( p ,2)-equations asymmetric at both zero and infinity

Abstract We consider a (p, 2) (p, 2) -equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a p -Laplacian and a Laplacian with p > 2 p>2. The reaction term is (p - 1) (p-1) -linear, but exhibits asymmetric behavior at ± ∞ and at 0 ± 0^{}. Using variational tools, together with truncation and comparison techniques and Morse theory, we prove two multiplicity theorems, one of them providing sign information for all the solutions (positive, negative, nodal).