Multiplicity results for some nonlinear elliptic problems with asymptotically ${{\varvec{p}}}$-linear terms

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 $1 < p \le N$.