## 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.

### 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

Zbl 1158.35046
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### References:

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