambda \in C_+\backslash\{0\}$$
  683. \end{proof}
  684.  
  685. Let $\rho = ||u_\vartheta||_\infty$ and $\hat{\xi}_\rho^\vartheta > 0$ as in $H_0$. We can always assume $\hat{\xi}_\rho^\vartheta > ||\xi||_\infty$. We have
  686.  
  687. \begin{align*}
  688. -&\Delta_pu_\lambda + [\xi(z) + \xi_\rho^\vartheta]u_\lambda^{p - 1} \\
  689. = \ \ &f(z, u_\lambda, \lambda) + g(z, u_\lambda) + \hat{\xi}_\rho^\vartheta u_\lambda^{p - 1} \\
  690. \leqslant \ \ &f(z, u_\vartheta, \lambda) + g(z, u_\vartheta) + \hat{\xi}_\rho^\vartheta u_\vartheta^{p - 1} \ (\mbox{see hypothesis } H_0 \mbox{ and recall } \lambda < \vartheta) \\
  691. = \ \ &f(z, u_\vartheta, \vartheta) + g(z, u_\vartheta) + \hat{\xi}_\rho^\vartheta u_\vartheta^{p - 1} - [f(z, u_\vartheta, \vartheta) - f(z, u_\vartheta, \lambda)] \\
  692. \leqslant \ \ &f(z, u_\vartheta, \lambda) + g(z, u_\vartheta) + \hat{\xi}_\rho^\vartheta u_\vartheta^{p - 1} - \tilde{\eta}_s \\
  693. &\mbox{with } 0 < s = \min_{\overline{\Omega}}\limits u_\vartheta \ (\mbox{recall } u_\vartheta \in D_+ \mbox{ and see hypothesis } H(f)(i\upsilon)) \\
  694. < \ \ &f(z, u_\vartheta, \vartheta) + g(z, u_\vartheta) + \hat{\xi}_\rho^\vartheta u_\vartheta^{p - 1}
  695. \end{align*}
  696. \begin{equation}
  697. =-\Delta_pu_\vartheta + [\xi(z) + \hat{\xi}_\rho^\vartheta]u_\vartheta^{p - 1} \mbox{ for almost all } z \in \Omega \ (\mbox{since } u_\vartheta \in S(\vartheta)).
  698. \label{eq: equationtwenty}
  699. \end{equation}
  700.  
  701. Since $\hat{\eta}_s > 0$ from (\ref{eq: equationtwenty} and Proposition 4), we infer that
  702.  
  703. $$u_\vartheta - u_\lambda \in intC_+.$$
  704.  
  705. Now let $\lambda > \lambda^*$. From Proposition 9 we know that $\lambda \in \mathcal{L}$. We show that problem $(P_\lambda)$ has at least two positive solutions
  706.  
  707. \begin{prop}
  708. If hypotheses $H(\xi), H(\beta), H(f), H(g), H_0$ hold and $\lambda > \lambda^*$, then problem $(P_\lambda)$ has at least two positive solutions
  709. $$u_0, \hat{u} \in D_+,u_0 \not= \hat{u}.$$
  710. \end{prop}
  711.  
  712. \begin{proof}
  713. As we already mentioned $\lambda \in \mathcal{L}$. Let $\lambda^* < \eta < \lambda < \vartheta$. We have $\eta, \vartheta \in \mathcal{L}$ (see Proposition 9). According to Proposition 11, there are $u_\vartheta \in S(\vartheta)D_+$ and $u_\mu \in S(\mu) \subseteq D_+$ such that
  714.  
  715. $$u_\vartheta - u_\mu \in intC_+.$$
  716. \end{proof}
  717.  
  718. We introduce the Caratheodory function $l_\lambda(z, x)$ defined by
  719.  
  720. \begin{equation}
  721. l_\lambda(z, x) =
  722. \begin{cases}
  723. f(z, u_\eta(z), \lambda) + g(z, u_\eta(z)) + \mu u_\eta(z)^{p - 1} & \mbox{if } x < u_\eta(z) \\
  724. f(z, x, \lambda) + g(z, x) + \mu x^{p - 1} & \mbox{if } u_\eta(z) \leqslant x \leqslant u_\vartheta(z) \\
  725. f(z, u_\vartheta(z), \lambda) + g(z, x) + \mu u_\vartheta(z)^{p - 1} & \mbox{if } u_\vartheta(z) < x.
  726. \end{cases}
  727. \label{eq: equationto}
  728. \end{equation}
  729.  
  730. Recall that $\mu > ||\xi||_\infty$. We set $L_\lambda(z, x) = \mathlarger{\int_0^x} l_\lambda(z, s)ds$ and consider the $C^1$-functional $\hat{\varphi}_\lambda : W^{1, p}(\Omega) \rightarrow \mathbb{R}$ defined by
  731.  
  732. $$\hat{\varphi}_\lambda(u) = \frac{1}{p}\gamma_p(u) + \frac{\mu}{p}||u||_p^p - \int_\Omega L_\lambda(z, u)dz \mbox{ for all } u \in W^{1, p}(\Omega).$$
  733.  
  734. Since $\mu > ||\xi||_\infty,$ from (\ref{eq: equationto}) it is cleat that $\hat{\varphi}_\lambda(\cdot)$ is coercive. Also, it is sequentially weakly lower semicountinuous. So, we can fin $u_0 \in W^{1, p}(\Omega)$ such that
  735.  
  736. \begin{align*}
  737. & \hat{\varphi}_\lambda(u_0) = inf[\hat{\varphi}_\lambda(u) : u \in W^{1, p}(\Omega)], \\
  738. \Rightarrow \ \ \ & \hat{\varphi}_\lambda'(u_0) = 0, \\
  739. \Rightarrow \ \ \ & \langle A(u_0), h\rangle + \int_\Omega[\xi(z) + \mu]|u_0|^{p - 2}u_0hdz + \int_{\partial\Omega}\beta(z)|u_0|^{p - 2}u_0hd\sigma =
  740. \end{align*}
  741.  
  742. \begin{equation}
  743. \int_\Omega l_\lambda(z, u_0)hdz \mbox{ for all } h \in W^{1, p}(\Omega).
  744. \label{eq: equationtt}
  745. \end{equation}
  746.  
  747. In (\ref{eq: equationtt}) first we choose $h = (u_0 - u_\vartheta)^+ \in W^{1, p}(\Omega)$. Then
  748.  
  749. \begin{align*}
  750. & \langle A(u_0),(u_0 - u_\vartheta)^+\rangle + \int_\Omega[\xi(z) + \mu]u_0^{p - 1}(u_0 - u_\vartheta)^+dz + \int_{\partial\Omega}\beta(z)u_0^{p - 1}(u_0 - u_\vartheta)^d\sigma \\
  751. = \ \ \ & \int{\Omega}[f(z, u_\vartheta, \lambda) + g(z, u_\vartheta) + \mu u_\vartheta^{p - 1}](u_0 - u_\vartheta)^+dz \ (\mbox{see}(\ref{eq: equationtt})) \\
  752. \leqslant \ \ \ & \int_\Omega[f(z, u_\vartheta, \vartheta) + g(z, u_\vartheta) + \mu u_\vartheta^{p - 1}](u_0 - u_\vartheta)^+dz \\
  753. &(\mbox{see hypothesis } H(f)(i\upsilon) \mbox{ and recall } \lambda < \vartheta) \\
  754. = \ \ \ & \langle A(u_\vartheta), (u_0 - u_\vartheta)^+\rangle + \int_\Omega[\xi(z) +\mu]u_\vartheta^{p - 1}(u_0 - u_\vartheta)^+dz + \int_{\partial\Omega}\beta(z)u_\vartheta^{p - 1}(u_0 - u_\vartheta)^+ d\sigma \\
  755. &(\mbox{since } u_\vartheta \in S(\vartheta)) \\
  756. \Rightarrow \ \ \ & u_0 \leqslant u_\vartheta \ (\mbox{see Proposition 2 and recall that } \mu > ||\xi||_\infty).
  757. \end{align*}
  758.  
  759. Similarity, if in (\ref{eq: equationtt}) we choose $h = (u_\eta - u_0)^+ \in W^{1, p}(\Omega)$, we show that
  760.  
  761. $$u_\eta \leqslant u_0$$
  762.  
  763. So, we have proved that
  764.  
  765. \begin{equation}
  766. u_0 \in [u_\eta, u_\vartheta]
  767. \label{eq: equationttr}
  768. \end{equation}
  769.  
  770. \medskip
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  831. National Technical University, Department of Mathematics, Zografou Campus, Athens \hspace*{-0.5cm}15780, Greece\\
  832. \textit{E-mail address}: \textbf{\small{npapg@@math.ntua.gr}}
  833.  
  834. University of Craiova, Department of Mathematics, Street A.I.Cuza 13, 200585 Craiova, \hspace*{-0.5cm}Romania, and Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, P.O. \hspace*{-0.5cm}Box 1-764, 014700 Bucharest, Romania\\
  835. \textit{E-mail address}: \textbf{\small{vicentiu.radulescu@imar.ro}}
  836.  
  837. Faculty of Education and Faculty of Mathematics and Physics, University of Ljublijiana, \hspace*{-0.5cm}Karadeljeva Ploscad 16, SI-1000 Ljubljana, SLOVENIA\\
  838. \textit{E-mail address}: \textbf{\small{dusan.repovs@guest.arnes.si}}
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  842. \end{document}
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