scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

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

2021 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Minimal Solution ◽  
Solution Map ◽  
Continuity Properties ◽  
Superlinear Perturbation

AbstractIn 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.

scholarly journals (p, q)-Equations with Singular and Concave Convex Nonlinearities

2020 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Nonlinear Dirichlet Problem ◽  
Concave And Convex Nonlinearities

Abstract We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian with $$1<q<p$$ 1 < q < p . The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.

scholarly journals Singular Dirichlet (p, q)-Equations

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Combined Effects ◽  
Type Result ◽  
Nonlinear Dirichlet Problem ◽  
Continuity Properties ◽  
Superlinear Perturbation

AbstractWe 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 ∗ .

Positive solutions for parametric semilinear Robin problems with indefinite and unbounded potential

2017 ◽  
Vol 121 (2) ◽  
pp. 263 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Positive Solution ◽  
Laplace Operator ◽  
Positive Solutions ◽  
Type Theorem ◽  
Robin Problem ◽  
Continuity Properties ◽  
Superlinear Growth ◽  
Unbounded Potential ◽  
Carathéodory Function ◽  
The Laplace Operator

We consider a parametric Robin problem driven by the Laplace operator plus an indefinite and unbounded potential. The reaction term is a Carathéodory function which exhibits superlinear growth near $+\infty $ without satisfying the Ambrosetti-Rabinowitz condition. We are looking for positive solutions and prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter. We also establish the existence of the minimal positive solution $u^*_{\lambda }$ and investigate the monotonicity and continuity properties of the map $\lambda \mapsto u^*_{\lambda }$.

scholarly journals Concave-Convex Problems for the Robin p-Laplacian Plus an Indefinite Potential

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 421 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Type Theorem ◽  
Convex Problems ◽  
Continuity Properties ◽  
Indefinite Potential

We consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, ( p − 1 ) -sublinear) term and of a convex (that is, ( p − 1 ) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ↦ u λ * .

Dirichlet problems with singular and superlinear terms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550056
Author(s):  
Sergiu Aizicovici ◽  
Nikolaos S. Papageorgiou ◽  
Vasile Staicu
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Truncation Techniques ◽  
Superlinear Perturbation

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.

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

2021 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Result ◽  
Variational Tools ◽  
Superlinear Perturbation

AbstractWe 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.

scholarly journals Positive solutions for nonlinear parametric singular Dirichlet problems

2019 ◽  
Vol 09 (03) ◽  
pp. 1950011 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Positive Solutions ◽  
Singular Term ◽  
Principal Eigenvalue ◽  
Type Theorem ◽  
Dirichlet Problems

We consider a nonlinear parametric Dirichlet problem driven by the [Formula: see text]-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ([Formula: see text])-linear near [Formula: see text]. The problem is uniformly nonresonant with respect to the principal eigenvalue of [Formula: see text]. We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter [Formula: see text].

scholarly journals Anisotropic Singular Neumann Equations with Unbalanced Growth

2021 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Neumann Problem ◽  
Positive Solutions ◽  
Singular Term ◽  
Combined Effects ◽  
Positive Parameter ◽  
Type Result ◽  
Unbalanced Growth ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Perturbation

AbstractWe 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 ∗ .

scholarly journals Anisotropic singular double phase Dirichlet problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rǎdulescu ◽  
Youpei Zhang
Keyword(s):  
Positive Solutions ◽  
Singular Term ◽  
Independent Interest ◽  
Phase Problem ◽  
Type Result ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools ◽  
Superlinear Perturbation ◽  
Double Phase Problem

<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>

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