Premium
Standing waves for a class of fractional p ‐Laplacian equations with a general critical nonlinearity
Author(s) -
Lou QingJun,
Mao AnMin,
You Jin
Publication year - 2020
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.6804
Subject(s) - mathematics , sobolev space , p laplacian , nonlinear system , fractional laplacian , class (philosophy) , order (exchange) , sequence (biology) , critical exponent , exponent , type (biology) , integer (computer science) , mathematical analysis , laplace operator , pure mathematics , geometry , physics , finance , quantum mechanics , artificial intelligence , scaling , biology , computer science , programming language , economics , genetics , boundary value problem , ecology , linguistics , philosophy
In this paper, we concern with the following fractional p ‐Laplacian equation with critical Sobolev exponentε p s− Δp s u + V ( x )up − 2u = λ f ( x )uq − 2 u +up s ∗ − 2 u in ℝ N ,u ∈ W s , pℝ N, u > 0 ,where ε > 0 is a small parameter, λ > 0 , N is a positive integer, and N > p s with s ∈ (0, 1) fixed, 1 < q ≤ p ,p s ∗ : = N p / N − p s . Since the nonlinearity h ( x , u ) : = λ f ( x )uq − 2 u +up s ∗ − 2 u does not satisfy the following Ambrosetti‐Rabinowitz condition:0 < μ H ( x , u ) : = μ ∫ 0 u h ( x , t ) d t ≤ h ( x , u ) u , x ∈ ℝ N , 0 ≠ u ∈ ℝ , with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p ‐Laplacian type.