Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity
Leszek Gasiński Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2009, 8(4): 1421-1437 doi: 10.3934/cpaa.2009.8.1421
We consider a nonlinear periodic problem driven by the scalar $p$-Laplacian and a nonlinearity that exhibits a $p$-superlinear growth near $\pm\infty$, but need not satisfy the Ambrosetti-Rabinowitz condition. Using minimax methods, truncations techniques and Morse theory, we show that the problem has at least three nontrivial solutions, two of which are of fixed sign.
keywords: Poincaré-Hopf formula. mountain pass theorem Morse theory Scalar p-Laplacian critical groups
Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian
Michael E. Filippakis Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2004, 3(4): 729-756 doi: 10.3934/cpaa.2004.3.729
We study nonlinear Dirichlet problems driven by the scalar $p$-Laplacian with a nonsmooth potential. First for the so-called "sublinear problem", under nonuniform nonresonance conditions, we establish the existence of at least one strictly positive solution. Then we prove two multiplicity results for positive solutions. The first concerns the "superlinear problem" and the second is for the sublinear problem. The method of proof is variational based on the nonsmooth critical point theory for locally Lipschitz functions. Our results complement the ones obtained by De Coster (Nonlin.Anal.23 (1995)).
keywords: multiple strictly positive solutions. generalized subdifferential Mountain Pass Theorem nonsmooth critical point theory nonsmooth Palais-Smale condition Locally Lipschitz function
Multiple solutions for nonlinear coercive Neumann problems
Sophia Th. Kyritsi Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2009, 8(6): 1957-1974 doi: 10.3934/cpaa.2009.8.1957
In this paper we deal with a nonlinear Neumann problem driven by the $p$--Laplacian and with a potential function which asymptotically at infinity is $p$--linear. Using variational methods based on critical point theory coupled with suitable truncation techniques, we prove a theorem establishing the existence of at least three nontrivial smooth solutions for the Neumann problem. For the semilinear case (i.e., $p=2$) using Morse theory, we produce one more nontrivial smooth solution.
keywords: p–Laplacian critical groups. Morse theory linking theorem three nontrivial smooth solutions local minimizer second deformation theorem
Nonlinear Neumann problems with indefinite potential and concave terms
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2015, 14(6): 2561-2616 doi: 10.3934/cpaa.2015.14.2561
In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.
keywords: bifurcation local minimizer. nonlinear maximum principle positive solutions; nodal solutions Harnack inequality Nonlinear regularity
Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential
Leszek Gasiński Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2013, 12(5): 1985-1999 doi: 10.3934/cpaa.2013.12.1985
We consider a semilinear Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a Carathéodory reaction term. Using variational methods based on the critical point theory, combined with Morse theory (critical groups), we prove two multiplicity theorems.
keywords: local minimizer multiplicity theorems. unique continuation property Indefinite and unbounded potential Harnack inequality mountain pass theorem critical groups
Positive solutions for p-Laplacian equations with concave terms
Sophia Th. Kyritsi Nikolaos S. Papageorgiou
Conference Publications 2011, 2011(Special): 922-930 doi: 10.3934/proc.2011.2011.922
We consider a nonlinear Dirichlet problem driven by the p-Laplacian diff erential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is p-linear and resonant with respect to $\lambda_1 > 0$ (the principal eigenvalue of (-$\Delta_p,W_0^(1,p)(Z)$)) at infi nity and the other when the perturbation is p-superlinear at infi nity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper-lower solutions and with suitable truncation techniques.
keywords: p-superlinear perturbation Concave nonlinearity critical point theory. truncation techniques p-linear perturbation upper-lower solutions
Nonlinear hemivariational inequalities with eigenvalues near zero
Leszek Gasiński Nikolaos S. Papageorgiou
Conference Publications 2005, 2005(Special): 317-326 doi: 10.3934/proc.2005.2005.317
In this paper we consider an eigenvalue problem for a quasilinear hemivariational inequality of the type $-\Delta_p x(z) -\lambda f(z,x(z))\in \partial j(z,x(z))$ with null boundary condition, where $f$ and $j$ satisfy ``$p-1$-growth condition''. We prove the existence of a nontrivial solution for $\lambda$ sufficiently close to zero. Our approach is variational and is based on the critical point theory for nonsmooth, locally Lipschitz functionals due to Chang [4].
keywords: p-Laplacian Palais-Smale condition Hemivariational inequality eigenvalue problem critical point theory Clarke subdi®erential mountain pass theorem.
Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems
Gabriele Bonanno Pasquale Candito Roberto Livrea Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2017, 16(4): 1169-1188 doi: 10.3934/cpaa.2017057

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet $p-$Laplacian. We consider three cases. In the first the perturbation is $(p-1)-$sublinear near $+\infty$, while in the second the perturbation is $(p-1)-$superlinear near $+\infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in (0, \widehat{\lambda}_1)$ -$\lambda>0$ is the parameter and $\widehat{\lambda}_1$ being the principal eigenvalue of $\left(-\Delta_p, W^{1, p}_0(\Omega)\right)$ -we have positive solutions, while for $\lambda\geq \widehat{\lambda}_1$, no positive solutions exist. In the "sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.

keywords: p-Laplacian first eigenvalue generalized Picone's identity nonlinear regularity nonlinear maximum principle variational methods
Periodic solutions for time-dependent subdifferential evolution inclusions
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu
Evolution Equations & Control Theory 2017, 6(2): 277-297 doi: 10.3934/eect.2017015

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

keywords: Convex subdifferential multivalued perturbation extremal solutions strong relaxation
Double resonance for Robin problems with indefinite and unbounded potential
Nikolaos S. Papageorgiou Patrick Winkert
Discrete & Continuous Dynamical Systems - S 2018, 11(2): 323-344 doi: 10.3934/dcdss.2018018

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity $f(x, s)$ is a Carathéodory function which is asymptotically linear as $ s\to ± ∞$ and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval $ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$. Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.

keywords: Indefinite potential Robin boundary condition regularity theory critical groups multiple nontrivial solutions double resonance

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