Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation
Résumé
In a bounded domain with smooth boundary, the authors pose the problem of finding a function that is strictly positive inside the domain, vanishing on the boundary of the domain and satisfies a quasilinear differential equation, inside the domain, whose left-hand side has the $p$-Laplace operator of the unknown function and whose right-hand side is a linear combination of the unknown function in positive and negative powers. The main result of the paper, which is a proof of the existence of at least two weak solutions of the above-mentioned problem in the corresponding Sobolev spaces, is obtained by a development of the well-known variational method. In addition, the authors prove that the weak solutions belong to Hölder spaces. They also establish a strong comparison principle.