On existence and concentration behavior of ground state solutions for a class of problems with critical growth

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1.
Departamento de Matematica, Universidade Federal da Paraiba, 58100-907 Campina Grande-PB
2.
Departamento de Matematica e Estatistica, Universidade Federal da Paraiba,58109-970

Received: August 31, 2001

Revised: January 31, 2002

Published: August 2002

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Abstract

In this paper, we study the existence and the concentration behavior of ground state for the problem

$-h^2\Delta u+V(z)u=\lambda u^q+u^{2^{ *} -1,\mathbb R^N $

$u(z)>0\quad $ for all $z\in \mathbb R^N \qquad\qquad\qquad\qquad\qquad\qquad\qquad (P_{h})$

where $h, \lambda >0$, 1<$q$ <$2^{ * -1$ $=\frac{N+2}{N-2}$, $N\geq 3$ and $V: \mathbb R^N\to \mathbb R$ is a positive function such that

0< $i nf_{z\in\mathbb R^N}V(z)$< $limi nf_{|z| \rightarrow \infty}V(z)=V_{\infty}.$

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Mathematics Subject Classification: 35A05, 35A15 and 35J20.

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