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 λ of the form−Δpu = λf (u) in Ω, u = 0 on ∂Ω.We present a novel variational approach that allows us to obtain multiplicity, regularityand 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 λ 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 E =Φ − λΨ on open sublevels Φ−1(] − ∞, r[), combined with comparison principles andthe sub-supersolution method. Moreover, variational and topological arguments, such asthe mountain pass theorem, in conjunction with truncation techniques are the main toolsfor the proof of sign-changing solutions.