关键词: Nehari流形, $p(x)$-Laplace算子, 奇数项, 变分法

Abstract:

In this paper, we study the existence and multiplicity of positive solutions for the following double singular problem with $p(x)$-Laplace operator

\begin{equation*}

\left\{

\begin{array}{ll}

-\Delta _{p(x)}u+V(x)|u|^{p(x)-2}u=\mu\frac{|u|^{s(x)-2}u}{|x|^{s(x)}}+\lambda h(x)u^{-\gamma(x)}&\quad \text{in}\quad \Omega,\\

u=0&\quad \text{on}\quad \partial\Omega.

\end{array}%

\right.

\end{equation*}

Due to the presence of singular term $u^{-\gamma(x)}$ and singular potential $|x|^{-s(x)}$ in the equation, it is more difficult to deal with the existence of positive solutions. By using the decomposition of Nehari manifold and some refined estimates, we show that there admits at least two positive solutions for the double singular problem.

Key words: Nehari manifold, $p(x)$-Laplace operator, Singular terms, Variational method