A multiplicity theorem for a variable exponent Dirichlet problem. (English) Zbl 1154.35041

If \(N\in\mathbb{N}\) and \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with smooth boundary, consider the problem
\[ \begin{aligned} -\text{div}(| \nabla u| ^{p(x)-2}\nabla u)= m(x)| u| ^{q-2}u + f(x,u),&\quad\text{in }\Omega,\\ u=0,&\quad\text{on }\partial\Omega, \end{aligned}\tag{1} \] where \(p\in C^1(\overline{\Omega})\), \(\min_{\overline{\Omega}}p>q>1\), \(m\in L^\infty(\Omega)\backslash\{0\}\), \(m\geq0\), and \(f\) is a Carathéodory function with subcritical growth in a suitable sense, with respect to the variable exponent \(p(x)\). The authors prove the existence of three classical solutions to (1) that are ordered, and such that one solution is negative and one positive. The method consists in a combination of the sub-supersolution technique and the Mountain Pass Theorem applied to a suitably truncated functional. One should compare this result with the recent article by J. Yao [Nonlinear Anal., Theory Methods Appl. 68, No. 5 (A), 1271–1283 (2008; Zbl 1158.35046)], where a similar result is achieved under Neumann boundary conditions and with a different set of hypotheses, including the Ambrosetti-Rabinowitz condition.


35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B50 Maximum principles in context of PDEs


Zbl 1158.35046
Full Text: DOI


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