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.

中文翻译：

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关键 p&q 方程的正解的存在和集中

我们展示了由 −divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u= 给出的一类 p & q 关键问题的存在和集中结果f(u)+|u|q⋆−2uinRN, $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\ nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+ |u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ 其中 u ∈ W 1, p (ℝ N ) ∩ W 1 , q (ℝ N ), ε > 0 是一个小参数，1 < p ≤ q < N , N ≥ 2 和 q * = Nq /( N - q )。势 V 为正，f 是 C 1 类的超线性函数。我们使用山口定理和 Del Pino 和 Felmer 引入的与 Lions 的集中和紧凑性原则相关的惩罚论点，以克服缺乏紧凑性的问题。