-
Previous Article
On the geometric dependence of Riemannian Sobolev best constants
- CPAA Home
- This Issue
-
Next Article
Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$
Multiplicity of solutions for elliptic systems via local Mountain Pass method
1. | Universidade Federal da Campina Grande, Departamento de Matemática, 58109-970, Campina Grande - PB, Brazil |
2. | Universidade Federal do Pará, Departamento de Matemática, 66075-100, Belém - PA, Brazil |
3. | Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília - DF |
$-\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)$ in $\mathbb{R}^N,$
$-\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)$ in $\mathbb{R}^N, $
$u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0$ for each $x \in \mathbb{R}^N, $
where $\varepsilon>0$, $W$ and $V$ are positive potentials and $Q$ is a homogeneous function with subcritical growth. We relate the number of solutions with the topology of the set where $W$ and $V$ attain their minimum values. In the proof we apply Ljusternik-Schnirelmann theory.
[1] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461 |
[2] |
Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145 |
[3] |
Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436 |
[4] |
Hongyu Ye. Positive solutions for critically coupled Schrödinger systems with attractive interactions. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 485-507. doi: 10.3934/dcds.2018022 |
[5] |
Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138 |
[6] |
Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 |
[7] |
Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036 |
[8] |
Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118 |
[9] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure and Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[10] |
Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure and Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033 |
[11] |
Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125 |
[12] |
Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4705-4736. doi: 10.3934/dcds.2021054 |
[13] |
Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure and Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413 |
[14] |
D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 |
[15] |
Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 849-861. doi: 10.3934/dcdsb.2016.21.849 |
[16] |
Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure and Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283 |
[17] |
Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 |
[18] |
Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations and Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233 |
[19] |
Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 |
[20] |
Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]