Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells

Sato, Yohei; Tanaka, Kazunaga

doi:10.1090/s0002-9947-09-04565-6

Cited by 35 publications

(21 citation statements)

References 28 publications

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“…We also refer to [5] for nonconstant b(x) > 0, where the authors prove the existence of k solutions that may change sign for any k and λ large enough. For other results related to Schrödinger equations with deep potential well, we may refer the readers to [10,19,18,21].…”

Section: Introductionmentioning

confidence: 99%

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

Alves

Figueiredo

Pimenta

2019

Bull Braz Math Soc, New Series

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In this work we prove the existence of ground state solutions for the following classwhere λ > 0, ∆ 1 denotes the 1−Laplacian operator which is formally defined by ∆ 1 u = div(∇u/|∇u|), V : R N → R is a potential satisfying some conditions and f : R → R is a subcritical and superlinear nonlinearity. We prove that for λ > 0 large enough there exists ground-state solutions and, as λ → +∞, such solutions converges to a ground-state solution of the limit problem in Ω = int(V −1 ({0})).

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Section: Introductionmentioning

confidence: 99%

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

Alves

Figueiredo

Pimenta

2019

Bull Braz Math Soc, New Series

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“…, u k ) ∈ H 1 ( 1 ) ⊕ · · · ⊕ H 1 ( k ) and ∈ (0, 1). It is showed in [15]. The key of the proof is the subsolution estimate in [16].…”

Section: Asymptotic Profile Of Solutions U (X)mentioning

confidence: 97%

“…In this section, we give our variational formulation of ( * ) , which is also used in [15] in different setting. In subsection (a), we give function spaces and norms which used throughout this paper.…”

Section: Variational Formulationmentioning

confidence: 99%

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

Sato

2007

Calc. Var.

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We study the nonlinear Schrödinger equations:where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has "critical frequency" inf x∈R N V(x) = 0. We also assume thatIn critical frequency cases, Byeon-Wang [5,6] showed the existence of singlepeak solutions which concentrating around global minimum of V(x). Their limiting profiles-which depend on the local behavior of the potential V(x)-are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u (x) as follows: rescaled function w (y) = g( ) 2 p−1 u (g( )y + x ) converges to a least energy solution of − w + V 0 (y)w = w p , w > 0 in 0 , w ∈ H 1 0 ( 0 ). Here g( ), V 0 (x) and 0 depend on the local behaviors of V(x). IntroductionIn this paper, we consider the following nonlinear Schrödinger equations:

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“…For other results related to Schrödinger equations with deepening potential well, we may refer the readers to Bartsch et al, Ding and Tanaka, Stuart and Zhou, Sato and Tanaka, and Wang and Zhou. [3][4][5][6][7] Taking into account the above commentaries, it is quite natural to ask if the results in Bartsch and Wang 1 still hold for problem (1.1). Unfortunately, we cannot draw a similar conclusion in a straight way, because of the fact that our nonlinearity is of the form (t) = t log t 2 , and that the energy functional associated to (1.1) is not of C 1 class.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the above results, Clapp and Ding studied a related problem where the nonlinearity has a critical growth. For other results related to Schrödinger equations with deepening potential well, we may refer the readers to Bartsch et al, Ding and Tanaka, Stuart and Zhou, Sato and Tanaka, and Wang and Zhou …”

Section: Introductionmentioning

confidence: 99%

On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

Alves

Filho

Figueiredo

2019

Math Methods in App Sciences

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In this work, we prove the existence of positive solution for the following class of problems −normalΔu+λVfalse(xfalse)u=ulogu2,1emx∈RN,u∈H1false(RNfalse), where λ>0 and V:RN→double-struckR is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C1 functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V−1({0})).

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Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells

Cited by 35 publications

References 28 publications

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

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