The existence of a nontrivial solution to the p&qp&q-Laplacian problem with nonlinearity asymptotic to up−1up−1 at infinity in RNRN

We consider the following elliptic problem: {−Δpu+m|u|p−2u−Δqu+n|u|q−2u=f(x,u),x∈RNu∈W1,p(RN)⋂W1,q(RN), where m,n>0m,n>0, N≥3N≥3 and 1