A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity

Abstract

Abstract. In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on R N . The main difficulties to overcome are the lack of a priori bounds for PalaisSmale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the "Problem at infinity" is autonomous, in contrast to just periodic, can be used in order to regain compactness.Mathematics Subject Classification. 35J60, 58E05.

Positive solutions for nonlinear Schrödinger equations with deepening potential well

“…If f (s) is sublinear, some existence results for (1.1) were obtained recently in [33]. Moreover, we mention that the asymptotically linear problem (1.1) with V λ (x) ≡ V (x) was also discussed by many authors, for example, Jeanjean and Tanaka [16,17], Liu and Wang [24,23] et al, even if V (x) vanishes at infinity [21]. As pointed out in [31], the results of [16,24] imply that (1.1) may have a positive solution for λ not necessarily large.…”

“…As pointed out in [31], the results of [16,24] imply that (1.1) may have a positive solution for λ not necessarily large. In our notation, the existence results for (1.1) proved in [16] and [24] are based on the following conditions:…”

“…Instead of using abstract critical point theorems, we seek critical points of I by simply using the following version of the Mountain Pass Theorem, applied in [8,16]. Proposition 1.1 (Mountain Pass Theorem).…”

“…Next, we try to use the properties (1.8) to get a contradiction via w. Motivated by [15], we seek a contradiction by applying the following variant of the concentration-compactness lemma [35, Lemma 2.1], essentially based on [20]. This kind of idea was used in many papers, for example, [16,18,14].…”

Radial solutions with a vortex to an asymptotically linear elliptic equation

“…Then v(x) should satisfy problem (1.1) with V (x) = K(x)−E. Especially, problem (1.1) has been studied when V (x) ∈ C(R N , R) and the nonlinearity f is asymptotically linear at infinity (see [3], [8], [12], [13] and references therein). In the physical point of view, asymptotically linear nonlinearity is said to have the saturation effect.…”