American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity
  • DCDS Home
  • This Issue
  • Next Article
    Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space
January  2008, 21(1): 69-89. doi: 10.3934/dcds.2008.21.69

Supercritical elliptic problems from a perturbation viewpoint

Manuel del Pino 1,

1. 

Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Received  March 2007 Published  February 2008

We review some recent existence results for the elliptic problem $\Delta u + u^p =0$, $u>0$ in an exterior domain, $\Omega = \R^N\setminus \D$ under zero Dirichlet and vanishing conditions, where $\D$ is smooth and bounded, and $p>\frac{N+2}{N-2}$. We prove that the associated Dirichlet problem has infinitely many positive solutions. We establish analogous results for the standing-wave supercritical nonlinear Schrödinger equation $\Delta u - V(x)u + u^p = 0 $ where $V\ge 0$ and $V(x) = o(|x|^{-2})$ at infinity. In addition we present existence results for the Dirichlet problem in bounded domains with a sufficiently small spherical hole if $p$ differs from certain sequence of resonant values which tends to infinity.
Keywords: supercritical elliptic problems, exterior domains., Critical Sobolev exponent.
Mathematics Subject Classification: Primary: 35J25, 35J2.
Citation: Manuel del Pino. Supercritical elliptic problems from a perturbation viewpoint. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 69-89. doi: 10.3934/dcds.2008.21.69
[1]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[2]

Satoshi Hashimoto, Mitsuharu Ôtani. Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 323-333. doi: 10.3934/dcds.2007.19.323
[3]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[4]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[5]

Riccardo Molle, Donato Passaseo. On the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 445-454. doi: 10.3934/dcds.1998.4.445
[6]

Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327
[7]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure and Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[8]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[9]

Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure and Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453
[10]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[11]

Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[12]

Seunghyeok Kim, Angela Pistoia. Supercritical problems in domains with thin toroidal holes. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4671-4688. doi: 10.3934/dcds.2014.34.4671
[13]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[14]

Xiaoli Chen, Jianfu Yang. Improved Sobolev inequalities and critical problems. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3673-3695. doi: 10.3934/cpaa.2020162
[15]

Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4297-4318. doi: 10.3934/dcds.2021037
[16]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[17]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[18]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[19]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy