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On a Class of Semilinear Elliptic Eigenvalue Problems in ℝ2

Part of: Partial differential equations Elliptic equations and systems Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  19 May 2016

Marcelo F. Furtado ,
Everaldo S. Medeiros  and
Uberlandio B. Severo
Marcelo F. Furtado
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília-DF, Brazil (mfurtado@unb.br)
Everaldo S. Medeiros
Affiliation:
Universidade Federal do Paraíba, Departamento de Matemática, 58051-900 João Pessoa-PB, Brazil (everaldo@mat.ufpb.br; uberlandio@mat.ufpb.br)
Uberlandio B. Severo
Affiliation:
Universidade Federal do Paraíba, Departamento de Matemática, 58051-900 João Pessoa-PB, Brazil (everaldo@mat.ufpb.br; uberlandio@mat.ufpb.br)
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Abstract

We consider the semilinear problem

where λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.

Keywords

eigenvalue problemTrudinger–Moser inequalityself-similar solutionscritical exponential growth equation

MSC classification

Secondary: 35J50: Variational methods for elliptic systems 35B33: Critical exponents 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 107 - 126
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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