Isolated singularity for semilinear elliptic equations

In this paper, we study a class of semilinear elliptic equations with the Hardy potential. By means of the super-subsolution method and the comparison principle, we explore the existence of a minimal positive solution and a maximal positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin. Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.

[1]  Yihong Du,et al.  Isolated singularities for weighted quasilinear elliptic equations , 2010 .

[2]  F. Cîrstea,et al.  On trichotomy of positive singular solutions associated with the Hardy-Sobolev operator ✩ , 2009 .

[3]  J. López-Gómez,et al.  Metasolutions in cooperative systems , 2008 .

[4]  B. Abdellaoui,et al.  Elliptic problems with a Hardy potential and critical growth in the gradient: Non-resonance and blow-up results☆ , 2007 .

[5]  Jorge García-Melián,et al.  Boundary behavior for large solutions to elliptic equations with singular weights , 2007 .

[6]  J. López-Gómez.  Optimal uniqueness theorems and exact blow-up rates of large solutions , 2006 .

[7]  Julián López-Gómez,et al.  The boundary blow-up rate of large solutions , 2003 .

[8]  M. Willem,et al.  Elliptic problems with critical exponents and Hardy potentials , 2003 .

[9]  Vicenţiu D. Rădulescu,et al.  EXISTENCE AND UNIQUENESS OF BLOW-UP SOLUTIONS FOR A CLASS OF LOGISTIC EQUATIONS , 2002 .

[10]  A. Tertikas,et al.  Optimizing Improved Hardy Inequalities , 2002 .

[11]  Yihong Du,et al.  Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow‐Up Solutions , 2001 .

[12]  J. M. Fraile,et al.  ELLIPTIC EIGENVALUE PROBLEMS AND UNBOUNDED CONTINUA OF POSITIVE SOLUTIONS OF A SEMILINEAR ELLIPTIC EQUATION , 1996 .

[13]  J. López-Gómez.  The Maximum Principle and the Existence of Principal Eigenvalues for Some Linear Weighted Boundary Value Problems , 1996 .

[14]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[15]  Yihong Du,et al.  Blow-Up Solutions for a Class of Semilinear Elliptic and Parabolic Equations , 1999, SIAM J. Math. Anal..