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Title: | Ground state solution for a generalized Choquard Schrodinger equation with vanishing potential in homogeneous fractional Musielak Sobolev spaces |
Authors: | Dwivedi, Gaurav |
Keywords: | Mathematics Partial differential equations Choquard equation Nonlinear analysis |
Issue Date: | Jan-2023 |
Abstract: | This paper aims to establish the existence of a weak solution for the following problem: (−Δ)sHu(x)+V(x)h(x,x,|u|)u(x)=(∫RNK(y)F(u(y))|x−y|λdy)K(x)f(u(x)) in RN, where N≥1, s∈(0,1),λ∈(0,N),H(x,y,t)=∫|t|0h(x,y,r)r dr, h:RN×RN×[0,∞)→[0,∞) is a generalized N-function and (−Δ)sH is a generalized fractional Laplace operator. The functions V,K:RN→(0,∞), non-linear function f:R→R are continuous and F(t)=∫t0f(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. |
URI: | https://arxiv.org/abs/2301.04393 http://dspace.bits-pilani.ac.in:8080/jspui/handle/123456789/17403 |
Appears in Collections: | Department of Mathematics |
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