American Institute of Mathematical Sciences

Advanced
October  1997, 3(4): 565-578. doi: 10.3934/dcds.1997.3.565

Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains

Takahiro Hashimoto 1, and  Mitsuharu Ôtani 2,

1. 

Department of Mathematical Sciences, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime, Japan 790-77
2. 

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555

Received  January 1997 Revised  June 1997 Published  July 1997

The nonexistence of positive solutions is discussed for $-\Delta_p u = a(x) u^{q-1}$ in $\Omega$, $u|_{\partial\Omega}= 0$, for the case where $a(x)$ is a bounded positive function and $\Omega$ is a strip-like domain such as $\Omega = \Omega_d \times \mathcal{R}^{N-d}$ with $\Omega_d$ bounded in $\mathcal{R}^{d}$. The existence of nontrivial solution of (E) is proved by Schindler for $q \in (p, p*)$ where $p*$ is Sobolev's critical exponent. Our method of proofs for nonexistence rely on the "Pohozaev-type inequality" (for $q \ge p*$); and on a new argument concerning the simplicity of the first eigenvalue for (generalized) eigenvalue problems combined with translation invariance of the domain (for $q \le p$).
Keywords: Quasilinear elliptic equations, positive solutions..
Mathematics Subject Classification: 35J20, 35J7.
Citation: Takahiro Hashimoto, Mitsuharu Ôtani. Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 565-578. doi: 10.3934/dcds.1997.3.565
[1]

Antonio Ambrosetti, Zhi-Qiang Wang. Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 55-68. doi: 10.3934/dcds.2003.9.55
[2]

Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[3]

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
[4]

Haiyan Wang. Positive radial solutions for quasilinear equations in the annulus. Conference Publications, 2005, 2005 (Special) : 878-885. doi: 10.3934/proc.2005.2005.878
[5]

Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
[6]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[7]

Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112
[8]

João Marcos do Ó, Sebastián Lorca, Justino Sánchez, Pedro Ubilla. Positive radial solutions for some quasilinear elliptic systems in exterior domains. Communications on Pure & Applied Analysis, 2006, 5 (3) : 571-581. doi: 10.3934/cpaa.2006.5.571
[9]

Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974
[10]

Jun Bao, Lihe Wang, Chunqin Zhou. Positive solutions to elliptic equations in unbounded cylinder. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1389-1400. doi: 10.3934/dcdsb.2016001
[11]

Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077
[12]

Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021214
[13]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[14]

Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831
[15]

Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157
[16]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349
[17]

Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016
[18]

Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943
[19]

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
[20]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy