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)).

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.

On p-Laplace equations with concave terms and asymmetric perturbations

2011 ◽  
Vol 141 (1) ◽  
pp. 171-192 ◽  
Author(s):  
D. Motreanu ◽  
V. V. Motreanu ◽  
N. S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Principle ◽  
Point Theory ◽  
Smooth Solutions ◽  
Precise Information ◽  
Nonlinear Dirichlet Problem ◽  
Laplace Equations ◽  
Asymmetric Behaviour ◽  
Truncation Techniques ◽  
Concave Term

We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)-superlinear on ℝ+ and (p − 1)-(sub)linear on ℝ−. Using variational methods based on the critical point theory together with truncation techniques, Ekeland's variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

2016 ◽  
Vol 16 (1) ◽  
pp. 51-65 ◽  
Author(s):  
Salvatore A. Marano ◽  
Sunra J. N. Mosconi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Multiple Solutions ◽  
First Eigenvalue ◽  
Reaction Term ◽  
The First Eigenvalue

AbstractThe existence of multiple solutions to a Dirichlet problem involving the ${(p,q)}$-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of ${-\Delta_{p}}$ in ${W^{1,p}_{0}(\Omega)}$ and exhibits a concave behavior near 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.

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.

Dirichlet problems with singular and superlinear terms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550056
Author(s):  
Sergiu Aizicovici ◽  
Nikolaos S. Papageorgiou ◽  
Vasile Staicu
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Truncation Techniques ◽  
Superlinear Perturbation

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.

scholarly journals Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

2013 ◽  
Vol 113 (2) ◽  
pp. 206 ◽  
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.

scholarly journals Existence and Multiplicity of Solutions for Resonant (p,2)-Equations

2018 ◽  
Vol 18 (1) ◽  
pp. 105-129 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Variational Methods ◽  
Existence Theorem ◽  
Elliptic Equations ◽  
Smooth Solution ◽  
Critical Groups ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Existence And Multiplicity ◽  
Multiplicity Theorem

Abstract We consider Dirichlet elliptic equations driven by the sum of a p-Laplacian {(2<p)} and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both {\pm\infty} and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.

Positive, Negative, and Nodal Solutions to Elliptic Differential Inclusions Depending on a Parameter

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Salvatore A. Marano ◽  
Dumitru Motreanu
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Homogeneous Dirichlet Problem ◽  
Partial Differential Inclusion ◽  
Truncation Techniques ◽  
Elliptic Differential Inclusions

AbstractThe homogeneous Dirichlet problem for a partial differential inclusion involving the p- Laplace operator and depending on a parameter λ > 0 is investigated. The existence of three smooth solutions, a smallest positive, a biggest negative, and a nodal one, is obtained for any λ sufficiently large by combining variational methods with truncation techniques.

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