CPAA
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
PROC
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.
PROC
Existence results for quasilinear hemivariational inequalities at resonance
Leszek Gasiński
Conference Publications 2007, 2007(Special): 409-418 doi: 10.3934/proc.2007.2007.409
In this paper we consider quasilinear hemivariational inequality at resonance. We obtain two existence theorems using a Landesman-Lazer type condition. The method of the proof is based on the nonsmooth critical point theory for locally Lipschitz functions.
keywords: Clarke subdifferential Landesman-Lazer type conditions first eigenvalue nonsmooth Cerami condition p-Laplacian nonsmooth Palais-Smile condition mountain pass theorem. strong resonance at infinity
DCDS
Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential
Leszek Gasiński
Discrete & Continuous Dynamical Systems - A 2007, 17(1): 143-158 doi: 10.3934/dcds.2007.17.143
In this paper we study boundary value problem with one dimensional $p$-Laplacian. Assuming complete resonance at $+\infty$ and partial resonance at $0^+$, an existence of at least one positive solution is proved. By strengthening our assumptions we can guarantee strict positivity of the obtained solution.
keywords: resonance generalized nonsmooth Palais-Smale condition first eigenvalue positive solution p-Laplacian generalized nonsmooth mountain pass theorem.
DCDS
Dirichlet $(p,q)$-equations at resonance
Leszek Gasiński Nikolaos S. Papageorgiou
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 2037-2060 doi: 10.3934/dcds.2014.34.2037
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
CPAA
Multiple solutions for a class of nonlinear Neumann eigenvalue problems
Leszek Gasiński Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2014, 13(4): 1491-1512 doi: 10.3934/cpaa.2014.13.1491
We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian. Using variational methods we show that when the parameter $\lambda > \widehat{\lambda}_1$ (where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., $p=2$), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.
keywords: flow invariance. critical groups Smooth solutions of constant sign first nonzero eigenvalue negative gradient flow mountain pass theorem nodal solutions
PROC
Optimal control problem of Bolza-type for evolution hemivariational inequality
Leszek Gasiński
Conference Publications 2003, 2003(Special): 320-326 doi: 10.3934/proc.2003.2003.320
In this paper we study an optimal control problem of Bolza-type described by evolution hemivariational inequality of second order. Sufficient conditions for obtaining an existence result for such problem are given.
keywords: Clarke's subdifferential. Bolza-type problem Hemivariational inequality
CPAA
A pair of positive solutions for $(p,q)$-equations with combined nonlinearities
Leszek Gasiński Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2014, 13(1): 203-215 doi: 10.3934/cpaa.2014.13.203
We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential operator, with a reaction consisting of a ``concave'' term perturbed by a $(p-1)$-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.
keywords: Positive solutions Cerami condition $p$-superlinearity nonlinear regularity q)$-differential operator. $(p multiplicity theorem strong maximum principle
CPAA
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
DCDS-B
Periodic solutions for nonlinear nonmonotone evolution inclusions
Leszek Gasiński Nikolaos S. Papageorgiou
Discrete & Continuous Dynamical Systems - B 2018, 23(1): 219-238 doi: 10.3934/dcdsb.2018015

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple $(X,H,X^*)$ of spaces. The operator $A(t,x)$ representing the spatial differential operator is not in general monotone. The reaction (source) term $F(t,x)$ is defined on $[0,b]× X$ with values in $2^{X^*}\setminus\{\emptyset\}$. Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

keywords: Evolution triple pseudomonotone map coercivity elliptic regularization parabolic inclusion

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