Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well

Du, Miao; Li, Tian; Wang, Jun; Zhang, Fubao

doi:10.1063/1.4941036

Journal of Mathematical Physics

2016

DOI: 10.1063/1.4941036

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Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well

Miao Du

Tian Li

Jun Wang

et al.

Abstract: In this paper, we are concerned with a class of Schrödinger-Poisson systems with the asymptotically linear or asymptotically 3-linear nonlinearity. Under some suitable assumptions on V , K, a, and f , we prove the existence, nonexistence, and asymptotic behavior of solutions via variational methods. In particular, the potential V is allowed to be sign-changing for the asymptotically linear case. C 2016 AIP Publishing LLC.

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Cited by 15 publications

(16 citation statements)

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“…Some other existent results on radial solutions can be found in [31] and references therein, where different methods are employed. Besides the works mentioned above, some other type of systems such as Schrödinger-Poisson system with steep potential well [6,11,32,34] and Schrödinger-Poisson system with critical growth [8,28] are also considered.…”

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confidence: 99%

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Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Cheng

Wang

2022

DCDS-B

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In the present paper we study a class of Schrödinger-Poisson equations<disp-formula><label/><tex-math id="FE0"> \begin{document}$\begin{equation} \begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases} \end{equation}\quad\quad\quad (1)$ \end{document}</tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> are of different forms on the half space, i.e. <inline-formula><tex-math id="M3">\begin{document}$ V(x) = V_{1}(x), a(x) = a_{1}(x) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ x_{1}>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ V(x) = V_{2}(x), a(x) = a_{2}(x) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ x_{1}<0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ V_{1},V_{2},a_{1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ a_{2} $\end{document}</tex-math></inline-formula> are periodic in each coordinate direction. By using a concentration compactness discussion, we establish the existence of surface gap soliton ground state of (1) for <inline-formula><tex-math id="M9">\begin{document}$ p\in [3,5) $\end{document}</tex-math></inline-formula>. We also give a Mountain-Pass type ground state of (1) for <inline-formula><tex-math id="M10">\begin{document}$ p\in (3,5) $\end{document}</tex-math></inline-formula>.

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“…Furthmore, if the surface gap soliton is ground state of ( 7), then we call it surface gap soliton ground state. We will still use such a terminology for system (6). Due to the physical background for system ( 6) and ( 7), we study nontrivial solutions of ( 6) and establish the existence of ground state of ( 6) in this paper.…”

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confidence: 99%

Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Cheng

Wang

2022

DCDS-B

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“…Later, steep potential well was first applied into Schrödinger-Poisson system in [17], and then has been widely applied to the study of Schrödinger-Poisson system. We refer the reader to [14,17,22,26,28] and the references therein.…”

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confidence: 99%

“…In particular, the potential V is allowed to be sign-changing for the case 4 < p < 6. Later, Du et al [14] considered system (1…”

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confidence: 99%

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An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well

Wang¹

2021

CPAA

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In this paper, we study an eigenvalue problem for Schrödinger-Poisson system with indefinite nonlinearity and potential well as follows: −∆u + µV (x)u + K(x)φu = λf (x)u + g(x)|u| p−2 u in R 3 , −∆φ = K(x)u 2 in R 3 , where 4 ≤ p < 6, the parameters µ, λ > 0, V ∈ C(R 3) is a potential well with the bottom Ω := {x ∈ R 3 : V (x) = 0}, and the functions f and g are allowed to be sign-changing. By establishing an approximate estimate between the positive principal eigenvalue of −∆u + µV (x)u = λf (x)u in R 3 and the positive principal eigenvalue λ 1 (f Ω) of −∆u = λf Ω (x)u in Ω, we prove that at least a positive solution exists in 0 < λ ≤ λ 1 (f Ω) while at least two positive solutions exist in λ > λ 1 (f Ω) and near λ 1 (f Ω), where f Ω := f | Ω. The results are obtained via variational method and steep potential. Furthermore, we also study the concentration of solutions as µ → ∞ and the decay rate of solutions at infinity.

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The number of positive solutions for Schrödinger–Poisson system under the effect of eigenvalue

Sun

Wang

2022

Math Methods in App Sciences

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We investigate the eigenvalue problem for the Schrödinger-Poisson system as follows:A bounded Cerami sequence is established by using the truncation technique, decomposition of Hilbert space and mountain pass theorem. Then exploiting the property of steep potential well, under some proper assumptions on K, we conclude that one positive solution exists for 0 < 𝜆 < 𝜆 1 (𝑓 Ω ) while two positive solutions exist forand 𝜙 1 > 0 is the corresponding principal eigenfunction. Moreover, an interesting phenomenon is that when ∫ Ω g𝜙 p 1 dx > 0, one positive solution is always present near 𝜆 1 (𝑓 Ω ) for 𝜆 < 𝜆 1 (𝑓 Ω ) unaffected by the non-local term, however large its norm might be.

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Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well

Cited by 15 publications

References 42 publications

Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well

The number of positive solutions for Schrödinger–Poisson system under the effect of eigenvalue

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