On the Schroedinger equation in $\mathbb{R}^{N}$ under the effect of a general nonlinear term

Azzollini, Antonio; Pomponio, Alessio

doi:10.1512/iumj.2009.58.3576

Cited by 69 publications

(72 citation statements)

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“…When α = 1, for example, we refer to [5,28]. When 0 < α < 1, we also have the Pohozaev identity (see (46)) and it might be useful to get a bounded Palais-Smale sequence in the case 0 < α < 1.…”

Section: Remark 12 (I)mentioning

confidence: 99%

Existence of solutions of scalar field equations with fractional operator

Ikoma

2016

J. Fixed Point Theory Appl.

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Abstract. In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation (1 − ∆) α u = f (u) in R N where 0 < α < 1, N ≥ 2 and f (s) is a Berestycki-Lions type nonlinearity. The characterization of the least energy by the mountain pass value is also considered and the existence of optimal path is shown. Finally, exploiting these results, the existence of positive solution for the equation (1 − ∆) α u = f (x, u) in R N is established under suitable conditions on f (x, s). IntroductionIn this paper, we are concerned with the existence of nontrivial solutions of (1) (where N ≥ 2 and 0 < α < 1. The fractional operator (1 − ∆) α u is defined byand H α (R N ) a fractional Sobolev space consisted by real valued functions, that is,Throughout this paper, we deal with a weak solution of (1), namely, a functionwhere a denotes the complex conjugate of a. The operator (1 − ∆) α is related to the pseudo-relativistic Schrödinger operator (m 2 − ∆) 1/2 − m (m > 0) and recently a lot of attentions are paid for equations involving them. Here we refer to [2-4, 12-17, 21, 23, 32, 34, 38] and references therein for more details and physical context of (1 − ∆) α . In these papers, the authors study the existence of nontrivial solution and infinitely many solutions for the equations with (m 2 − ∆) α and various nonlinearities. This paper is especially motivated by two papers [23] and [34]. In [23], the existence of positive solution of (1) is proved under the following conditions on f (x, s):(vi) There exist continuous functionsf (s) and a(x) such thatf satisfies (i)-(v) and 0On the other hand, in [34], the author obtains a nontrivial solution of (1) with f (x, s) = λb(x)|u| p−1 u + c(x)|u| q−1 u under different conditions on b(x), c(x), p, q where λ > 0 is a constant. Among other things, under 1 < p, q < 2 * α − 1 and some strict inequality for the mountain pass value (the infimum of the functional on the 2010 Mathematics Subject Classification. 35J60, 35S05.

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Section: Remark 12 (I)mentioning

confidence: 99%

Existence of solutions of scalar field equations with fractional operator

Ikoma

2016

J. Fixed Point Theory Appl.

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“…In this work we intend to address a question posed by Azzollini and Pomponio in [3] on the existence of a solution for the problem…”

Section: Introductionmentioning

confidence: 99%

“…The first part of [3] is dedicated to prove that (P ) has at least a radially symmetric positive solution under the assumptions that V (x) ≥ 0, lim |x|→∞ V (x) = 0 and V is radially symmetric, among other natural hypotheses. Furthermore, in case V is not necessarily symmetric they obtained a nonexistence result of a ground state solution of problem (P ) but in this more general setting, they did not show existence of a positive solution.…”

Section: Introductionmentioning

confidence: 99%

“…There is a problem of lack of compactness due to the fact that the problem is studied in an unbounded domain in R N . In general, this difficulty is circumvent by assuming symmetry properties of V , for example that V is radially symmetric as in [3,7] and [24] among many others. Our objective is to prove the existence of a positive solution of (P ) with V (x) V ∞ , lim |x|→∞ V (x) = V ∞ , verifying suitable decay assumptions, but not requiring any symmetry properties.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, one is motivated to use the more suitable projections on the set of points which satisfy the Pohozaev identity [19], the so-called the Pohozaev manifold of (P ), instead. In recent years the use of Pohozaev manifold was shown very effective when treating nonlinearities which do not satisfy Ambrosetti-Rabinowitz superquadraticity condition [2] and the monotonicity condition f (s)/s increasing [12]; this idea is performed in [3,16,18] and [23] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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