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Existence of solution to singular Schrödinger systems involving the fractional p ‐Laplacian with Trudinger–Moser nonlinearity in ℝ N
Author(s) -
Nguyen Thin Van
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7208
Subject(s) - mathematics , lemma (botany) , mountain pass theorem , nonlinear system , laplace transform , compact space , laplace operator , fractional laplacian , mathematical analysis , pure mathematics , quantum mechanics , ecology , physics , poaceae , biology
In this paper, we study the existence of weak solution for singular fractional Schrödinger system inℝ Ninvolving Trudinger–Moser nonlinearity as follows:( − Δ ) p s u + | u | p − 2 u =H u ( x , u , v ) | x | γ( − Δ ) p s v + | v | p − 2 v =H v ( x , u , v ) | x | γ, where N  ≥ 1, 0 <  s  < 1,  N  =  p s ,  γ  ∈ [0,  N ) , and H has exponential growth and does not satisfy the Ambrosetti–Rabinowitz condition. Note that our problem is the lack of compactness. First, we give a version of vanishing lemma due to Lions; using that result and a version of Mountain pass theorem without ( P S ) condition, we obtain the existence of nontrivial solution to the above system. When H satisfies the Ambrosetti–Rabinowitz condition, motivated by the work of Chen et al., we study the existence of nontrivial solution to singular Schrödinger system involving the fractional ( p 1 ,  p ) ‐Laplace and Trudinger–Moser nonlinearity. In our best knowledge, this is the first time the above problems are studied.

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