September/October 2023 Existence of positive solutions for $p\&q$ equations involving vanishing potentials with exponential decay
Gustavo S. Costa, Giovany M. Figueiredo
Differential Integral Equations 36(9/10): 859-876 (September/October 2023). DOI: 10.57262/die036-0910-859

Abstract

We show the existence of positive solutions for a class of $p\&q$ problems given by $$ \begin{cases} -div\left(a\left(|\nabla u|^p\right) |\nabla u|^{p-2}\nabla u\right)+V(z)b \left(|u|^p\right)|u|^{p-2}u = f(u)\ \ \mbox{in $\mathbb{R}^N,$} \\ u \in D^{1,p}(\mathbb{R}^{N})\cap D^{1,q}(\mathbb{R}^{N}), \end{cases} $$ $N\geq 3$ and $1 < p\leq q < N$. The potential $V$ can vanish at infinity with exponential decay and $f$ is a function with subcritical growth of class $C^1$. We use Del Pino and Felmer's arguments [18] in order to overcome the lack of compactness.

Citation

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Gustavo S. Costa. Giovany M. Figueiredo. "Existence of positive solutions for $p\&q$ equations involving vanishing potentials with exponential decay." Differential Integral Equations 36 (9/10) 859 - 876, September/October 2023. https://doi.org/10.57262/die036-0910-859

Information

Published: September/October 2023
First available in Project Euclid: 25 May 2023

Digital Object Identifier: 10.57262/die036-0910-859

Subjects:
Primary: 35J10 , 35J20 , 35J60

Rights: Copyright © 2023 Khayyam Publishing, Inc.

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Vol.36 • No. 9/10 • September/October 2023
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