A NONEXISTENCE RESULT OF SINGLE PEAKED SOLUTIONS TO A SUPERCRITICAL NONLINEAR PROBLEM

Abstract

This paper is concerned with the nonlinear elliptic problem (P ε ): -Δu = u p+ε , u > 0 in Ω; u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ n , n ≥ 3, p + 1 = 2n/(n - 2) is the critical Sobolev exponent and ε is a small positive parameter. In contrast with the subcritical problem (P - ε ) studied by Han [11] and Rey [17], we show that (P ε ) has no single peaked solution for small ε.