On a p(x)-biharmonic problem with Navier boundary condition

Zheng Zhou

doi:10.1186/s13661-018-1071-2

Abstract

In this paper, we study a p(x)-biharmonic equation with Navier boundary condition \t\t\t{Δp(x)2u+a(x)|u|p(x)−2u=λf(x,u)+μg(x,u)in Ω,u=Δu=0on ∂Ω.\\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}$$ \\textstyle\\begin{cases} \\Delta^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u= \\lambda f(x,u)+\\mu g(x,u)\\quad \\text{in } \\Omega, \\\\ u=\\Delta u=0 \\quad \\text{on } \\partial\\Omega. \\end{cases} $$\\end{document} Here Omegasubsetmathbb{R}^{N} (Ngeq1) is a bounded domain with smooth boundary ∂Ω, Delta^{2}_{p(x)}u is a p(x)-biharmonic operator with p(x) in C(overline{Omega}), p(x)>1. lambda,muinmathbb{R}, ain L^{infty}(Omega) such that inf_{xinOmega}a(x)=a^{-}>0. By variational methods, we establish the results of existence and non-existence of solutions.

Highlights

In recent years, the study on variational problems with variable exponent is an interesting topic, which arises from nonlinear electrorheological fluids and elastic mechanics
In this paper, we study a p(x)-biharmonic equation with Navier boundary condition
1 Introduction In recent years, the study on variational problems with variable exponent is an interesting topic, which arises from nonlinear electrorheological fluids and elastic mechanics

Summary

Introduction

The study on variational problems with variable exponent is an interesting topic, which arises from nonlinear electrorheological fluids and elastic mechanics (see [1–3]). Notice that the work space X = W 2,p(x)( ) ∩ W01,p(x)( ) is a separable, reflexive Banach space, we apply the fountain theorem and the dual fountain theorem to obtain infinitely many solutions In this part, we suppose ⊆ RN (N ≥ 1) is a bounded domain with smooth boundary, and assume (H0) f : × R → R is a Carathéodory function such that f (x, t) ≤ c1|t|q(x)–1, g(x, t) ≤ c2|t|γ (x)–1, ∀(x, t) ∈ × R, where c1, c2 > 0, q(x), γ (x) ∈ C( ) and 1 < q(x), γ (x) < p∗2(x), ∀x ∈ ; (H1) ∃l > 0, μ > p+, 0 < μF(x, s) ≤ f (x, s)s for |s| ≥ l and x ∈ ;.

The Lebesgue space with variable exponent is defined by

It is clear that