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).

NODAL AND MULTIPLE SOLUTIONS FOR NONLINEAR PERIODIC PROBLEMS WITH COMPETING NONLINEARITIES

2013 ◽  
Vol 15 (03) ◽  
pp. 1350001 ◽  
Author(s):  
SERGIU AIZICOVICI ◽  
NIKOLAOS S. PAPAGEORGIOU ◽  
VASILE STAICU
Keyword(s):  
Multiple Solutions ◽  
Periodic Problem ◽  
Point Theory ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Periodic Problems ◽  
Truncation Techniques ◽  
Special Case ◽  
Concave And Convex Terms

We consider a nonlinear periodic problem drive driven by a nonhomogeneous differential operator which incorporates as a special case the scalar p-Laplacian, and a reaction which exhibits the competition of concave and convex terms. Using variational methods based on critical point theory, together with suitable truncation techniques and Morse theory (critical groups), we establish the existence of five nontrivial solutions, two positive, two negative and the fifth nodal (sign-changing). In the process, we also prove some auxiliary results of independent interest.

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.

On the Existence of Three Nontrivial Solutions for Periodic Problems Driven by the Scalar p-Laplacian

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Francesca Papalini
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Periodic Problem ◽  
Point Theory ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Nonsmooth Critical Point Theory ◽  
Periodic Problems ◽  
Nonsmooth Critical Point ◽  
Nonsmooth Potential

AbstractWe consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential function. First we establish an alternative minimax expression for the first nonzero eigenvalue for the negative periodic scalar p-Laplacian and then using it we prove the existence of three nontrivial solutions, two of which have constant sign. Our approach is variational, based on the nonsmooth critical point theory.

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

scholarly journals ON THE MULTIPLICITY OF SOLUTIONS FOR NON-LINEAR PERIODIC PROBLEMS WITH THE NON-LINEARITY CROSSING SEVERAL EIGENVALUES

2009 ◽  
Vol 52 (2) ◽  
pp. 271-302 ◽  
Author(s):  
SOPHIA TH. KYRITSI ◽  
NIKOLAOS S. PAPAGEORGIOU
Keyword(s):  
Finite Number ◽  
Critical Point Theory ◽  
Variational Approach ◽  
Periodic Problem ◽  
Point Theory ◽  
Constant Sign ◽  
Smooth Potential ◽  
Periodic Problems ◽  
Non Linear ◽  
Asymmetric Behaviour

AbstractIn this paper we consider a non-linear periodic problem driven by the scalar p-Laplacian and with a non-smooth potential. We assume that the multi-valued right-hand-side non-linearity exhibits an asymmetric behaviour at ±∞ and crosses a finite number of eigenvalues as we move from −∞ to +∞. Using a variational approach based on the non-smooth critical-point theory, we show that the problem has at least two non-trivial solutions, one of which has constant sign. For the semi-linear (p = 2), smooth problem, using Morse theory, we show that the problem has at least three non-trivial solutions, again one with constant sign.

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 SOLUTIONS FOR DOUBLY RESONANT NONLINEAR NON-SMOOTH PERIODIC PROBLEMS

2005 ◽  
Vol 48 (1) ◽  
pp. 199-211 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point Theory ◽  
Periodic Problem ◽  
Point Theory ◽  
Minimax Principle ◽  
Spectral Interval ◽  
Primary 34B15 ◽  
Locally Lipschitz Functions ◽  
Smooth Potential ◽  
Periodic Problems ◽  
Locally Lipschitz

AbstractWe study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

scholarly journals A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM

2008 ◽  
Vol 50 (2) ◽  
pp. 335-349 ◽  
Author(s):  
NIKOLAOS S. PAPAGEORGIOU ◽  
EUGÉNIO M. ROCHA
Keyword(s):  
Dirichlet Problem ◽  
Variable Exponent ◽  
Point Theory ◽  
Critical Groups ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Multiplicity Theorem ◽  
Noncoercive Problems ◽  
Truncation Techniques

AbstractWe consider a nonlinear Dirichlet problem driven by thep(ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

Solutions and multiple solutions for superlinear perturbations of the periodic scalar p-Laplacian

2013 ◽  
Vol 56 (3) ◽  
pp. 805-827
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O'Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Methods ◽  
Existence Theorem ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Periodic Problem ◽  
Point Theory ◽  
Superlinear Growth ◽  
Multiplicity Theorem ◽  
Truncation Techniques

AbstractWe consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

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}}.

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