Boundary blow-up solutions for a class of quasilinear elliptic equations

Abstract

For a given bounded domain Ω in R ^N with smooth boundary ∂Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ∇u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.

Keywords:

• quasilinear elliptic equation
• blow-up solutions
• lower and upper solutions
• fixed point theory
• weak comparison principle

AMS Subject Classifications::

• 35J25
• 35J65

Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant No. 10871060) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005).