CPAA
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
CPAA
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
CPAA
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
PROC
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
PROC
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.
CPAA
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
CPAA
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
CPAA
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.
DCDS
Dirichlet $(p,q)$-equations at resonance
Leszek Gasiński Nikolaos S. Papageorgiou
We consider a parametric nonlinear Dirichlet equation driven by the sum of a $p$-Laplacian and a $q$-Laplacian ($1 < q < p < +\infty$, $p ≥ 2$) and with a Carathéodory reaction which at $\pm\infty$ is resonant with respect to the principal eigenvalue $\widehat{\lambda}_1(p) > 0$ of $(-\Delta_p, W^{1,p}_0(\Omega))$. Using critical point theory, truncation and comparison techniques and critical groups (Morse theory), we show that for all small values of the parameter $\lambda>0$, the problem has at least five nontrivial solutions, four of constant sign (two positive and two negative) and the fifth nodal (sign-changing).
keywords: critical groups. nonlinear strong maximum principle Constant sign and nodal solutions nonlinear regularity
DCDS
Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term
Sophia Th. Kyritsi Nikolaos S. Papageorgiou
We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator. We assume that the Carathéodory reaction term $f(z,x)$ exhibits an asymmetric behavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is $(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near $+\infty$, but without satisfying the well-known Ambrosetti--Rabinowitz condition (AR-condition). Combining variational methods based on critical point theory, with suitable 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, the other negative).
keywords: radial retraction. Nonlinear maximum principle superlinear reaction critical groups strong deformation retract homotopy invariance

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