Existence of radial solutions for a p ( x ) $p(x)$ -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 − Δ p ( x ) u + R ( x ) u p ( x ) − 2 u = a ( x ) | u | q ( x ) − 2 u − b ( x ) | u | r ( x ) − 2 u $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ with Dirichlet boundary condition in the unit ball in R N $\mathbb{R}^{N}$ (for N ≥ 3 $N \geq 3$ ), where a, b, R are radial functions.

Keywords

• Radial solution
• p ( x ) $p(x)$ -Laplacian
• Dirichlet boundary condition
• Variational principle