Existence, Multiplicity and $$C^1$$ -Regularity for Singular Parametric Problems Driven by the Sum of Three Distinct Anisotropic Operators

Abstract

In this paper, we focus on a problem with parameter dependence both in the leading term and in the reaction term. Such problem is driven by the sum of three anisotropic operators with distinct variable exponents and has in the reaction the combined effects of a singular term and of concave and convex nonlinearities. Under very general assumptions, we produce positive solutions for this problem as well as we establish the precise dependence of the set of such solutions on the parameter \(\lambda >0\) (which appears in the reaction) as the latter varies. Our approach is based on the use of variational tools together with truncation and comparison techniques. We stress that here we also describe the asymptotic behavior of specific positive solutions as both parameters in the leading term and in the reaction term vary in an appropriate range.