Nonlinear Parametric Robin Problems with Combined Nonlinearities

AbstractWe consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.

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Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15570 ◽

2013 ◽

Vol 113 (2) ◽

pp. 206 ◽

Cited By ~ 3

Author(s):

Leszek Gasiński ◽

Nikolaos S. Papageorgiou

Keyword(s):

Dirichlet Problem ◽

Multiple Solutions ◽

Critical Case ◽

Critical Value ◽

Constant Sign ◽

Smooth Solutions ◽

Dirichlet Problems ◽

Truncation Techniques ◽

Semilinear Problem ◽

Concave Term

We consider a nonlinear parametric Dirichlet problem with parameter $\lambda>0$, driven by the $p$-Laplacian and with a concave term $\lambda|u|^{q-2}u$, $1<q<p$ and a Carathéodory perturbation $f(z,\zeta)$ which is asymptotically $(p-1)$-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value $\lambda^*>0$ of the parameter such that for $\lambda\in (0,\lambda^*)$ the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case ($p=2$), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case $\lambda=\lambda^*$, we show that the nonlinear problem ($p\ne 2$) still has two nontrivial constant sign smooth solutions and the semilinear problem ($p=2$) has three nontrivial smooth solutions, two of which have constant sign.

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Asymmetric Robin Problems with Indefinite Potential and Concave Terms

Advanced Nonlinear Studies ◽

10.1515/ans-2018-2022 ◽

2019 ◽

Vol 19 (1) ◽

pp. 69-87 ◽

Cited By ~ 2

Author(s):

Nikolaos S. Papageorgiou ◽

Vicenţiu D. Rădulescu ◽

Dušan D. Repovš

Keyword(s):

Variational Methods ◽

Morse Theory ◽

Critical Groups ◽

Robin Problem ◽

Perturbation Techniques ◽

Smooth Solutions ◽

Asymptotically Linear ◽

Unbounded Potential ◽

Indefinite Potential ◽

Concave Term

Abstract We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing four and five, respectively, nontrivial smooth solutions when the parameter {\lambda>0} is small.

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Three Solutions of Constant Sign for a System of Discrete Equations

Acta Applicandae Mathematicae ◽

10.1007/s10440-004-2585-4 ◽

2004 ◽

Vol 84 (2) ◽

pp. 121-162 ◽

Cited By ~ 2

Author(s):

Ravi P. Agarwal ◽

Donal O’Regan ◽

Patricia J. Y. Wong

Keyword(s):

Constant Sign ◽

Three Solutions ◽

Discrete Equations ◽

Solutions Of Constant Sign

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Multiplicity theorems for semilinear Robin problems

Advances in Calculus of Variations ◽

10.1515/acv-2013-0026 ◽

2015 ◽

Vol 8 (3) ◽

Author(s):

Nikolaos S. Papageorgiou ◽

Vicenţiu D. Rădulescu

Keyword(s):

Variational Methods ◽

Growth Condition ◽

Morse Theory ◽

Robin Problem ◽

Perturbation Techniques ◽

Nontrivial Solutions ◽

Dirichlet Problems ◽

Parametric Problem ◽

Special Case ◽

Global Growth

AbstractWe consider a semilinear Robin problem driven by the Laplacian with a reaction which does not satisfy a global growth condition, only a local one. Using variational methods coupled with truncation and perturbation techniques and Morse theory, we prove two multiplicity theorems producing four and three nontrivial solutions respectively, all with precise sign. Also, we show that our results incorporate as a special case a semilinear parametric problem which has been considered primarily in the context of Dirichlet problems.

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Three Solutions for a Neumann Partial Differential Inclusion Via Nonsmooth Morse Theory

Set-Valued and Variational Analysis ◽

10.1007/s11228-016-0387-2 ◽

2016 ◽

Vol 25 (2) ◽

pp. 405-425 ◽

Cited By ~ 2

Author(s):

Francesca Colasuonno ◽

Antonio Iannizzotto ◽

Dimitri Mugnai

Keyword(s):

Differential Inclusion ◽

Morse Theory ◽

Three Solutions ◽

Partial Differential ◽

Partial Differential Inclusion ◽

Nonsmooth Morse Theory

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000044 ◽

2018 ◽

Vol 61 (4) ◽

pp. 943-959 ◽

Cited By ~ 2

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.

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Superlinear Robin problems with indefinite linear part

Science Technology and Innovation ◽

10.5604/01.3001.0012.0898 ◽

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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A variational approach to nonlinear logistic equations

Communications in Contemporary Mathematics ◽

10.1142/s0219199714500217 ◽

2015 ◽

Vol 17 (03) ◽

pp. 1450021 ◽

Cited By ~ 7

Author(s):

Leszek Gasiński ◽

Donal O'Regan ◽

Nikolaos S. Papageorgiou

Keyword(s):

Positive Solutions ◽

Variational Approach ◽

Continuous Dependence ◽

Type Equation ◽

Logistic Equation ◽

Extremal Solutions ◽

Constant Sign ◽

Nontrivial Solutions ◽

Logistic Equations ◽

Solutions Of Constant Sign

We consider a nonlinear logistic type equation. For all big values of the parameter, we show that the problem admits nontrivial solutions of constant sign and in fact we establish the existence of extremal constant sign solutions. Using these extremal solutions, we produce a nodal (sign-changing) solution. We also investigate the uniqueness and continuous dependence on the parameter of positive solutions. Finally, we study the degenerate p-logistic equation.

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