Annales Academię Scientiarum Fennicę

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_{p}*u* - \Delta_{q}*u*
= *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_{p}*u* - \Delta_{q}*u*
= *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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