Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

2021/07/09 by Rawat, Sushmita, Sreenadh, K.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.04249

Abstract

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: (-Δ)^s u · amp;= -λ|u|^q-2u + au + b( ∫_Ω \frac(u⁺(y))^2^*_μ,s|x-y|^ μ dy) (u⁺)^2^*_μ,s-2u \quadin Ω, u · amp;= 0 in ℝ^N\backslashΩ, where Ω is open bounded domain of ℝ^N with C² boundary, N > 2s and s ∈ (0,1). Here (-Δ)^s is the fractional Laplace operator, λ> 0 is a real parameter, q ∈ (1, 2), a > 0 and b> 0 are given constants, and 2^*_μ,s = (2N-μ)/(N-2s) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation u⁺ = max \u, 0\. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

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