Existence of radial solutions for a $p(x)$-Laplacian Dirichlet problem

Abstract

Abstract In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized $p(x)$ p(x) -Laplacian problem $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ −Δp(x)u+R(x)up(x)−2u=a(x)|u|q(x)−2u−b(x)|u|r(x)−2u with Dirichlet boundary condition in the unit ball in $\mathbb{R}^{N}$ RN (for $N \geq 3$ N≥3 ), wherea,b,Rare radial functions.