On a semilinear Schrödinger equation with critical Sobolev exponent

Chabrowski, J.; Szulkin, Andrzej

doi:10.1090/s0002-9939-01-06143-3

Cited by 70 publications

(42 citation statements)

References 15 publications

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“…There are many results on the existence of bound states and ground state solutions. For example, see. The problem (NLS) with periodic potential has been widely investigated for both its importance and mathematical interests.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Zhang

2014

Math Methods in App Sciences

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Zhang

2014

Math Methods in App Sciences

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“…By using the generalized linking theorem of Kryszewski-Szulkin and Bartsch-Ding, the author obtained infinitely many geometrically distinct weak solutions of (16). Chabrowski and Szulkin [3] investigated the semilinear Schrödinger equation with indefinite linear part…”

Section: Introduction Consider the Choquard Equationmentioning

confidence: 99%

“…However, to our knowledge, there seem to be no results on nontrivial solutions for (1) with strongly indefinite linear part and upper critical exponent. Motivated by [3,26] and aforementioned works, in the present paper, we deal with the case when 0 lies in a gap of the spectrum σ(A) and p = N +α N −2 . In addition to (G 1 )-(G 2 ), we also assume that (G 3 ) G(s) := 1 2 g(s)s − G(s) > 0 if s = 0, and there exist c 0 > 0, σ ∈ (0, 1) and r 0 > 0 such that…”

Section: Introduction Consider the Choquard Equationmentioning

confidence: 99%

“…We will overcome this difficulty by estimating the norm of Ψ 1 (u n ) (see(23)), similar to that of Ackermann [1]. Secondly, since the equation with critical growth brings many obstacles for recovering the compactness, it becomes very hard to prove that the Cerami sequence {u n } is non-vanishing; we can't apply the arguments in [3] directly because of the absorption of the convolution term. We shall introduce some new tricks to overcome this difficulty.…”

Section: Introduction Consider the Choquard Equationmentioning

confidence: 99%

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Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent

Guo¹,

Tang²,

Zhang³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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This paper is dedicated to studying the Choquard equation −∆u + V (x)u = (Iα * |u| p)|u| p−2 u + g(u), x ∈ R N , u ∈ H 1 (R N), where N ≥ 4, α ∈ (0, N), V ∈ C(R N , R) is sign-changing and periodic, Iα is the Riesz potential, p = N +α N −2 and g ∈ C(R, R). The equation is strongly indefinite, i.e., the operator −∆+V has infinite-dimensional negative and positive spaces. Moreover, the exponent p = N +α N −2 is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on g, we obtain the existence of nontrivial solutions for this equation.

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“…The Schrödinger equations have been extensively studied by many authors. We refer, for instance, to and the references therein. Recently, the existence of infinitely many solutions has been studied by many authors, see .…”

Section: Introductionmentioning

confidence: 99%