Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient

Abstract

We deal with the existence of constant sign solutions for the following variable exponent system Neumann boundary value problem: $-\text{div}({|\nabla u|}^{p(x)-\mathrm{2}}\nabla u)+\lambda {|u|}^{p(x)-\mathrm{2}}u=F_u(x,u,v)\mathrm{}\mathrm{}\hspace{0.17em}$ in $\hspace{0.17em}\mathrm{\Omega },\hspace{0.17em}-\text{div}({|\nabla v|}^{q(x)-\mathrm{2}}\nabla v)+\lambda {|v|}^{q(x)-\mathrm{2}}v=F_v(x,u,v)$ in $\mathrm{\Omega },\hspace{0.17em}\partial u/\partial \gamma =\mathrm{0}=\partial v/\partial \gamma \hspace{0.17em}$ on $\hspace{0.17em}\partial \mathrm{\Omega }$. We give several sufficient conditions for the existence of the constant sign solutions, when $F(x,·,·)$ satisfies neither sub-($p(x),q(x)$) growth condition, nor Ambrosetti-Rabinowitz condition (subcritical). In particular, we obtain the existence of eight constant sign solutions.