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 − Δ p u = λ f ( u ) in  Ω , u = 0 on  ∂ Ω . 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 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 − λ Ψ on open sublevels Φ − 1 ( ] − ∞ , 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.