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

Abstract

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the AmbrosettiRabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞.

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

“…Instead, the Trudinger-Moser inequality [27,33] states that H 1 is continuously embedded into an Orlicz space defined by the Young function φ(t) = e αt 2 − 1. In [1,14,15,25], with the help of TrudingerMoser embedding, problems in a bounded domain were investigated, when the nonlinear term f behaves at infinity like e αs 2 for some α > 0. We refer the reader to [13] for a recently survey on this subject.…”

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

“…First of all, in dimension two we have recently established in [19] the existence of nontrivial nonnegative solutions to the Laplacian equation (i.e., p = 2) when the nonlinear term f has the subcritical or critical exponential growth of order exp(αu 2 ) but without satisfying the Ambrosetti-Rabinowitz condition. These results in dimension two in [19] extend those of [16] to the case when f does not have the (AR) condition. Second, there have been many works in the literature in which the (AR) condition was replaced by other alternative conditions when f has the polynomial growth.…”

Existence of Solution for an Elliptic Problem with a Sublinear Term

“…Then, by Maximum Principle, we have v > 0 in Ω, that is, v is a solution of the problem (6). For each n ∈ N, let us denote by v n the solution of (6).…”