Superlinear Ambrosetti–Prodi problem for the p-Laplacian operator

Abstract

Abstract. Based on a new Liouville theorem, we study a superlinear Ambrosetti-Prodi problem for the p-Laplacian operator, 1 < p < N. For this, we use the sub and supersolution method, blow up technique and the Leray-Schauder degree theory.

An Ambrosetti–Prodi-type problem for the (p,q)-Laplacian operator

“…). Finally, we prove an a priori bound on eventual solutions and an upper bound for the parameter t. Initially, arguing as [4,20], we will prove the limitation in norm L ∞ (Ω) of negative part of an eventual solution of (P t,µ ). Lemma 3.1.…”

“…Motivated by the pioneering paper by Ambrosetti and Prodi [3], many generalizations and applications of the results in [3] have been achieved, such as the known results in the semilinear case [7,11,21,22]. In particular, the Ambrosetti-Prodi problem for the p-Laplace operator has been intensely studied, we can cite among them, the works of [1,4,18] that study the p-linear case, [5,20] that work with the p-superlinear case, [17] that study systems and [13,24] in which consider the Ambrosetti-Prodi problem with different boundary conditions. For the (p, q)-Laplace operator we do not know any result about this kind of problem.…”

Quasilinear elliptic equations with critical growth involving jumping nonlinearities

“…These situations can be treated variationally by the mountain pass theorem and were studied in [10] and [18] when has subcritical growth. For the p ‐Laplacian version, one can find related works in [2, 4, 5, 16]. Following these papers, critical cases for were investigated.…”

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

“…The results in [3] opened the door to many generalizations, and further investigation of problems of this type, but in different frameworks ( structures and boundary conditions). In particular, the Ambrosetti-Prodi problem for the p-Laplace operator and Dirichlet boundary conditions has been considered in [2,4,5,25,37,39,40], among many others. It is important to mention that for the Dirichlet problem, the regularity theory is not addressed, since the domain is assumed to be smooth (and thus the regularity results are standard and known).…”