Infinitely many solutions for quasilinear Schrödinger equation with critical exponential growth in

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• 作者：Hongxue Song^a ; ^b ; ^{songhx@njupt.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Caisheng Chen^a ; ^{cshengchen@hhu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Qinglun Yan^b
• 关键词：Quasilinear elliptic equations ; N-Laplacian ; Trudinger&ndash ; Moser inequality ; Schwarz symmetrization ; Mountain-pass theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：439
• 期：2
• 页码：575-593
• 全文大小：429 K

文摘

This paper shows the existence of infinitely many solutions for the quasilinear equations of the form where △_Nu${△}_{N}u$ is the N  -Laplacian operator, N≥3$N\ge 3$, λ≥0$\lambda \ge 0$, 1<q<N$1<q<N$, K∈L^θ(R^N)$K\in L^{\theta }({\mathbb{R}}^N)$, θ=N/(N−q)$\theta =N/(N-q)$ and h   is an odd continuous function having critical exponential growth. The potential function V(x)∈C(R^N)$V(x)\in C({\mathbb{R}}^N)$ and 0<inf_x∈R^N⁡V(x)≤sup_x∈R^N⁡V(x)<∞$0<{\inf }_{x\in {\mathbb{R}}^N}V(x)\le {\sup }_{x\in {\mathbb{R}}^N}V(x)<\infty$. Using mountain-pass theorem and some special techniques, we demonstrate that there exists λ₀>0${\lambda }_0>0$ such that problem (0.1) admits infinitely many high-energy solutions provided that λ∈[0,λ₀]$\lambda \in [0,{\lambda }_0]$.