Topological Methods in Nonlinear Analysis

Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in $\mathbb{R}^{n}$

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of weak solutions for nonhomogeneous quasilinear elliptic equations of the form $$-\Delta_n u + a(x)|u|^{n-2}u= b(x)|u|^{n-2}u+g(x)f(u)+\varepsilon h \quad \mbox{in }\mathbb{R}^n ,$$% where $n \geq 2$, $\Delta_n u \equiv {\rm div}(|\nabla u|^{n-2}\nabla u)$ is the $n$-Laplacian and $\varepsilon$ is a positive parameter. Here the function $g(x)$ may be unbounded in $x$ and the nonlinearity $f(s)$ has critical growth in the sense of Trudinger-Moser inequality, more precisely $f(s)$ behaves like $e^{\alpha_0 |s|^{n/(n-1)}}$ when $s\to+\infty$ for some $\alpha_0 > 0$. Under some suitable assumptions and based on a Trudinger-Moser type inequality, our results are proved by using Ekeland variational principle, minimization and mountain-pass theorem.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 45, Number 2 (2015), 615-639.

Dates
First available in Project Euclid: 30 March 2016

https://projecteuclid.org/euclid.tmna/1459343998

Digital Object Identifier
doi:10.12775/TMNA.2015.029

Mathematical Reviews number (MathSciNet)
MR3408838

Zentralblatt MATH identifier
1376.35071

Citation

de Souza, Manassés; Marcos do Ó, João; Silva, Tarciana. Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in $\mathbb{R}^{n}$. Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 615--639. doi:10.12775/TMNA.2015.029. https://projecteuclid.org/euclid.tmna/1459343998

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