W01,p versus C1 local minimizers for a singular and critical functional

Let Omega subset of R-N be a bounded smooth domain, 1 < p < +infinity, 0 < delta < 1, f : (Omega) over bar R -> R be a C-1 function with f(x, s) >= 0. for all(x, s) is an element of Omega x R+ and sup(x is an element of Omega) f(x, s) <= C(1 + s)(q), for all(S) is an element of R+. for some 0 <= q satisfying q <= p* - 1 if p < N. Lot g : R+ -> R+ continuous on (0, + infinity) nonincreasing and satisfying c(1) = lim inf(t -> 0+) g(t)t(delta) <= lim sup(t -> 0+) g(t)t(delta) = c(2). for some c(1), c(2) > 0. Consider the singular functional I : W-0(1,p)(Omega) -> R defined as I(u) =(def) 1/p parallel to u parallel to(p)(w01,p)((Omega)) -integral F-Omega(x, u(+)) - integral(Omega) G(u(+)) where F(x, u) = integral(s)(0) f(x, s) ds, G(u) = integral(s)(0) g(s) ds. Theorem 1.1 proves that if u(0) is an element of C-1((Omega) over bar) satisfying u(0) >= eta dist(x, delta 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-0(1,p)(Omega) topology. This result is useful for proving multiple solutions to the associated Euler-Lagrange equation (P) defined below. Theorem 1.1 generalises some results in Giacomoni, Schindler and Takac (2007) [17] and due to the new proof given in the present paper can be also extended to more general quasilinear elliptic equations. (C) 2009 Elsevier Inc. All rights reserved.