Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

Abstract

We investigate the problemin a bounded smooth domain Ω ⊂ R N . Using a Kazdan-Kramer change of variable we reduce this problem to a quasilinear one without gradient term and therefore approachable by variational methods. In this way we come to some new and interesting problems for quasilinear elliptic equations which are motivated by the need to solve (P ). Among other results, we investigate the validity of the Ambrosetti-Rabinowitz condition according to the behavior of g and f . Existence and multiplicity results for (P ) are established in several situations.

Existence of Solutions for a Quasilinear System with Gradient Terms

“…At this point, we quote that, to the best of our knowledge, it is the first time that the system (1) is considered under the conditions (H 1 )-(H 2 ). This manuscript also completes the study done in [2] due to the fact that a system version of the problem in [2] is studied and the papers [1,[3][4][5][6][7][8] in the sense that different hypotheses can be considered to study systems with convection terms and involving the p-Laplacian operator.…”

“…In the last decades, elliptic problems involving gradient terms have been attracting the attention of several researchers due to interesting difficulties which arise when one intends to consider this kind of problem, see for instance [1][2][3].…”

“…In [3], it is considered the problem of a Kazdan-Kramer change of variable, the equation is reduced to a quasilinear one without gradient term which is approachable by variational methods. Such a method allowed the authors to obtain several existence and multiplicity results.…”

Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

“…It implies that I λ does not satisfies an Ambrosetti-Rabinowitz-type condition and proving that Palais-Smale or Cerami sequences are bounded may be challenging. In the case of the Laplacian, when p = 2, dealing with this issue is now relatively standard but for elliptic problems with a p-Laplacian things are more complex and we refer to [18,27,28,34] in that direction. Note however that in these last works, it is always assumed a kind of homogeneity condition which is not available here.…”

A type of Brézis–Oswald problem to $$\Phi $$-Laplacian operator with strongly-singular and gradient terms

“…This property is crucial in variational methods to show the boundedness of Palais-Smale sequences. For instance see [4,12,16,30] and references therein.…”