Elliptic Problems with Nonlinearities Indefinite in Sign

Abstract

Let 0/R N , N 3, be a bounded open set with smooth boundary and * # R. We study the Dirichlet problem,with 1<q<p and h, k # L (0) nonnegative functions. We prove existence, nonexistence and multiplicity results depending on * and according to the integrability properties of the ratio k p&1 Âh q&1 .

Existence of entire solutions for a class of quasilinear elliptic equations

“…Here, we solve the problem under conditions (1.1)-(1.4). Sections 2-4 of the paper are devoted to the proof of the following main existence By the comments above it is clear that Theorem A extends Theorem 1.1 of [2] to quasilinear elliptic equations in R N . Theorem 1.2 of [2] is a complete statement when 2 < q < 2 * and w satisfies (1.3) for p = 2, rather than Theorem 1.1 of [2].…”

“…Sections 2-4 of the paper are devoted to the proof of the following main existence By the comments above it is clear that Theorem A extends Theorem 1.1 of [2] to quasilinear elliptic equations in R N . Theorem 1.2 of [2] is a complete statement when 2 < q < 2 * and w satisfies (1.3) for p = 2, rather than Theorem 1.1 of [2]. Hence, it still remains an open problem the extension of Theorem 1.2 of [2] to quasilinear elliptic equations in R N , that is when (1.3) is replaced by the weaker condition w(w/h) (q−p)/(r−q) ∈ L N/p (R N ).…”

“…Later, Alama and Tarantello in [2] studied a related semilinear Dirichlet problem in a bounded domain, with weighted nonlinear terms. In [2] also solvability and multiplicity were proved under various assumptions on the weights and on the parameter λ ∈ R. The famous results of [3] were partially extended by De Figueiredo et al to indefinite nonlinearities for the semilinear case in [10] and for the p-Laplacian operator in [11]. For recent contributions on related semilinear Dirichlet problems in bounded domains we refer to [8,18] and on equations in the entire R N to [19], and to the references therein.…”

“…Theorem 1.2 of [2] is a complete statement when 2 < q < 2 * and w satisfies (1.3) for p = 2, rather than Theorem 1.1 of [2]. Hence, it still remains an open problem the extension of Theorem 1.2 of [2] to quasilinear elliptic equations in R N , that is when (1.3) is replaced by the weaker condition w(w/h) (q−p)/(r−q) ∈ L N/p (R N ). In any case, Theorem A extends the first part of Theorem 1.2 of [2] under condition (1.4).…”

Multiple positive solutions for a superlinear problem: A topological approach

“…Existence or multiplicity results of positive solutions for (1.3) in the superlinear indefinite case have been obtained in [2,3,8,9] (see also [33] for a more complete list of references concerning different aspects related to the study of superlinear indefinite problems, including the case of non-positive oscillating solutions). Typically, the right-hand side of (1.3) takes the form f (x, s) = λs + a(x)g(s), with λ a real parameter.…”