带凹凸项的分数阶Laplace方程解的存在性

Abstract

Abstract:

In this paper, we investigate the existence of solutions for the following equation with the fractional Laplacian (-Δ)^sand
concave-convex nonlinearities,
{(-Δ)^s u=λa(x)|u|^q-2u+b(x)|u|^p-2u inΩ,
u=0 inRⁿ\Ω.
where s∈ (0,1), q∈(1,2), p∈(2,2_s^*] and 2_s^*=2n/n-2s, n>2s, λ>0, Ω is a bounded domain of Rⁿ, a(x) and b(x) are bounded continuous with b(x)≥0 and a(x) changes signs. We not only prove the existence of nontrivial nonnegative solutions by the Mountain Pass Lemma when p is subcritical and critical, but also, by using Fountain Theorem, we obtain infinitely many solutions for the subcritical case.

Key words: Concave-Convex nonlinearity, Fractional Laplacian, Critical exponent, Infinitely many solutions

CLC Number:

35A15

LI Ben-Niao, CHEN Xiao-Li. Existence of Solutions for Fractional Laplace Equation with Concave-Convex Term[J].Acta mathematica scientia,Series A, 2014, 34(5): 1228-1235.

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Existence of Solutions for Fractional Laplace Equation with Concave-Convex Term

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