Standing waves for semilinear Schrödinger equations with discontinuous dispersion
Abstract: We consider here semilinear Schrödinger equations with a non standard dispersion that is discontinuous at x = 0. We first establish the existence and uniqueness of standing wave solutions for these equations. We then study the orbital stability of these standing wave into a subspace of the energy space that where classical methods as the concentration-compactness method of P.L. Lions can be used.
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 3 publications
References 17 publications
In this paper, we construct the solutions to the following nonlinear Schrödinger system $$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ - ϵ 2 Δ u + P ( x ) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q ( x ) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where $$3< p<+\infty $$ 3 < p < + ∞ , $$N\in \{1,2\}$$ N ∈ { 1 , 2 } , $$\epsilon >0$$ ϵ > 0 is a small parameter, the potentials P, Q satisfy $$0<P_{0} \le P(x)\le P_{1}$$ 0 < P 0 ≤ P ( x ) ≤ P 1 and Q(x) satisfies $$0<Q_{0} \le Q(x)\le Q_{1}$$ 0 < Q 0 ≤ Q ( x ) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When $$x_{0}$$ x 0 is a local maximum point of the potentials P and Q and $$P(x_{0})=Q(x_{0})$$ P ( x 0 ) = Q ( x 0 ) , we construct k spikes concentrating near the local maximum point $$x_{0}$$ x 0 . When $$x_{0}$$ x 0 is a local maximum point of P and $$\overline{x}_{0}$$ x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point $$x_{0}$$ x 0 and m spikes of v concentrating at the local maximum point $$\overline{x}_{0}$$ x ¯ 0 when $$x_{0}\ne \overline{x}_{0}.$$ x 0 ≠ x ¯ 0 . This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case $$N=3$$ N = 3 , $$p=3$$ p = 3 .
In this paper, we construct the solutions to the following nonlinear Schrödinger system $$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ - ϵ 2 Δ u + P ( x ) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q ( x ) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where $$3< p<+\infty $$ 3 < p < + ∞ , $$N\in \{1,2\}$$ N ∈ { 1 , 2 } , $$\epsilon >0$$ ϵ > 0 is a small parameter, the potentials P, Q satisfy $$0<P_{0} \le P(x)\le P_{1}$$ 0 < P 0 ≤ P ( x ) ≤ P 1 and Q(x) satisfies $$0<Q_{0} \le Q(x)\le Q_{1}$$ 0 < Q 0 ≤ Q ( x ) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When $$x_{0}$$ x 0 is a local maximum point of the potentials P and Q and $$P(x_{0})=Q(x_{0})$$ P ( x 0 ) = Q ( x 0 ) , we construct k spikes concentrating near the local maximum point $$x_{0}$$ x 0 . When $$x_{0}$$ x 0 is a local maximum point of P and $$\overline{x}_{0}$$ x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point $$x_{0}$$ x 0 and m spikes of v concentrating at the local maximum point $$\overline{x}_{0}$$ x ¯ 0 when $$x_{0}\ne \overline{x}_{0}.$$ x 0 ≠ x ¯ 0 . This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case $$N=3$$ N = 3 , $$p=3$$ p = 3 .
<abstract><p>This paper is devoted to considering the attainability of minimizers of the $ L^2 $-constraint variational problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ m_{\gamma, a} = \inf \, \{J_{\gamma}(u):u\in H^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}} \vert u\vert^2 dx = a^2 \} {, } $\end{document} </tex-math></disp-formula></p> <p>where</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ J_{\gamma}(u) = \frac{\gamma}{2}\int_{\mathbb{R}^{N}} \vert\Delta u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} \vert\nabla u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} V(x)\vert u\vert^2 dx-\frac{1}{2\sigma+2}\int_{\mathbb{R}^{N}} \vert u\vert^{2\sigma+2} dx, $\end{document} </tex-math></disp-formula></p> <p>$ \gamma > 0 $, $ a > 0 $, $ \sigma\in(0, \frac{2}{N}) $ with $ N\ge 2 $. Moreover, the function $ V:\mathbb{R}^{N}\rightarrow [0, +\infty) $ is continuous and bounded. By using the variational methods, we can prove that, when $ V $ satisfies four different assumptions, $ m_{\gamma, a} $ are all achieved.</p></abstract>
Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential
<abstract><p>In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.</p></abstract>
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
