Electron. J. Differential Equations, Vol. 2018 (2018), No. 31, pp. 1-20.

Solutions for p(x)-Laplace equations with critical frequency

Xia Zhang, Chao Zhang, Huimin Gao

Abstract:
This article concerns the p(x)-Laplace equations with critical frequency
$$ -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)|u|^{p(x)-2}u=f(x,u)\quad \text{in } \mathbb{R}^N, $$
where $1<p_{-}\leq p(x)\leq p_{+}<N$. We study this equation with the potentials being zero. By using variational method, we obtain the existence of nonnegative solutions. Moreover, if f(x,t) is odd in t, for any $m\in\mathbb{N}$ we derive m pairs of nontrivial solutions.

Submitted September 28, 2017. Published January 19, 2018.
Math Subject Classifications: 35J60.
Key Words: Variable exponent space; p(x)-Laplace; critical frequency; weak solution.

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Xia Zhang
Department of Mathematics
Harbin Institute of Technology, China
email: zhangxia@hit.edu.cn
Chao Zhang
Department of Mathematics
Harbin Institute of Technology, China
email: czhangmath@hit.edu.cn
  Huimin Gao
Department of Mathematics
Harbin Institute of Technology, China
email: 1576055240@qq.com

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