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).

Arbitrarily small nodal solutions for nonhomogeneous Robin problems

2021 ◽  
pp. 1-15
Author(s):  
Shengda Zeng ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Variational Tools

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).

Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Differential Operator ◽  
Robin Problem ◽  
Reaction Term ◽  
Nonhomogeneous Differential Operator ◽  
Superlinear Reaction ◽  
Nonlinear Nonhomogeneous Differential Operator

AbstractWe consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is

Infinitely Many Nodal Solutions for Nonlinear Nonhomogeneous Robin Problems

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Differential Operator ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator

AbstractWe consider a nonlinear Robin problem driven by a nonhomogeneous differential operator which incorporates the

scholarly journals Positive solutions for nonlinear nonhomogeneous parametric Robin problems

2018 ◽  
Vol 30 (3) ◽  
pp. 553-580 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Differential Operator ◽  
Positive Solution ◽  
Positive Solutions ◽  
Type Theorem ◽  
Sobolev Norm ◽  
Robin Problem ◽  
Reaction Term ◽  
Nonhomogeneous Differential Operator ◽  
Nonlinear Nonhomogeneous Differential Operator

AbstractWe study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter {\lambda>0} approaches zero, we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally, we show that for every admissible parameter value, there is a smallest positive solution {u^{*}_{\lambda}} of the problem, and we investigate the properties of the map {\lambda\mapsto u^{*}_{\lambda}}.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

2018 ◽  
Vol 61 (4) ◽  
pp. 943-959 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Reaction Function ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Potential Term ◽  
Symmetric Mountain Pass Theorem ◽  
Indefinite Potential ◽  
Concave Term

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

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.

A Multiplicity Theorem for p-Superlinear Neumann Problems with a Nonhomogeneous Differential Operator

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Giuseppina Barletta ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Critical Point Theory ◽  
Point Theory ◽  
Smooth Solutions ◽  
Neumann Problems ◽  
Nonhomogeneous Differential Operator ◽  
Nonlinear Neumann Problem ◽  
Multiplicity Theorem ◽  
Truncation Techniques ◽  
Special Case

AbstractWe consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator (special case is the p-Laplacian) with a (p − 1)-superlinear Carathéodory reaction term, which need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory coupled with suitable truncation techniques, we show that the problem has at least five nontrivial smooth solutions.

Multiple Solutions for Nonlinear Periodic Problems

2013 ◽  
Vol 56 (2) ◽  
pp. 366-377 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Periodic Problem ◽  
Point Theory ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Periodic Problems ◽  
Superlinear Growth ◽  
Special Case ◽  
Nonlinear Nonhomogeneous Differential Operator

Abstract We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term f (t; x) that exhibits a (p – 1)-superlinear growth in x 2 R near 1 and near zero. A special case of the differential operator is the scalar p-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

scholarly journals On a Parametric Nonlinear Dirichlet Problem with Subdiffusive and Equidiffusive Reaction

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Mean Curvature ◽  
Nonlinear Eigenvalue Problem ◽  
Nodal Solution ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Nonhomogeneous Differential Operator

AbstractWe study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.

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