In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem -Delta u = beta(u)vertical bar del u vertical bar(2) + lambda f(x) in Omega, u = 0 on partial derivative Omega where beta is a continuous nondecreasing positive function and f belongs to some suitable Lebesgue spaces. (c) 2005 Elsevier Inc. All rights reserved

This paper deals with the existence and nonexistence results for quasilinear elliptic equations of the formwhere ∆ p := div (|∇u| p−2 ∇u), p > 1, and the solutions are understood in the sense of renormalized or, equivalently, entropy solutions. In particular we prove nonexistence results in the case f(x, u) = u p |x| − p , that is related to a classical Hardy inequality.

For 1 < p < N and −∞ < γ < N −p p we obtain the following improved Hardy-Sobolev Inequalities Ω

We study the existence of different types of positive solutions to problem, and 2 * = 2N N −2 is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0. B. Abdellaoui and I.

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problemsin Ω,Lipschitz boundary such that 0 ∈ Ω and N > 2s. We will mainly consider the solvability in two cases:(1) The linear problem, that is, f (x, t) = f (x), where according to the summability of the datum f and the parameter λ we give the summability of the solution u.(2) The problem with a nonlinear term f (x, t) = h(x) t σ for t > 0. In this case, existence and regularity will depend on the value of σ and on the summability of h. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.

Let 0 < s < 1 and p > 1 be such that ps < N . Assume that Ω is a bounded domain containing the origin. Starting from the ground state inequality by R. Frank and R. Seiringer in [16] to obtain:(1) The following improved Hardy inequality for p 2: For all q < p, there exists a positive constant C ≡ C(Ω, q, N, s) such that

This paper is devoted to the study of the elliptic problems with a critical potential,where N > 3, ¶ > 0 and 0 < q < 1 < p 6 (N + 2)=(N ¡ 2). Existence, multiplicity, behaviour in x = 0 and bifurcation are considered under some hypotheses in h and g.

Abstract. In this work we study the problemThe main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case α = 1, 1 + γ > 0; (ii) the nonexistence of solutions if α > 1, 1 + γ > 0; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions in the case 0 < α ≤ 1.