Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity

We prove that the quasilinear equation $-\Delta_p u=\lambda V |u|^{p-2}u+g(x,u)$, with $g$ subcritical and $p$-superlinear at $0$ and at infinity, admits a nontrivial weak solution $u\in W^{1,p}_0(\Omega)$ for any $\lambda\in\R$. A minimax approach, allowing also an estimate of the corresponding critical level, is used. New linking structures, associated to certain variational eigenvalues of $-\Delta_p u=\lambda V |u|^{p-2}u$, are recognized, even in absence of any direct sum decomposition of $W^{1,p}_0(\Omega)$ related to the eigenvalue itself.