Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

Quasilinear asymptotically periodic Schrödinger equations with critical growth

“…Proof Considering that the other limits have already been proven when we deal with the problem in the subcritical case (see [23]), we shall establish the second limit in Lemma 9. Given δ > 0, since w ∈ L 2 * (R N ), we find 0 < ε < δ such that, for every measurable set A ⊂ R N satisfying |A| < ε, A |w| 2 * < δ.…”

“…By using the Sobolev space H 1 (R N ), they proved the existence of solutions based on classical results given by Berestycki and Lions [3] when N = 1 or N ≥ 3, and Berestycki et al [4] when N = 2. In a recent article [23], the authors generalized the earlier results for the subcritical case by supposing (V ), (g 1 ), (g 2 ) and (g 5 )(ii) (for N ≥ 3), and a version of (g 3 ). We should also mention the article [18] where the authors used the Nehari method and considered a more general quasilinear problem including the ones to which the change of variables does not apply.…”

Existence of solutions for fractional Schrodinger equation with asymptotica- lly periodic terms

“…Motivated by above results, in this paper we study non-trivial solution and ground state solution to problem (1.1) under asymptotically periodic case of V and f at infinity. In the context about asymptotic periodic, we refer the reader to [1,12,23,24,31,32].…”

Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin

“…The same method was also used in [3], but the usual Sobolev space H 1 (R N ) framework was used as the working space. We refer the reader to [5,6,9,20,21] for more results. Usually, the authors consider the case that the function g(x, t) is sublinear at the origin and superlinear at infinity.…”

“…As it is well known, this type of condition provides the boundedness of the Palais-Smale sequences of the associated functional. More generally, under suitable extra assumptions, it is possible to deal with the condition lim |t|→+∞ G(x, t)/t 4 = +∞ (see [20,24]). …”