An existence result for a quasilinear degenerate problem in $\mathbb{R}^N$
In this paper we study an existence result of the quasilinear problem $-\mbox{div}[\phi{'}(|\nabla u|^2)\nabla u] + a(x)|u|^{\alpha-2}u = |u|^{\gamma-2}u + |u|^{\beta-2}u$ in $\mathbb{R}^N (N\geq 3)$, where $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $a$ is a pos...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2017-01-01
|
| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5423 |
| Summary: | In this paper we study an existence result of the quasilinear problem $-\mbox{div}[\phi{'}(|\nabla u|^2)\nabla u] + a(x)|u|^{\alpha-2}u = |u|^{\gamma-2}u + |u|^{\beta-2}u$ in $\mathbb{R}^N (N\geq 3)$, where $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $a$ is a positive potential, $1<p<q<N$, $1<\alpha\leq p^{*}q{'}/p{'}$ and $\max\left\{\alpha, q\right\}<\gamma<\beta<p^{*}=pN/(N-p)$, with $p{'}$ and $q{'}$ the conjugate exponents of $p$, respectively $q$. Our main result is the proof of the existence of a weak solution, based on the mountain pass theorem. |
|---|---|
| ISSN: | 1417-3875 |