scholarly journals Resonant Anisotropic (p,q)-Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1332
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Source Term ◽  
Direct Method ◽  
Principal Eigenvalue ◽  
Critical Groups ◽  
Negative Solution ◽  
Smooth Solutions ◽  
Carathéodory Function ◽  
Variational Tools ◽  
Truncation Techniques ◽  
Solutions Of Constant Sign

We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

Three non-trivial solutions for not necessarily coercive p-Laplacian equations

2009 ◽  
Vol 52 (3) ◽  
pp. 679-688
Author(s):  
Shouchuan Hu ◽  
Nikolas S. Papageorgiou
Keyword(s):  
Upper And Lower Solutions ◽  
Elliptic Problems ◽  
Critical Groups ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Truncation Techniques ◽  
Laplacian Equations ◽  
Coercive Problems

AbstractWe consider the existence of three non-trivial smooth solutions for nonlinear elliptic problems driven by the p-Laplacian. Using variational arguments, coupled with the method of upper and lower solutions, critical groups and suitable truncation techniques, we produce three non-trivial smooth solutions, two of which have constant sign. The hypotheses incorporate both coercive and non-coercive problems in our framework of analysis.

scholarly journals Anisotropic double-phase problems with indefinite potential: multiplicity of solutions

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Dongdong Qin ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Maximum Principle ◽  
Critical Groups ◽  
Phase Problem ◽  
Smooth Solutions ◽  
Double Phase ◽  
Multiplicity Theorem ◽  
Variational Tools ◽  
Indefinite Potential ◽  
Double Phase Problem ◽  
Strong Comparison Principle

AbstractWe consider an anisotropic double-phase problem plus an indefinite potential. The reaction is superlinear. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we prove a multiplicity theorem producing five nontrivial smooth solutions, all with sign information and ordered. In this process we also prove two results of independent interest, namely a maximum principle for anisotropic double-phase problems and a strong comparison principle for such solutions.

On a class of critical Robin problems

2020 ◽  
Vol 32 (1) ◽  
pp. 95-109 ◽  
Author(s):  
Salvatore Leonardi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
The Other ◽  
Critical Groups ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Variational Tools

AbstractWe consider a nonlinear parametric Robin problem. In the reaction, there are two terms, one critical and the other locally defined. Using cut-off techniques, together with variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least three nontrivial smooth solutions all with sign information, which converge to zero in {C^{1}(\bar{\Omega})} as the parameter {\lambda\to 0^{+}}.

Existence of Multiple Solutions for Nonlinear Dirichlet Problems with a Nonhomogeneous Differential Operator

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Linear Differential Operator ◽  
Principal Eigenvalue ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Multiplicity Results ◽  
Nonlinear Elliptic ◽  
Nonlinear Differential Operator ◽  
Linear Differential

AbstractWe consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.

scholarly journals CONSTANT SIGN AND NODAL SOLUTIONS FOR NONLINEAR ROBIN EQUATIONS WITH LOCALLY DEFINED SOURCE TERM

2020 ◽  
Vol 25 (3) ◽  
pp. 374-390
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Differential Operator ◽  
Source Term ◽  
Constant Sign ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Special Cases ◽  
Variational Tools ◽  
Nonlinear Nonhomogeneous Differential Operator

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).

scholarly journals Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term

2020 ◽  
Vol 10 (1) ◽  
pp. 76-101 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Youpei Zhang
Keyword(s):  
Nonlinear Elliptic Equation ◽  
Point Theory ◽  
Boundary Term ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Variational Tools ◽  
Indefinite Potential ◽  
Concave Boundary

Abstract We consider a nonlinear elliptic equation driven by the (p, q)–Laplacian plus an indefinite potential. The reaction is (p − 1)–superlinear and the boundary term is parametric and concave. Using variational tools from the critical point theory together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information and which are linearly ordered.

scholarly journals A multiplicity theorem for parametric superlinear (p,q)-equations

2020 ◽  
Vol 40 (1) ◽  
pp. 131-149
Author(s):  
Florin-Iulian Onete ◽  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro
Keyword(s):  
Critical Groups ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Multiplicity Theorem ◽  
Variational Tools

We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

scholarly journals Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Potential Term ◽  
Indefinite Potential

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When $$q=2$$ q = 2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.

Semilinear Robin problems resonant at both zero and infinity

2018 ◽  
Vol 30 (1) ◽  
pp. 237-251
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Robin Boundary Condition ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Robin Boundary

Abstract We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at {\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at {\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

Superlinear Robin problems with indefinite linear part

2018 ◽  
Vol 2 (1) ◽  
pp. 74-94
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Existence Theorem ◽  
Smooth Solution ◽  
Linear Part ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Unbounded Potential ◽  
Multiplicity Theorem ◽  
Variational Tools ◽  
Superlinear Reaction

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools we prove two theorems. An existence theorem producing a nontrivial smooth solution and a multiplicity theorem producing a whole unbounded sequence of nontrivial smooth solutions.

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