Abstract
In a bounded domain Omega subset of R-N, N >= 2, we study the solvability of the boundary value problem
-Delta(p)u = H(x, u, del u) in Omega, u =0 on partial derivative Omega
where -Delta(p)u = -div (vertical bar del u vertical bar p(-2) del u) is the p-Laplace operator with 1 < p < N and the nonlinearity H satisfies vertical bar H(x, u, del u)vertical bar <= a vertical bar del u vertical bar(q) + b vertical bar u vertical bar(r) + f(x) with a, b >= 0 and q, r > 0. Assuming f is an element of L-m (Omega) for a suitable exponent m > 1, we prove the existence of solutions for different sets of values (q, r). To this end, we apply the classical Schauder fixed point theorem, relying on some well-known uniqueness, regularity estimates and stability results for renormalized and weak solutions. In the last section we make some examples to illustrate the applicability of our results to a large class of equations.