Abstract
Let Omega subset of R-N, (N >= 2) be a bounded smooth domain, p is Holder continuous on (Omega) over bar,
1 < p(-) : = inf(Omega)p(x) <= p(+) =sup(Omega)p(x) < infinity,
and f : Omega xR -> R be a C-1 function with f(x,s)>= 0, for all(x,s)is an element of Omega xR(+) and sup(x is an element of Omega)f(x,s)<= C(1+s)(q(x)), for all s is an element of R+, for all x is an element of Omega for some 0<q(x)is an element of C(<(Omega)over bar>) satisfying 1<p(x)<q(x)<= p* (x)-1, for all x is an element of(Omega) over bar and 1<p(-)<= p(+) < q(-) <= q(+). As usual, p*(x) = Np(x)/N-p(x) if p(x) < N and p*(x) = infinity if p(x)>= N. Consider the functional I: W-0(1,p(x))(Omega) -> R defined as
I(u)=(def)integral Omega 1/p(x)vertical bar del u vertical bar(p(x))dx - integral F-Omega(x,u(+))dx, for all u is an element of W-0(1,p(x))(Omega),
where F (x,u) - f(0)(s)f(x,s) ds. Theorem 1.1 proves that if u(0)is an element of C-1 ((Omega) over bar) 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(x))(Omega) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below.