Uniqueness of positive solutions for a boundary blow-up problem

Abstract In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem { Δ u = f ( u ) in Ω , u = ∞ on ∂ Ω , where Ω is a C 2 bounded domain in R N , under the hypotheses that f ( t ) is nondecreasing in t > 0 and f ( t ) / t p is increasing for large t and some p > 1 . We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a ( x ) .

[1]  Zhijun Zhang,et al.  Boundary behavior of solutions to some singular elliptic boundary value problems , 2008 .

[2]  Zhijun Zhang,et al.  The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems ✩ , 2005 .

[3]  J. García-Melián.  Uniqueness for boundary blow-up problems with continuous weights , 2007 .

[4]  Hongjie Dong,et al.  On Uniqueness of Boundary Blow-Up Solutions of a Class of Nonlinear Elliptic Equations , 2007, 0705.2287.

[5]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[6]  J. Keller.  On solutions of δu=f(u) , 1957 .

[7]  Ahmed Mohammed,et al.  Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values , 2007 .

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

[9]  L. Bieberbach.  Δu=eu und die automorphen Funktionen , 1916 .

[10]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[11]  Vicentiu D. Rădulescu,et al.  Asymptotics for the Blow-Up Boundary Solution of the Logistic Equation with Absorption , 2003 .

[12]  J. García-Melián.  Nondegeneracy and uniqueness for boundary blow-up elliptic problems , 2006 .

[13]  Richard von Mises,et al.  Die Differential- und Integralgleichungen der Mechanik und Physik , 1925 .

[14]  Corina Cîrstea,et al.  Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach , 2005, Asymptot. Anal..

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

[16]  Yihong Du,et al.  General Uniqueness Results and Variation Speed for Blow‐Up Solutions of Elliptic Equations , 2005 .

[17]  Vicenţiu D. Rădulescu,et al.  Uniqueness of the blow-up boundary solution of logistic equations with absorbtion , 2002 .

[18]  Jorge García-Melián,et al.  Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up , 2001 .

[19]  A note on boundary blow-up problem of ∆u = u p , 2002 .

[20]  Serge Dumont,et al.  Back to the Keller-Osserman Condition for Boundary Blow-up Solutions , 2007 .

[21]  Robert Osserman,et al.  On the inequality $\Delta u\geq f(u)$. , 1957 .

[22]  Martin Chuaqui,et al.  Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights , 2004 .

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

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