September  2009, 8(5): 1607-1618. doi: 10.3934/cpaa.2009.8.1607

Entire large solutions of semilinear elliptic equations of mixed type

1. 

Department of Mathematics and Statistics, Air Force Institute of Technology/ ENC, 2950 Hobson Way, Wright Patterson AFB, OH 45433-7765, United States

2. 

Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, United States

Received  October 2008 Revised  January 2009 Published  April 2009

We prove existence and nonexistence of nonnegative entire large solutions for the semilinear elliptic equation $\Delta u = p(x)f(u) + q(x)g(u)$ in which $f$ and $g$ are nondecreasing and vanish at the origin. The locally Hölder continuous functions $p$ and $q$ are nonnegative. We extend results previously obtained for special cases of $f$ and $g$.
Citation: Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607
[1]

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

[2]

Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3525-3533. doi: 10.3934/dcdss.2020247

[3]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[4]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381

[5]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

[6]

Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure and Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719

[7]

Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051

[8]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

[9]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707

[10]

Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13

[11]

Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1369-1393. doi: 10.3934/dcdss.2020077

[12]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure and Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197

[13]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[14]

Philip Korman. On uniqueness of positive solutions for a class of semilinear equations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 865-871. doi: 10.3934/dcds.2002.8.865

[15]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801

[16]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393

[17]

Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905

[18]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155

[19]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

[20]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]