Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Abstract

We show that the critical nonlinear elliptic Neumann problem Δu - μu + u7/3 = 0 in Ω, u > 0 in Ω, ∂u/∂v = 0 on ∂Ω, where Ω is a bounded and smooth domain in ℝ5, has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K, there exists μK > 0 such that for 0 < μ< μK, the above problem has a nontrivial solution which blows up at K interior points in Ω, as μ → 0. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed. © European Mathematical Society 2005.