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 ‾ ).

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).

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

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.

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 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}}.

Coercive and noncoercive nonlinear Neumann problems with indefinite potential

2016 ◽  
Vol 28 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Differential Operator ◽  
Neumann Problems ◽  
Nonhomogeneous Differential Operator ◽  
Indefinite Potential ◽  
Nonlinear Neumann Problems

AbstractWe consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is (

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