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

Abstract. In this paper, we consider the regularity of weak solutions u ∈ W (R ) ∩ W (R ) of the elliptic partial differential equation −∆pu−∆qu = f(x), x ∈ R , where 1 < q < p < N . We prove that these solutions are locally in C and decay exponentially at infinity. Furthermore, we prove the regularity for the solutions u ∈ W (R ) ∩W (R ) of the following equations −∆pu−∆qu = f(x, u), x ∈ R , where N ≥ 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.