Positive Solutions for Singular Anisotropic (p, q)-Equations

In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

Singular Dirichlet (p, q)-Equations

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 ∗ .

On a class of singular anisotropic (p, q)-equations

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.

Anisotropic Singular Neumann Equations with Unbalanced Growth

We consider a nonlinear parametric Neumann problem driven by the anisotropic (p, q)-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies. We also show the existence of minimal positive solutions $u_{\lambda }^{*}$ u λ ∗ and determine the monotonicity and continuity properties of the map $\lambda \mapsto u_{\lambda }^{*}$ λ ↦ u λ ∗ .

Anisotropic singular double phase Dirichlet problems

<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

Dirichlet problems with singular and superlinear terms

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.