关键词: 凹凸项, 分数阶Laplace, 临界指数, 无穷多解

Abstract:

In this paper, we investigate the existence of solutions for  the following equation with the fractional Laplacian (-Δ)^s and
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