A quasilinear problem without the Ambrosetti–Rabinowitz-type condition

Abstract

We study the existence of a positive solution of a p-superlinear equation involving the p-Laplacian operator. The main difficulty here is that the nonlinearity considered does not necessarily verify the well-known Ambrosetti–Rabinowitz condition. As an application, by performing an adequate change of variables we obtain an existence result of a quasilinear equation depending on the gradient.

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

“…Now, concerning (P λ ) with p = 2, the only results in the case λc 0 are, up to our knowledge, presented in [1,27]. In [27] the case c constant and h ≡ 0 is covered and in [1], the model equation is −∆ p u = |∇u| p + λf (x)(1 + u) b , b ≥ p − 1 and f 0.…”

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

“…Finally, our condition (L3) can be implied by the following stronger condition, which is assumed in many other papers, as in [19,32]:…”

“…The above condition is often assumed in many works; see [19], for example. Finally, our condition (L3) can be implied by the following stronger condition, which is assumed in many other papers, as in [19,32]:…”

“…For example, Miyagaki and Souto [32] studied a problem with parameter λ and adapted some monotonicity arguments as in [41,42]. In [19,20,36,42,45], the authors used a more suitable version of the Mountain Pass Theorem to overcome this difficulty. Nevertheless, the authors of these papers treated the equations with subcritical polynomial growth of the nonlinearity f (x, u):…”

Existence of multiple solutions to an elliptic problem with measure data

“…Thus g is p-superlinear at infinity, in the sense that lim |t|→∞ G(x,t) |t| p = ∞. Of late, the problem in (1.1) has been tackled without the AR condition by [7,14,16,19,20,21,23,24] and the references therein. Miyagaki [21] studied (1.1) with a Laplacian by using the following condition on g: ∃t 0 > 0 such that g(x,t) t is increasing for t ≥ t 0 and decreasing for t ≤ −t 0 ∀x ∈ Ω.…”