Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains

Abstract

By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities.

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

“…Even so, much more can be said about the properties of u. Indeed, according to the next proposition, whose proof may be found in [32], if a(.) decays sufficiently fast to zero at infinity, then u is essentially bounded on compact subsets of Ω and, by virtue of Harnack's inequality, it is strictly positive in Ω.…”

“…However, the present work differs essentially from [32] since the equation considered here does not involve any supercritical terms and our primary interest lies on demonstrating Liouville-type theorems.…”

Solutions and multiple solutions for p(x)-Laplacian equations with nonlinear boundary condition

“…The author showed the existence result when f 1 (x, u) and f 2 (x, u) are superlinear and satisfy the AmbrosettiRabinowitz type condition, and got the multiplicity of solutions when one of f 1 (x, u) and f 2 (x, u) is sublinear and the other one is superlinear. For other relative results to the problem (1.3), we refer to [14][15][16][17] and the references therein.…”

Indefinite Quasilinear Neumann Problem on Unbounded Domains

“…The coefficients a and b are allowed to change signs while c is assumed to be nonnegative and measurable on Ω. This problem was considered in the paper [9]. The authors of that paper established the existence of a nonnegative nontrivial solution assuming that c ∈ L ∞ (Ω), b ≥ 0 and the coefficients h, a and b converge to 0 at a certain rate as |x| → ∞.…”

“…The authors of that paper established the existence of a nonnegative nontrivial solution assuming that c ∈ L ∞ (Ω), b ≥ 0 and the coefficients h, a and b converge to 0 at a certain rate as |x| → ∞. In this paper we consider this problem under different assumptions than those in [9]. More specifically, we assume that …”