Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations
Let \(\Omega\subseteq\mathbb R^N\) be a bounded domain with a \(C^2\)-boundary \(\partial\Omega\). This article is devoted to a nonlinear parametric Dirichlet eigenvalue problem \[ \left\{ {-\Delta_p u(z)-\Delta u(z) = \lambda f(z,u(z)) \qquad \text{in} \; \Omega},\atop {u|_{\partial\Omega}=0,\quad u>0,\quad \lambda>0, \quad 2<p<+\infty.} \right. \eqno(P)_{\lambda} \] Here, \(-\Delta_p\) denotes the \(p\)-Laplacian differential operator defined by \[ -\Delta_p u(z)=\text{div}(\|\nabla u(z)\|^{p-2}\nabla u(z))\quad \forall u\in W_0^{1,p}(\Omega). \] The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter \(\lambda>0\). The hypotheses H on the reaction \(f\) are the following: \(f:\Omega\times\mathbb R\longrightarrow\mathbb R\;\) is a Carathéodory function such that \(\,f(z,0)=0\;\) for almost all \,\(z\in\Omega\), \(f(z,\zeta)\geq 0\;\) for almost all \(z\in\Omega\) and all \(\zeta\geq 0\), and (i) \,\, for every \(\varrho>0\), there exists \(a_{\varrho}\in L^{\infty}(\Omega)\), such that \[ f(z,\zeta)\leq a_{\varrho}(z)\quad \text{for almost all } z\in\Omega, \text{ all } \zeta\in [0,\varrho]; \] (ii) \, \(\lim_{\zeta\to+\infty}{f(z,\zeta) \over \zeta^{p-1}}=0\) uniformly for almost all \(z\in\Omega\); (iii) \(\lim_{\zeta\to 0^+}{f(z,\zeta) \over \zeta^{p-1}}=0\) uniformly for almost all \(z\in\Omega\); (iv) for every \(\varrho>0\), there exists \(\;\xi_{\varrho}>0\;\) such that for almost all \(z\in\Omega\), the map \(\zeta\mapsto f(z,\zeta)+\xi_{\varrho}\zeta^{p-1}\;\) is nondecreasing on \([0,\varrho]; \) (v) \, \, if \[ F(z,\zeta)= \int_0^{\zeta} f(z,s)\, ds, \] \(\qquad \quad \) then there exists \(c\in R\) such that \[ F(z,c)>0\quad \text{for almost all } z\in\Omega . \] Let \[ C_+=\{ u\in C_0^1(\overline{\Omega}): u(z)\geq 0 \text{ for all } z\in \overline{\Omega} \}, \] \[ {\mathcal Y}=\{\lambda>0 : \text{problem \((P)_{\lambda}\) has a nontrivial positive solution} \} \] and let \(S(\lambda)\) be the set of solutions of \((P)_{\lambda}\). We set \[ \lambda_{\star} =\inf {\mathcal Y}. \] The main results are divided into Propositions 3.3-3.7: if the hypotheses H hold, then (3.3) \(S(\lambda)\subseteq \text{int}\,C_+ \) and \(\lambda_{\star}>0\), (3.4) \({\mathcal Y}\not=\emptyset\), (3.5) if in addition \(\lambda\in{\mathcal Y}\), then \([\lambda,+\infty)\subseteq{\mathcal Y}\), (3.6) for every \(\lambda>\lambda_{\star}\) problem \((P)_{\lambda}\) has at least two positive solutions \(u_0, \hat u\in \text{int} C_+, \;u_0\not= \hat u \). (3.7) \(\lambda_{\star}\in {\mathcal Y}\). To prove these results the authors use a variational approach based on critical point theory, combined with suitable truncation and comparison techniques.
nonlinear parametric Dirichlet eigenvalue problem
scientific article; zbMATH DE number 6234139