Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles

We study a quasilinear elliptic problem depending on a parameter $\lambda$ of the form $-\Delta_p u=\lambda f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters$\lambda$ for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form $\Phi-\lambda\Psi$ on open sublevels $\Phi^{-1}(]-\infty,r[)$, combined with comparison principles and the sub-supersolution method. Moreover, variational and topological arguments, such as the mountain pass theorem, in conjunction with truncation techniques are the main tools for the proof of sign-changing solutions. © 2012 Elsevier Ltd