Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations

Abstract

We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda>0$ of the form $$ -\Delta_pu=\lambda f(u)\quad\mbox{in }\ \Omega,\quad u=0\quad\mbox{on }\ \partial\Omega, $$ where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the $p$-Laplacian, and $f: \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontrivial constant-sign as well as nodal (sign-changing) solutions. In our approach, a crucial role plays an abstract critical point result for functionals whose critical points are attained in certain open level sets. To the best of our knowledge, the novelty of this paper is twofold. First, neither an asymptotic condition for $f$ at zero nor at infinity is required to ensure multiple constant-sign solutions. Second, only by imposing some $\liminf$ and $\limsup$ condition of $f$ at zero, the existence of at least three nontrivial solutions including one nodal solution can be proved.