Multiple normalized solutions for quasi-linear Schrödinger equations

Jeanjean, Louis; Luo, Tingjian; Wang, Zhi–Qiang

doi:10.1016/j.jde.2015.05.008

Cited by 82 publications

(51 citation statements)

References 30 publications

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“…Also the second solution u − corresponds to a critical point of mountain-pass type for F on S(c). The existence of two critical points on S(c), one being a local minimizer and the second one of mountain-pass type is reminiscent of recent works [7,16,19,26] where a similar structure have been observed for prescribed norm problems.…”

Section: Introductionmentioning

confidence: 61%

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

Cingolani

Jeanjean²

2019

SIAM J. Math. Anal.

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The paper deals with the existence of standing wave solutions for the Schrödinger-Poisson system with prescribed mass in dimension N = 2. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namelywhere c > 0 is a given real number. Under different assumptions on γ ∈ R, a ∈ R, p > 2, we prove several existence and multiplicity results. With respect to the related higher dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer and it forces the implementation of new ideas to catch the normalized solutions.

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Section: Introductionmentioning

confidence: 61%

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

Cingolani

Jeanjean²

2019

SIAM J. Math. Anal.

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“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Li¹,

Luo²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper, we study the existence and multiplicity of solutions with a prescribed L 2 -norm for a class of nonlinear Chern-Simons-Schrödinger equations in RTo get such solutions we look for critical points of the energy functional, we prove a sufficient condition for the nonexistence of constrain critical points of I on S r (c) for certain c and get infinitely many minimizers of I on S r (8π). For the value p ∈ (4, +∞) considered, the functional I is unbounded from below on S r (c). By using the constrained minimization method on a suitable submanifold of S r (c), we prove that for certain c > 0, I has a critical point on S r (c). After that, we get an H 1 -bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on S r (c) for any c ∈ 0,.

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“…On the other hand, when q ∈ (0, 2 + 4 N ) there holds that m(c) ∈ (−∞, 0]. Especially, if the energy is strictly less than zero, namely, [14] recently discovered that there existsĉ ∈ (0, c(q, N )), such that functional (1.7) admits a local minimum on the manifold {u ∈ X : |u| 2 L 2 = c} for all c ∈ (ĉ, c(q, N )) and q ∈ ( 4 N , 2 + 4 N ). Furthermore, mountain pass type critical point of (1.7) was also obtained therein for all c ∈ (ĉ, ∞), which is different from the minimum solution.…”

Section: Introductionmentioning

confidence: 99%

Existence and Asymptotic Behavior for the Ground State of Quasilinear Elliptic Equations

Zeng

Zhang

2018

Advanced Nonlinear Studies

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In this paper, we are concerned with the existence and asymptotic behavior of minimizers for a minimization problem related to some quasilinear elliptic equations. Firstly, we proved that there exist minimizers when the exponent q equals to the critical case q * = 2 + 4 N , which is different from that of [6]. Then, we proved that all minimizers are compact as q tends to the critical case q * when a < a * is fixed. Moreover, we studied the concentration behavior of minimizers as the exponent q tends to the critical case q * for any fixed a > a * .

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Multiple normalized solutions for quasi-linear Schrödinger equations

Cited by 82 publications

References 30 publications

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Existence and Asymptotic Behavior for the Ground State of Quasilinear Elliptic Equations

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