SOLUTIONS OF ELLIPTIC-EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS WITH NEUMANN BOUNDARY-CONDITIONS

We consider the problem -Δu=|u| p-1u+λu in Ω with {Mathematical expression} on δΩ, where Ω is a bounded domain in R N, p=(N+2)/(N-2) is the critical Sobolev exponent, n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball in R N, then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω in R 3, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial solution. © 1990 Springer-Verlag.