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

Papageorgiou, Nikolaos S.; Rocha, Eugénio M.

Glasg. Math. J. 50, No. 2, 335-349 (2008).

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.

Reviewer: Nils Ackermann (México)

MSC:

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

Keywords:

varying exponent; Dirichlet boundary values; subcritical nonlinear equation; sub- and supersolution

Citations:

Zbl 1158.35046

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