Singular Dirichlet (p, q)-Equations

Abstract

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $$(p-1)$$ ( p - 1 ) -superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $$\lambda >0$$ λ > 0 varies. Moreover, we prove the existence of a minimal positive solution $$u^*_\lambda $$ u λ ∗ and study the monotonicity and continuity properties of the map $$\lambda \rightarrow u^*_\lambda $$ λ → u λ ∗ .

Some recent results on singular p-Laplacian equations

“…Obviously, a first attempt might be considering equations driven by the p-Laplacian, and this section aims to provide a short account of the nowadays literature. However, further possibly non-homogeneous operators have been considered; see, e.g., [14,15,26,27,28,29,20,30,31].…”

“…We end this section by pointing out two very recent works, namely [49], which deals with possibly non-monotone singular reactions (see also [50,51], essentially based on sub-super-solution methods) and [31], devoted to singular equations driven by the (p, q)-Laplace operator u → ∆ p u + ∆ q u.…”

Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

“…( [1]) proved the existence of two solutions of (4) using the technique of sub-super solutions. In a series of papers ( [11], [12], [13]), Papageorgiou and Winkert have explored bifurcation type results describing the changes in the set of positive solutions of the problem as the parameter λ varies. The reaction term considered in their papers has the combined effects of the singular term as well as a superdiffusive growth term.…”

An existence result for singular Finsler double phase problems

“…Finally, we mention recent papers which are very close to our topic. We refer to the works of Bahrouni-Rȃdulescu-Winkert [1], Marino-Winkert [33], Papageorgiou-Rȃdulescu-Repovš [37], Papageorgiou-Winkert [39,40], Zeng-Bai-Gasiński-Winkert [48,47] and the references therein.…”