Abstract
Let Omega subset of R-N, be a bounded domain with smooth boundary. Let g : R+ -> R+ be a continuous function on (0, +infinity) non-increasing and satisfying
c(1) = lim(t -> 0+) inf g(t)t(delta) <= lim(t -> 0+) supg(t)t(delta) = c(2),
for some c(1),c(2) > 0 and 0 < delta < 1. Let f (x, s) = h(x, s)e(bsN/N-1), b > 0 is a constant. Consider the singular functional I : W-1,W-N(Omega) -> R defined as
I(u) =(def) 1/N parallel to u parallel to(N)(W1,N) (Omega) - integral(Omega) G(u(+))dx - integral(Omega) F(x, u(+))dx
-1/q+1 parallel to u parallel to(q+1)(Lq+1) (partial derivative Omega)
where f(x, u) = integral(s)(0) f(x, s) ds, G(u) = integral(s)(0) g(s) ds. We show that if u(0) is an element of C-1 ((Omega) over bar) satisfying u(0) >= eta dist(x, partial derivative Omega), for some 0 < eta, is a local minimum of I in the C-1((Omega) over bar) boolean AND C-0((Omega) over bar) topology, then it. is also a local minimum in W-1,W-N (Omega) topology. This result is useful to prove the multiplicity of positive solutions to critical growth problems with co-normal boundary conditions.