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Volume 10, Number 5, 2020, Pages 2024-2035                                                                DOI:10.11948/20190319
Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent
Chang-Mu Chu,Haidong Liu
Keywords:$p(x)$-Laplacian, variable exponent, infinitely many solutions, Clark''s theorem, symmetric mountain pass lemma.
Abstract:
      This paper is concerned with the $p(x)$-Laplacian equation of the form $$ \left\{\begin{array}{ll} -\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\u=0, &\mbox{on}\ \partial \Omega, \end{array}\right. \eqno{0.1} $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $1p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma.
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