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

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals J λ , λ ∈ I ⊂ R. Under suitable assumptions, we locate an open subinterval of values λ in I for which J λ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problemwhere « is a bounded open domain in R N with smooth boundary. Under suitable assumptions on the matrix A(x; s), and depending on the behaviour of the function f near u = 0 and near u = +1 , we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in di® erent directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

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In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem isHere Ω is an open bounded set of ℝ N , γ(x) is a positive continuous function and f(x) is a positive function that belongs to a certain Lebesgue space. We prove that there exists a solution to this problem in the natural energy space H (Ω) when γ(x) ≤ in a strip around the boundary. For another case, we prove that the solution belongs to H loc (Ω) and that it is zero on the boundary in a suitable sense.

In this paper we analyze the existence, non-existence and uniqueness of positive solutions of some nonlinear elliptic equations containing singular terms and natural growth in the gradient. We use an adequate sub-supersolution method to prove the existence of solutions, different arguments for the nonexistence and results from Arcoya-Segura de León for the uniqueness.

Abstract. For a continuous function g ≥ 0 on (0, +∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ∇u, −∆u + g(u)|∇u| 2 , with a power type nonlinearity, λu p + f 0 (x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.

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