scholarly journals Second order estimates for boundary blow-up solutions of elliptic equations

2007 ◽  
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Second Order

A second-order estimate for blow-up solutions of elliptic equations

2011 ◽  
Vol 74 (6) ◽  
pp. 2342-2350 ◽  
Author(s):  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Shengzhi Zhang ◽  
Jinhua Xi
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Second Order ◽  
Order Estimate

Multiple boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

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