Abstract
In this paper, we treat the existence of solutions for a class of general elliptic problems whose prototype is the following:
{-Delta(p)u + h(x)vertical bar u vertical bar(q-1) u = beta vertical bar del u vertical bar(p) + lambda f(x) in Omega
u = 0 on partial derivative Omega.
where Omega is a bounded open subset of R-N with N > 1, 1 < p < N, q >= 1, lambda is an element of R, beta is an element of R, h is an element of L-1 (Omega) with h >= 0 and f is an element of L-1 (Omega). Assuming that the source term f satisfies
lambda(1) (f) = inf{integral(Omega) vertical bar del w vertical bar(p)dx/integral(Omega)vertical bar f vertical bar vertical bar w vertical bar(p)dx: w is an element of W-0(1,p) (Omega) \ {0}} > 0,
we obtain the existence of a solution u is an element of W-0(1,p) (O) when vertical bar lambda vertical bar is sufficiently small.