Existence and concentration of positive solutions for a critical p & q equation

Advances in Nonlinear Analysis

article

Gustavo S. Costa¹  Giovany M. Figueiredo¹ 

[1]Universidade de Brasília

关键词: Critical exponent;    p&q Laplacian operator;    Variational methods;   

DOI  :  10.1515/anona-2020-0190

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We show existence and concentration results for a class of p & q critical problems given by −divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u=f(u)+|u|q⋆−2uinRN, $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1, p (ℝ N ) ∩ W 1, q (ℝ N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N , N ≥ 2 and q * = Nq /( N − q ). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.

【 授权许可】

CC BY