Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
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 .
In this paper, we study a fractional Schrödinger–Poisson system with p-Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.
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.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.