Existence results for a class of (p,q) Laplacian systems

G A A Afrouzi,M Mirzapour

doi:10.15388/na.15.4.14311

Abstract

We establish the existence of a nontrivial solution for inhomogeneous quasilinear elliptic systems:−∆pu = λ a(x) u |u|γ−2 + α (α + β)–1 b(x) u |u|α−2 |v|β + f in Ω,−∆qv = µ d(x) v |v|γ−2 + β (α + β)–1 b(x) |u|α v |v|β−2 + g in Ω,(u,v) ∈ W01,p(Ω) × W01,q(Ω).Our result depending on the local minimization.

Highlights

In this paper we deal with the nonlinear elliptic system α α+β b(x)u|u|α−2|v|β f in Ω,β α+β b(x)|u|αv|v|β−2 g in Ω, (1)W01,q (Ω), where 1 < p, q < N and Ω is a regular set of RN, N ≥ 3, α > 0, β > 0, λ and μ are positive parameters, functions a(x), b(x) and d(x) ∈ C(Ω) are smooth functions with change sign on Ω, we assume here that 1 < γ < min(p, q), γ < α + β, α + β > max(p, q) and α/p + β/q = 1
For p ≥ 1 ∆pu is the p-Laplacian defined by ∆pu = div(|∇u|p−2∇u) and W01,p(Ω) is the closer of C0∞(Ω) equipped by the norm u 1,p := ∇u p, where . p represent the norm of Lebesgue space Lp(Ω)
Let p be the conjugate to p, W0−1,p (Ω) is the dual space to W01,p(Ω) and we denote by . −1,p its norm

Summary

Introduction

In this paper we deal with the nonlinear elliptic system α α+β b(x)u|u|α−2|v|β f in Ω,. P represent the norm of Lebesgue space Lp(Ω). Let us define X = W01,p(Ω) × W01,q(Ω) equipped with the norm (u, v) X = u 1,p + v 1,q and It is clear that problem (1) has a variational structure It is well known if the Euler function φ is bounded below and φ has a minimizer on X, this minimizer is a critical point of φ. The Euler function φ(u, v), associated with the problem (1), is not bounded below on the whole space X, but is bounded on an appropriate subset, and has a minimizer on this set (if it exists) which gives rise to solution to (1). The critical points of φ are the weak solutions of problem (1). J1(u, v), (w, z) X := |∇u|p−2∇u∇w, J2(u, v), (w, z) X := |∇v|q−2∇v∇z, D1(u, v), (w, z) X := a(x)|u|γ−2uw, D2(u, v), (w, z) X := d(x)|v|γ−2vz, B1(u, v), (w, z) X := b(x)|u|α−2|v|βuw, B2(u, v), (w, z) X := b(x)|u|α|v|β−2vz

Main results

Proof of the Theorem 1