| Abstract: | Taking any \(p > 1\), we consider the asymptotically p-linear problem $$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\partial \Omega , \end{array} \right. \end{aligned}$$ where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in 14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\). |
