In this paper we deal with the existence of critical points of functionals defined on the Sobolev space W-0(1,p)(Omega), p > 1, by J(u) = (Omega)integral S(x, u, Du) dx, where Omega is a bounded, open subset of R(N). Even for very simple examples in R(N) the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem

In this work we study the existence of a solution for the problem − p u = f u + t x + h x , with homogeneous Dirichlet boundary conditions. Here the nonlinear term f u is a so-called jumping nonlinearity. In the proofs we use topological arguments and the sub-supersolutions method, together with comparison principles for the p-Laplacian.

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We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with Singular lower order terms that have natural growth with respect to the gradient, whose model is {-Delta u + vertical bar del u vertical bar(2)/u(gamma) = f in Omega. u = 0 on partial derivative Omega. where Omega is an open bounded subset of R, gamma > 0 and f is a function which is strictly positive on every compactly contained subset of Omega. As a consequence of our main results, we prove that the condition gamma < 2 is necessary and sufficient for the existence of solutions in H(0)(1) (Omega) for every sufficiently regular f as above. (C) 2009 Elsevier Inc. All rights reserved

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