Article ID Journal Published Year Pages File Type
4615237 Journal of Mathematical Analysis and Applications 2015 19 Pages PDF
Abstract

In this study, we consider the following coupled elliptic system with a Sobolev critical exponent:equation(PP){−Δu1+λ1u1=ν1u1p1−1+μ1u12⁎−1+βu12⁎2−1u22⁎2,x∈Ω,−Δu2+λ2u2=ν2u2p2−1+μ2u22⁎−1+βu12⁎2u22⁎2−1,x∈Ω,u1,u2≥0inΩ,u1=u2=0on∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain, N≥5N≥5, 20μj>0 for j=1,2j=1,2, and λ1(Ω)λ1(Ω) is the first eigenvalue of −Δ with the Dirichlet boundary condition. We demonstrate the existence of a positive ground state solution for problem (PP) when the coupling parameter β≥−μ1μ2. Under some other conditions, we show the nonexistence of positive solutions for (PP) when N≥3N≥3. We also construct multiple nontrivial solutions and sign-changing solutions for the following system:{−Δu1+λ1u1=μ1u13+βu22u1,x∈Ω,−Δu2+λ2u2=μ2u23+βu12u2,x∈Ω,u1=u2=0on∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain and N≤4N≤4.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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