    • Previous Article
From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation
• DCDS Home
• This Issue
• Next Article
Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient
July  2010, 28(3): 1083-1099. doi: 10.3934/dcds.2010.28.1083

## On least energy solutions to a semilinear elliptic equation in a strip

 1 Ecole des hautes etudes en sciences sociales, CAMS, 54, boulevard Raspail, F - 75006 - Paris, France 2 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  March 2010 Revised  April 2010 Published  April 2010

We consider the following semilinear elliptic equation on a strip:

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$
$u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L))$

where $1< p\leq \frac{N+2}{N-2}$. When $1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L * >0 such that for L $\leq$L * , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L * , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L * < L ** such that the least energy solution is trivial when L $\leq$L *, the least energy solution is nontrivial when L $\in$(L *,L **], and the least energy solution does not exist when L >L **. A connection with Delaunay surfaces in CMC theory is also made.

Citation: Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
  Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527  Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41  Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907  Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237  Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103  Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357  Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795  Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126  Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007  Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991  Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018  Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024  Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376  Takahiro Hashimoto, Mitsuharu Ôtani. Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 565-578. doi: 10.3934/dcds.1997.3.565  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  Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033  M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233  Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383  Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361  Qiuping Lu, Zhi Ling. Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2411-2429. doi: 10.3934/cpaa.2015.14.2411

2018 Impact Factor: 1.143