Multiple solutions for nonlinear coercive Neumann problems
Sophia Th. Kyritsi Nikolaos S. Papageorgiou
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
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
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
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
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
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

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
Nonlinear Dirichlet problems with a crossing reaction
Shouchuan Hu Nikolaos S. Papageorgiou
We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.
keywords: Nonlinear regularity critical groups. resonance nonlinear maximum principle
Nonlinear Neumann equations driven by a nonhomogeneous differential operator
Shouchuan Hu Nikolaos S. Papageorgiou
We consider a nonlinear Neumann problem driven by a nonhomogeneous nonlinear differential operator and with a reaction which is $(p-1)$-superlinear without necessarily satisfying the Ambrosetti-Rabinowitz condition. A particular case of our differential operator is the $p$-Laplacian. By combining variational methods based on critical point theory with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).
keywords: nonlinear regularity Morse relation Moser iteration method. Mountain Pass theorem critical group C-condition
Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter
Salvatore A. Marano Nikolaos S. Papageorgiou
A nonlinear elliptic equation with $p$-Laplacian, concave-convex reaction term depending on a parameter $\lambda>0$, and homogeneous boundary condition, is investigated. A bifurcation result, which describes the set of positive solutions as $\lambda$ varies, is obtained through variational methods combined with truncation and comparison techniques.
keywords: $p$-Laplacian Concave-convex nonlinearities positive solutions.

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