Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 337-371

THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR SCALAR FIELD ELLIPTIC EQUATIONS CONTAINING p&q-LAPLACIANS

Chengjun He and Gongbao Li

Chinese Academy of Sciences, Wuhan Institute of Physics and Mathematics
Wuhan, 430071, China; cjhe 'at' wipm.ac.cn

Central China Normal University, School of Mathematics and Statistics
Wuhan, 430079, China; ligb 'at' mail.ccnu.edu.cn

Abstract. In this paper, we consider the regularity of weak solutions u \in W^1,p(R^N) \cap W^1,q(R^N) of the elliptic partial differential equation

-\Delta_pu - \Delta_qu = f(x), x \in R^N,

where 1 q p N$. We prove that these solutions are locally in C^1,\alpha and decay exponentially at infinity. Furthermore, we prove the regularity for the solutions u \in W^1,p(R^N) \cap W^1,q(R^N) of the following equations

-\Delta_pu - \Delta_qu = f(x,u), x \in R^N,

where N \geq 3, 1 q p N, and f(x,u) is of critical or subcritical growth about u. As an application, we can show that the solution we got in [8] has the same regularity.

2000 Mathematics Subject Classification: Primary 35B65, 35D10.

Key words: Regularity, weak Solutions, p&q-Laplacians.

Reference to this article: C. He and G. Li: The regularity of weak solutions to nonlinear scalar field elliptic equations containing p&q-Laplacians. Ann. Acad. Sci. Fenn. Math. 33 (2008), 337-371.

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