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Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

  • P. T. Thuy
  • N. M. Tri
Article

Abstract

In this paper we study boundary value problems for semilinear equations involving strongly degenerate elliptic differential operators. Via a Pohozaev’s type identity we show that if the nonlinear term grows faster than some power function then the boundary value problem has no nontrivial solution. Otherwise when the nonlinear term grows slower than the same power function, by establishing embedding theorems for weighted Sobolev spaces associated with the strongly degenerate elliptic equations, then applying the theory of critical values in Banach spaces, we prove that the problem has a nontrivial solution, or even infinite number of solutions provided that the nonlinear term is an odd function.

Mathematics Subject Classification (2010)

Primary 35B33 Secondary 35B38 35D30 35H99 

Keywords

Semilinear strongly degenerate elliptic equations Boundary value problems Critical exponents Critical values Nontrivial solutions Embedding theorems Pohozaev’s type identities 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

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