Ground state solution for a generalized Choquard Schrodinger equation with vanishing potential in homogeneous fractional Musielak Sobolev spaces

2023/01/11 by Shilpa Gupta, Gupta, Shilpa, Gaurav Dwivedi +1
Mathematics · #35J20 #35J62 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.04393

openalex publication_date 2023/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper aims to establish the existence of a weak solution for the following problem: (-Δ)^s_Hu(x) +V(x)h(x,x,|u|)u(x)=(∫_ℝ^N\dfracK(y)F(u(y))|x-y|^λdy ) K(x)f(u(x)) \hboxin ℝ^N, where N≥ 1, s∈(0,1), λ∈(0,N), H(x,y,t)=∫₀^|t| h(x,y,r)r dr, h:ℝ^N×ℝ^N× [0,∞)→[0,∞) is a generalized N-function and (-Δ)^s_H is a generalized fractional Laplace operator. The functions V,K:ℝ^N→ (0,∞), non-linear function f:ℝ→ ℝ are continuous and F(t)=∫₀^tf(r)dr. First, we introduce the homogeneous fractional Musielak-Sobolev space and investigate their properties. After that, we pose the given problem in that space. To establish our existence results, we prove and use the suitable version of Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces together with variational technique based on the mountain pass theorem. We also prove the existence of a ground state solution by the method of Nehari manifold.

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