On superlinear Schrödinger equations with periodic potential

Liu, Shibo

doi:10.1007/s00526-011-0447-2

Cited by 91 publications

(55 citation statements)

References 13 publications

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“…Assume lim n→∞ sup y∈Z 3 B(y,r) |v n | 2 dx = 0 for some fixed r > √ 3. By Lemma 4.5, lim n→∞ R 3 Φ(|v n |) dx = 0, and arguing similarly as Liu [23], we obtain a contradiction. More precisely, recalling J ′ (v n , w n )[w n ] = 0, Proposition 4.1 with t n = s/ v n D and ψ n = −t n w n implies that for every s > 0,…”

mentioning

confidence: 59%

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Mederski

Schino

Szulkin

2019

Arch Rational Mech Anal

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We look for ground states and bound states E : R 3 → R 3 to the curl-curl problemwhich originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of ∇ × (∇ × ·).The growth of the nonlinearity f is controlled by an N -function Φ :We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in R 3 and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations. 2 2010 Mathematics Subject Classification. Primary: 35Q60; Secondary: 35J20, 78A25.In general J ′ is not (sequentially) weak-to-weak * continuous, however we show the weakto-weak * continuity of J ′ for sequences on the topological manifold M. Obviously, the same regularity holds for E ′ and M E .for any (φ, ψ) ∈ V × W.

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confidence: 59%

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Mederski

Schino

Szulkin

2019

Arch Rational Mech Anal

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“…As we know, the above inequality holds under the conditions (F 1 ), (S 2 ) and (S 3 ) or (S 4 ) (see also [23,24]). However, this inequality is no longer valid under the conditions we considered in this paper, hence their arguments are collapsed in this case.…”

Section: Remark 13mentioning

confidence: 82%

“…To our knowledge, there is no work devoted to this kind of problems under weaker assumptions, hence our result is new. As a motivation we recall that there are a large number of literatures devoted to the study of the existence of ground state solutions for Schrödinger equation or system, we refer readers to [25,24,26,27,23,22,[28][29][30][31][32][33][34][35][36][37][38] and the references therein.…”

Section: Remark 13mentioning

confidence: 99%

Ground state solutions for a diffusion system

Zhang

Tang

Zhang

2015

Computers & Mathematics with Applications

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“…Conditions (G1), (G2), (G4) are standard and considered e.g. in [1,10,17,21,32] and in the references therein. For large |u|, (G3) describes super-quadratic behaviour of G in K whereas (G5) provides quadratic behaviour of G outside K. If K = R N , (G5) is redundant and there is a series of results concerning the existence and multiplicity of solutions.…”

Section: Introductionmentioning

confidence: 99%

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Bieganowski¹,

Mederski²

2019

Applied Mathematics Letters

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We prove the existence results for the Schrödinger equation of the formwhere g is superlinear and subcritical in some periodic set K and linear in R N \ K for sufficiently large |u|. The periodic potential V is such that 0 lies in a spectral gap of −∆ + V . We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that g is odd in u.Recently it has been shown that materials with large range of prescribed properties can be created ([12, 20, 24, 26, 27]) with different linear and nonlinear effects. Our aim is to model a wide range of nonlinear phenomena that allow to consider a composite of materials with different nonlinear polarization. In our case, the polarization g(x, ·) may be linear for some x ∈ R N \K (for sufficiently large |u|) and nonlinear outside of it, where K is a given Z N -periodic subset of R N . We admit Date: May 14, 2019.Lemma 6.1. Let β ≥ c N and suppose that K has a finite number of distinct orbits. If (u n ), (v n ) ⊂ X + are two Cerami sequences for J • m such thatand lim inf n→∞ u n − v n < κ, then lim n→∞ u n − v n = 0.

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On superlinear Schrödinger equations with periodic potential

Cited by 91 publications

References 13 publications

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Ground state solutions for a diffusion system

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

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