May  2007, 19(2): 447-467. doi: 10.3934/dcds.2007.19.447

On domains and their indexes with applications to semilinear elliptic equations

1. 

Department of Applied Mathematics, Hsuan Chuang University, Hsinchu

Received  May 2005 Revised  October 2005 Published  July 2007

Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $N\geq1$, and $2^$∗$=\infty$ if $N=1,2$, $2^$∗$=\frac{2N}{N-2}$ if $N>2$, $2 < p < 2^$∗. Consider the semilinear elliptic equation $ -\Delta u+u=|u|^{p-2}u\text{ in }\Omega; u\in H_{0}^{1}(\Omega). $ The existence, the nonexistence, and the multiplicity of positive solutions of the equation are affected by the geometry and the topology of the domain $\Omega$. In the article, we first present various analyses and use them to characterize which domain $\Omega$ is a ground state domain or a non-ground state domain. Secondly, for a $y$-symmetric domain $\Omega$, we study their index $\alpha(\Omega)$ and $y$-symmetric index $\alpha_{s}(\Omega)$. We determine whether $\alpha(\Omega)=\alpha_{s}(\Omega)$ or $\alpha (\Omega)<\alpha_{s}(\Omega)$. In case that $\alpha(\Omega)<\alpha_{s}(\Omega)$ and that both $\alpha(\Omega)$ and $\alpha_{s}(\Omega)$ admits ground state solutions, then we obtain that in $\Omega$, the equation has three positive solutions, of which one is $y$-symmetric and other two are not $y$-symmetric.
Citation: Hwai-Chiuan Wang. On domains and their indexes with applications to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447
[1]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[2]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17

[3]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329

[4]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987

[5]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

[6]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393

[7]

Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123

[8]

Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126

[9]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108

[10]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[11]

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

[12]

Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943

[13]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure and Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813

[14]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195

[15]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure and Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71

[16]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[17]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[18]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[19]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure and Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030

[20]

Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]