Multiple Solutions for -Laplacian Equations With Nonlinearity Sublinear at Zero

Abstract

We consider the Dirichlet problem for $p(x)$ -Laplacian equations of the form $$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$ The odd nonlinearity $f(x,u)$ is $p(x)$ -sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $ . Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$ .