Opuscula Math. 40, no. 1 (2020), 71-92
https://doi.org/10.7494/OpMath.2020.40.1.71

Opuscula Mathematica

# Nonhomogeneous equations with critical exponential growth and lack of compactness

Giovany M. Figueiredo
Vicenţiu D. Rădulescu

Abstract. We study the existence and multiplicity of positive solutions for the following class of quasilinear problems $-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,$ where $$\epsilon$$ is a positive parameter. We assume that $$V:\mathbb{R}^N \to \mathbb{R}$$ is a continuous potential and $$f:\mathbb{R}\to\mathbb{R}$$ is a smooth reaction term with critical exponential growth.

Keywords: exponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iteration.

Mathematics Subject Classification: 35J62, 35A15, 35B30, 35B33, 58E05.

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• Vicenţiu D. Rădulescu (corresponding author)
• https://orcid.org/0000-0003-4615-5537
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Department of Mathematics, University of Craiova, 200585 Craiova, Romania
• "Simion Stoilow" Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei, 010702 Bucharest, Romania
• Communicated by Marius Ghergu.
• Accepted: 2019-02-27.
• Published online: 2020-02-17.