scholarly journals Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

2021 ◽  
Vol 6 (6) ◽  
pp. 5837-5850
Author(s):  
Yile Wang ◽  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Nonlinear Schrodinger Equation ◽  
Inverse Power ◽  
Power Type ◽  
Nonlinear Schrödinger

scholarly journals Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

2022 ◽  
Vol 7 (4) ◽  
pp. 5957-5970
Author(s):  
Yali Meng ◽  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Nonlinear Schrodinger Equation ◽  
Inverse Power ◽  
Concentration Compactness ◽  
Nonlinear Schrödinger ◽  
Concentration Compactness Principle ◽  
Compactness Principle

<abstract><p>In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ 0 &lt; \gamma &lt; \infty $, $ 0 &lt; \sigma &lt; 2 $ and $ \frac{4}{N} &lt; \alpha &lt; \frac{4}{N-2} $. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a $ \gamma_0 &gt; 0 $ sufficiently small such that $ 0 &lt; \gamma &lt; \gamma_0 $ and for any $ a\in(0, a_0) $, there exist stable standing waves for the problem in the $ L^2 $-supercritical case. Our results are complement to the result of Li-Zhao in <sup>[<xref ref-type="bibr" rid="b23">23</xref>]</sup>.</p></abstract>

scholarly journals Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential

2006 ◽  
Vol 2006 ◽  
pp. 1-7
Author(s):  
Guanggan Chen ◽  
Jian Zhang ◽  
Yunyun Wei
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Variational Calculus ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Unbounded Potential

This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential iϕt=−Δϕ+V(x)ϕ−μ|ϕ|p−1ϕ−λ|ϕ|q−1ϕ, x∈ℝN, t≥0, where μ>0, λ>0, and 1<p<q<1+4/N. The potential V(x) is bounded from below and satisfies V(x)→∞ as |x|→∞. From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.

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