SHARP SOBOLEV INEQUALITY INVOLVING A CRITICAL NONLINEARITY ON A BOUNDARY

Abstract. We consider the solvability of the Neumann problem for the equation −∆u+ λu = 0, ∂u ∂ν = Q(x)|u|q−2u on ∂Ω, where Q is a positive and continuous coefficient on ∂Ω, λ is a pa-rameter and q = 2(N − 1)/(N − 2) is a critical Sobolev exponent for the trace embedding of H1(Ω) into Lq(∂Ω). We investigate the joint effect of the mean curvature of ∂Ω and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter λ interferes with the spectrum of − ∆ with the Neu-mann boundary conditions. We apply a min-max principle based on the topological linking. 1