On a class of singular Trudinger‐Moser type inequalities and its applications

This paper deals with an improvement of a class of the Trudinger‐Moser inequality with a singular weight associated to the embedding of the standard Sobolev space H 1 0 (Ω) into Orlicz spaces for any smooth domain \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega \subset \mathbb {R}^2$\end{document} , in particular for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega = \mathbb {R}^2$\end{document} . As an application of this result, using the Ekeland variational principle and the mountain‐pass theorem we establish sufficient conditions for the existence and multiplicity of weak solutions for the following class of problems\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ -\Delta u+ V(x)u=\frac{f(u)}{|x|^a}+h(x)\quad \mbox{in}\quad \mathbb {R}^2, $$ \end{document} where a ∈ [0, 2), V ( x ) is a continuous positive potential bounded away from zero and which can be “large” at the infinity, the nonlinearity f ( s ) behaves like \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$e^{\alpha s^2}$\end{document} when | s | → +∞ and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$h \in (H^1(\mathbb {R}^2))^*$\end{document} is a small perturbation. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim