• Previous Article
    On the Poincaré mapping and periodic solutions of nonautonomous differential systems
  • CPAA Home
  • This Issue
  • Next Article
    Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights
June  2007, 6(2): 531-540. doi: 10.3934/cpaa.2007.6.531

Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case

1. 

Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China, China

Received  January 2006 Revised  December 2006 Published  March 2007

In this paper, we study positive solution of the following system of quasilinear elliptic equations

div$(|\nabla u|^{p-2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$

div$(|\nabla v|^{q-2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1)

where $m_1>p-1,n_2>q-1, m_2,n_1>0$, and $\Omega\subset R^N$ is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: $u=\lambda, v=\mu$ or $u=v=+\infty$ or $u=+\infty, v=\mu$ on $\partial\Omega$, where $\lambda, \mu>0$. Under several hypotheses on the parameters $m_1,n_1,m_2,n_2$, we show that the existence of positive solutions. We further provide the asymptotic behaviors of the solutions near $\partial\Omega$. Some more general related problems are also studied.

Citation: Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure and Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531
[1]

Juntang Ding, Chenyu Dong. Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021222

[2]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013

[3]

Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022075

[4]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure and Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523

[5]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[6]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182

[7]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[8]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[9]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

[10]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

[11]

Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113

[12]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267

[13]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

[14]

Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549

[15]

Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691

[16]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

[17]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[18]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[19]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134

[20]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]