  We consider a double phase problem driven by the sum of the $p$-Laplace
operator and a weighted $q$-Laplacian ($q<p$), with a weight function which is
not bounded away from zero. The reaction term is $(p-1)$-superlinear. Employing
the Nehari method, we show that the equation has a ground state solution of
constant sign and a nodal (sign-changing) solution.
