Pairs of positive solutions for pp-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition

We consider a nonlinear Dirichlet problem driven by the pp-Laplacian differential. The right-hand-side nonlinearity, exhibits a (p−1)(p−1)-sublinear term of the form m(z)|x|r−2xm(z)|x|r−2x, r