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

Maria Alessandra Ragusa,Abdolrahman Razani,Farzaneh Safari

doi:10.1186/s13662-021-03369-x

Abstract

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized p(x)-Laplacian problem −Δp(x)u+R(x)up(x)−2u=a(x)|u|q(x)−2u−b(x)|u|r(x)−2u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \\vert u \\vert ^{q(x)-2} u- b(x) \\vert u \\vert ^{r(x)-2} u $$\\end{document} with Dirichlet boundary condition in the unit ball in mathbb{R}^{N} (for N geq 3), where a, b, R are radial functions.

Highlights

The study of differential equations and variational problems with nonstandard p(x)growth conditions (or nonstandard (p, q)-growth conditions) is an attractive topic and has been the object of considerable attention in recent years
The aim of this paper is to prove the existence of at least one positive radial solution belonging to the space W01,p(·)(B) ∩ Lqa(·)(B) ∩ Lrb(·)(B) for the problem
(2021) 2021:215 where p(x) := div(|∇|p(x)–2∇), B is the unit ball centered at the origin in RN, N ≥ 3, p, q, r ∈ C+(B), R is a positive radial function, and a(x) = θ |x| and b(x) = ξ |x|, (2)

Summary

Introduction

The study of differential equations and variational problems with nonstandard p(x)growth conditions (or nonstandard (p, q)-growth conditions) is an attractive topic and has been the object of considerable attention in recent years (see [1]) The reasons for such an interest are as follows: 1) Physically, it relies on the fact that they model phenomena arising from various fields such as the motion of electrorheological fluids, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field, the thermo-convective flows of non-Newtonian fluids, and the image processing; 2) Mathematically, it relies on the fact that the standard mathematical techniques are not adequate to study these problems and they need new techniques.

By the Poincaré inequality we have u

Consider the functional

Findings

Th erefore