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

We consider a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>)</m:mo> </m:mrow> </m:math> {(p,2)} -equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a p -Laplacian and a Laplacian with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p>2} . The reaction term is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(p-1)} -linear, but exhibits asymmetric behavior at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>±</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {±∞} and at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>0</m:mn> <m:mo>±</m:mo> </m:msup> </m:math> {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).