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critical exponential growth equation

Proceedings of the Edinburgh Mathematical Society (1)

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On a Class of Semilinear Elliptic Eigenvalue Problems in ℝ2
Part of
Marcelo F. Furtado, Everaldo S. Medeiros, Uberlandio B. Severo
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 1 / February 2017

Published online by Cambridge University Press:

19 May 2016, pp. 107-126
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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.