Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents

Zhang, Han-Ming and Liao, Jia-Feng (2023) Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (20). pp. 1-24. ISSN 1417-3875

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Abstract

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents: −∆u =|u|2∗−2u+ α 2∗|u|α−2|v|βu + λ|u|q−2u, in Ω, −∆v =|v|2∗−2v+ β 2∗|u|α|v|β−2v + µ|v|q−2v, in Ω, u, v > 0, in Ω, u =v=0, on ∂Ω, where Ω ⊂ RN(N ≥ 3) is a bounded domain with smooth boundary, λ, µ > 0, 1 < q < 2, α > 1, β > 1, α+β = 2∗ = 2N N−2. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.

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