Sign Changing Solutions for Coupled Critical Elliptic Equations

Xin Wang,Xiaorui Yue

doi:10.1155/2020/9841640

Abstract

In this paper, we consider the coupled elliptic system with a Sobolev critical exponent. We show the existence of a sign changing solution for problem P for the coupling parameter −μ1μ2<β<0. We also construct multiple sign changing solutions for the symmetric case.

Highlights

In this paper, we consider the following coupled elliptic system with a Sobolev critical exponent:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ (P)⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩− Δu1 + λ1u1 ]1􏼌􏼌􏼌􏼌u1􏼌􏼌􏼌􏼌p1− 2u1 + β􏼌􏼌􏼌􏼌u1􏼌􏼌􏼌􏼌(2∗/2)− 2u1􏼌􏼌􏼌􏼌u2􏼌􏼌􏼌􏼌2∗/2,− Δu2 + λ2u2 ]2􏼌􏼌􏼌􏼌u2􏼌􏼌􏼌􏼌p2− 2u2 + β􏼌􏼌􏼌􏼌u1􏼌􏼌􏼌􏼌2∗/2􏼌􏼌􏼌􏼌u2􏼌􏼌􏼌􏼌(2∗/2)− 2u2, u1 u2 0 on zΩ,+ μ1􏼌􏼌􏼌􏼌u1􏼌􏼌􏼌􏼌2∗−in Ω, + μ2􏼌􏼌􏼌􏼌u2􏼌􏼌􏼌􏼌2∗− in Ω, (1)where Ω ⊂ RN is a bounded smooth domain, N ≥ 6, 2 < pj < 2∗, λj ∈ (− λ1(Ω), 0), ]j, μj > 0 for j 1, 2, and λ1(Ω) is the first eigenvalue of − Δ with the Dirichlet boundary condition.In recent years, the following coupled elliptic system has attracted much interest:⎧⎪⎪⎨ − Δu1 + λ1u1 μ1u31 + βu22u1, x ∈ Ω
En, naturally and interestingly, we guess that the more general system corresponding to system (5) has similar result of the existence of sign changing solutions. at is, we consider the more general critical elliptic system as follows:
We study a coupled nonlinear Schrodinger system with critical exponents which arise in many physical problems

Summary

Introduction

We consider the following coupled elliptic system with a Sobolev critical exponent:. E authors in [29] show that when N ≥ 6, − λ1(Ω) < λj < 0, and μj > 0, equation (7) has sign changing solutions for j 1, 2. N ≥ 4, ]j, μj > 0, positive ground and λj ∈ (− λ1(Ω), state solutions v1, 0), problem (8) has v2 ∈ C2(Ω) ∩ C(Ω ) with the energy:. En, naturally and interestingly, we guess that the more general system corresponding to system (5) has similar result of the existence of sign changing solutions. Let v and w be the sign changing solutions of (12) when N ≥ 6, λ ∈ (− λ1(Ω), 0), and N ≥ 4, λ ≤ − λ1(Ω), respectively (the latter case really exists and we shall show it later). We shall show eorems 2 and 3 in Sections 3 and 4, respectively

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Conclusion