Solutions for Quasilinear Elliptic Problems via Critical Points in Open Sublevels and Truncation Principles

We investigate a quasilinear elliptic problem depending on a parameter. Our main goal is to establish 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 the nonlinearity involved that are prescribed only near zero. More precisely, we describe an interval of the parameter 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 for some specifically constructed functionals on open sublevels of these functionals, 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.