Nonminimizing Positive Solutions for Equations with Critical Exponents in the Half-Space

Abstract

This paper is concerned with the existence of positive solutions of the nonlinear elliptic problem $-\Delta u+ a(x)u=u^{(N+2)/(N-2)}$, $a(x)\ge 0$, with Neumann boundary conditions in a half-space $\Pi \, \s \, {\Bbb R}^N $, $N \ge 3$. The main feature of the problem is a double lack of compactness due to the unboundedness of the domain and the presence of the critical Sobolev exponent. The solutions are searched using variational methods, although the functional related to the problem does not satisfy the Palais--Smale compactness condition. We observe that the problem considered has no solutions if $a(x)$ is a positive constant; conditions on $a(x)$ are given sufficient to guarantee existence and multiplicity of positive solutions.