Concentration behavior of solutions for quasilinear elliptic equations with steep potential well

2022 ◽  
Vol 132 (1) ◽  
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Pingying Ling
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Potential Well ◽  
Steep Potential Well ◽  
Concentration Behavior ◽  
Quasilinear Elliptic

Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth

2017 ◽  
Vol 8 (1) ◽  
pp. 339-371 ◽  
Author(s):  
Kaimin Teng ◽  
Xiaofeng Yang
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Elliptic Equations ◽  
Nonnegative Function ◽  
Quasilinear Elliptic Equations ◽  
Sobolev Critical Exponent ◽  
Concentration Behavior ◽  
Quasilinear Elliptic

Abstract In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent -\varepsilon^{p}\Delta_{p}u-\varepsilon^{p}\Delta_{p}(u^{2})u+V(x)\lvert u% \rvert^{p-2}u=h(u)+\lvert u\rvert^{2p^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplace operator, {p^{*}=\frac{Np}{N-p}} ( {N\geq 3} , {N>p\geq 2} ) is the usual Sobolev critical exponent, the potential {V(x)} is a continuous function, and the nonlinearity {h(u)} is a nonnegative function of {C^{1}} class. Under some suitable assumptions on V and h, we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category.

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