Multiplicity of Solutions for a Class of Elliptic Problem of P-Laplacian Type with a P-Gradient Term

We consider the following problem: -Delta(p) mu= c (x)vertical bar u vertical bar(q-1) u +mu vertical bar del mu vertical bar(p) + h(x) in Omega, mu = 0 on partial derivative Omega, where O is a bounded set in R.. (N >= 3) with a smooth boundary, 1 <p < N, q > 0, mu is an element of R*, and.. and h belong to L-k (Omega) for some k > N/P. In this paper, we assume that c >= 0 a.e. in Omega and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that parallel to c parallel to(k) and parallel to h parallel to are suitably small. For this purpose, we use theMountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity.