Carlos Alberto Santos

Infinite many blow-up solutions for a Schrödinger quasilinear elliptic problem with a non-square diffusion term

Abstract In this paper, we consider existence of positive solutions for the quasilinear elliptic Schrödinger problem where and are non-negative and continuous functions with g being non-decreasing as well, , and . By a dual approach we establish sufficient conditions for existence and multiplicity of solutions for this problem.

Existence and non-existence of blow-up solutions for a non-autonomous problem with indefinite and gradient terms

We deal with existence and non-existence of non-negative entire solutions that blow-up at infinity for a quasilinear problem depending on a non-negative real parameter. Our main objectives in this paper are to provide far more general conditions for existence and non-existence of solutions. To this end, we explore an associated μ-parameter convective ground state problem, sub and super solutions method combined with approximation arguments to show existence of solutions. To show the result of non-existence of solutions, we follow an idea due to Mitidieri–Pohozaev.

On the existence and asymptotic behaviour of bounded positive entire solutions for quasilinear elliptic problems

We establish new results concerning the existence and asymptotic behaviour of solutions for the nonlinear elliptic problem where Δ p u = div(|∇u| p−2∇u), with 1 < p < N, denotes the p-Laplacian operator and f : ℝ N  × (0, ∞) → ℝ is a suitable continuous function. The principal aim of this article is to study the case 0 < l < ∞, because the extreme cases l = 0 and l = ∞ have been intensely studied in recent years. The main tools we use to prove the principal results are the method of lower and upper solutions, an argument of penalization and a technique of monotonization–regularization of the nonlinearity f.We establish new results concerning the existence and asymptotic behaviour of solutions for the nonlinear elliptic problem where Δ p u = div(|∇u| p−2∇u), with 1 < p < N, denotes the p-Laplacian operator and f : ℝ N  × (0, ∞) → ℝ is a suitable continuous function. The principal aim of this article is to study the case 0 < l < ∞, because the extreme cases l = 0 and l = ∞ have been intensely studied in recent years. The main tools we use to prove the principal results are the method of lower and upper solutions, an argument of penalization and a technique of monotonization–regularization of the nonlinearity f.