Positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:427

Article

Yue, Xiaorui

关键词: Critical exponent;    Elliptic system;    Multiple solutions;    Positive ground state solution;   

DOI  :  10.1016/j.jmaa.2015.01.073

来源: Elsevier

【 摘 要 】

In this study, we consider the following coupled elliptic system with a Sobolev critical exponent: {Delta u(1) + lambda(1)u(1) = nu(1)u(1)(p1-1) + mu(1)u(1)(2 star-1) + beta u(1)(2 star/2-1) u(2)(2 star/2), x is an element of Omega, -Delta u(2) + lambda(2)u(2) = nu(2)u(2)(p2-1) + mu(2)u(2)(2 star - 1) + beta u(1)(2 star/2) u(2)(2 star/2 - 1), x is an element of Omega, (P) u(1), u(2) >= 0 in Omega, u(1) = u(2) = 0 on partial derivative Omega, where Omega subset of R-N is a bounded smooth domain, N >= 5, 2 < p(1),p(2) < 2(star), lambda(j) is an element of (-lambda(1)(Omega), mu(j) > 0 for j = 1,2, and lambda(1)(Omega) is the first eigenvalue of -Delta with the Dirichlet boundary condition. We demonstrate the existence of a positive ground state solution for problem (P) when the coupling parameter beta >= -root mu(1)mu(2). Under some other conditions, we show the nonexistence of positive solutions for (P) when N >= 3. We also construct multiple nontrivial solutions and sign-changing solutions for the following system: {-Delta u(1) + lambda(1)u(1) = mu(1)u(1)(3) + beta u(2)(2)u(1), x is an element of Omega, -Delta u(2) + lambda(2)u(2) = mu(2)u(2)(3) + beta u(1)(2)u(2), x is an element of Omega, u(1) = u(2) = 0 on partial derivative Omega, where Omega subset of R-N is a bounded smooth domain and N <= 4. (C) 2015 Elsevier Inc. All rights reserved.

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