Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation

In this paper, we consider the nonlinear Neumann problem (P-epsilon): -Delta u+V(x)u= K(x)u((n+2)/(n-2)-epsilon, )u > 0 in Omega, partial derivative u/partial derivative v =0 on partial derivative Omega, where Omega is a smooth bounded domain in R-n, n >= 4, epsilon is a small positive real, and V and K are non-constant smooth positive functions on Omega. First, we study the asymptotic behavior of solutions for (P-epsilon) which blow up at interior points as epsilon moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where k equivalent to 1, problem (P-epsilon) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (P-epsilon) which allow us to provide multiplicity results for (P-epsilon). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.