Tohoku Mathematical Journal
2005
September
SECOND SERIES VOL. 57, NO. 3

Tohoku Math. J.
57 (2005), 427-445

Title POSITIVE SOLUTIONS OF ELLIPTIC AND PARABOLIC EQUATIONS WITH CONVEX-CONCAVE NONLINEARITIES

Author Qiuyi Dai and Yonggeng Gu

(Received October 23, 2003, revised July 21, 2004)
Abstract. We consider, respectively, the Dirichlet problem and the initial-boundary value problem of elliptic and parabolic equations with two power nonlinearities. We find that these problems are closely related to the so-called quenching problem. We obtain the existence and nonexistence of positive solutions to these problems on bounded and unbounded domains, by using the results of quenching problem and sub-super solution method.

2000 Mathematics Subject Classification. Primary 35J25; Secondary 35K20.
Key words and phrases. Elliptic equation, parabolic equation, quenching problem, Dirichlet problem, initial-boundary value problem, positive solution.

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