scholarly journals Asymmetric Robin Problems with Indefinite Potential and Concave Terms

2019 ◽  
Vol 19 (1) ◽  
pp. 69-87 ◽  
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.

scholarly journals Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-36 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Unbounded Potential ◽  
Indefinite Potential ◽  
Truncation Techniques ◽  
Concave Term

We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.

scholarly journals Noncoercive resonant (p,2)-equations with concave terms

2018 ◽  
Vol 9 (1) ◽  
pp. 228-249 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Chao Zhang
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Concave Term

Abstract We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

Multiplicity theorems for semilinear Robin problems

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.

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.

Nonlinear Parametric Robin Problems with Combined Nonlinearities

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Growth Condition ◽  
Morse Theory ◽  
Constant Sign ◽  
Robin Problem ◽  
Three Solutions ◽  
Multiplicity Theorem ◽  
Solutions Of Constant Sign ◽  
Semilinear Problem ◽  
Concave Term

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.

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.

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.

Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Methods ◽  
Neumann Problem ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Solutions ◽  
Asymmetric Reaction ◽  
Neumann Problems ◽  
Nonlinear Neumann Problem ◽  
Asymmetric Behaviour ◽  
Nonlinear Neumann Problems

AbstractWe consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

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.

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