Abstract
We study the effect of the coefficient of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for , , and , we should prove that for sufficiently small , there are at least positive solutions of the semilinear elliptic systems , , where is a bounded domain, , , and for .
Highlights
Introduction and Main ResultsFor N ≥ 3, α > 1, β > 1, and 1 ≤ q < 2 < α + β = 2∗ = 2N/(N − 2), consider the semilinear elliptic systems{{{{{−Δu {{{{{−ΔV = = λf μg (x) (x)|u|q−2u + |V|q−2V + α α + β h (x)|u|α−2 u|V|β β h (x) |u|α|V|β−2V α+β{u = V = 0 in Ω, in Ω, on ∂Ω,(Pλ,μ) where λ, μ > 0, Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω.Let f, g, and h satisfy the following conditions
We study the effect of the coefficient h(x) of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems
Under assumptions (H1)-(H2), she has showed that there are at least k positive solutions of the problem (Pλ,μ) for sufficiently small λ, μ > 0
Summary
Let f, g, and h satisfy the following conditions. Recent studies [1,2,3,4,5,6,7,8,9,10] have investigated the elliptic systems with subcritical or critical exponents and have proved the existence of a ground state solution or the existence of at least two positive solutions for these problems. Under assumptions (H1)-(H2), she has showed that there are at least k positive solutions of the problem (Pλ,μ) for sufficiently small λ, μ > 0. Under assumptions (H1)-(H2), we should prove that there exist at least k + 1 positive solutions of the problem (Pλ,μ) for sufficiently small λ, μ > 0.
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