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

Abstract

Let Ω be a bounded domain in R N . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type:when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N −Laplacian in R N when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami [11,12]. This approach can also be used to yield an existence result for the p-Laplacian equation (1 < p < N ) in the subcritical polynomial growth case.1991 Mathematics Subject Classification. 35B38, 35J92, 35B33, 35J62.

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

“…Indeed, observe that function H (x, s) is a "quasi-monotonic" function, and also if H is a monotonic function in s < 0 and s > 0, or a convex function in R, then it satisfies (L3) with θ = 1. Finally, we remark that existence and multiplicity results of nontrivial nonnegative solutions to N -Laplacian equations in R N with nonlinear terms of critical exponential growth exp(α|u| N N−1 ) and without the AR condition have been recently established by the authors in [21,22]. Existence of nontrivial and nonnegative solutions to polyharmonic equations of exponential growth and without the AR condition has also been carried out in [23].…”

Existence results for asymmetric fractional ‐Laplacian problem

“…Our work is to study asymmetric problem (1.1) without the (AR)‐condition in the positive semi‐axis . Our results have an essential difference compared with since our nonlinearities have different growth behavior at and In fact, this condition was studied by Liu and Wang in in the case of Laplacian (i.e., ) by the Nehari manifold approach. In this paper, by using the mountain pass theorem and its suitable version, we try to get the nontrivial solutions to problem (1.1) with .…”

“…In this paper, by using the mountain pass theorem and its suitable version, we try to get the nontrivial solutions to problem (1.1) with . Our results and the proof of compactness condition are different from those in .…”

“…Under almost all of the works mentioned above and the motivation of Lam and Lu , our first main results will be to study problem (1.1) in the improved subcritical polynomial growth uniformly for all which is weaker than usual subcritical growth condition, i.e. there is a constant such that uniformly for all Note that in this case, we do not have the Sobolev compact embedding anymore.…”

“…Under almost all of the works mentioned above and the motivation of Lam and Lu [14], our first main results will be to study problem (1.1) in the improved subcritical polynomial growth…”