The regularity of weak solutions to nonlinear scalar field elliptic equations containing p&q-laplacians

In this paper, we consider the regularity of weak solutions u is an element of W-1,W-p(R-N)boolean AND W-1,W-q(R-N) of the elliptic partial differential equation -Delta(p)u-Delta(q)u=f(x), x is an element of R-N, where 1 < q < p < N. We prove that these solutions are locally in C-1,C-alpha and decay exponentially at infinity. Furthermore, we prove the regualrity for the solutions u is an element of W-1,W-p(R-N)boolean AND W-1,W-q(R-N) of the following equations. -Delta(p)u-Delta(q)u=f(x,u), x is an element of R-N, where N >= 3, 1 < q < p < N, and f(x,u) is critical or subcritical growth about u. As an application, we can show that the solution we got in vertical bar 8 vertical bar has the same regularity.