Two positive nontrivial solutions for a class of semilinear elliptic variational systems

We obtain existence and localization results of positive nontrivial solutions for a class of semilinear elliptic variational systems. The proof is based on variants of Schechter's localized critical point theorems for Hilbert spaces not identified to their duals and on the technique of inverse-positive matrices. The Leray-Schauder boundary condition is also involved.     

Theory and computation for multiple positive solutions of non-local problems at resonance

Resonance non-positone and non-isotone problems for first order differential systems subjected to non-local boundary conditions are reduced to the non-resonance positone and isotone case by changes of variables. This allows us to prove the existence of multiple positive solutions. The theory is illustrated by two examples for which three positive numerical solutions are obtained using the Mathematica shooting program.     

Nash-Type Equilibria for Systems of Szulkin Functionals

In this paper the solutions of some systems of variational inequalities are obtained as Nash-type equilibria of the corresponding systems of Szulkin functionals. This is achieved by an iterative scheme based on Ekeland’s variational principle, whose convergence is proved via the vector technique involving inverse-positive matrices. An application to periodic solutions for a system of two second order ordinary differential equations with singular ?-Laplacians is included.     

Fixed Point Theorems Under Combined Topological and Variational Conditions

The new idea is to replace part of the conditions on the operator involved in the classical fixed point theorems of Schauder, Krasnoselskii, Darbo and Sadovskii, by assumptions upon the associated functional, in case that the fixed point equation has a variational form. Fixed points minimizing the associated functionals are obtained via Ekeland’s variational principle and the Palais–Smale compactness condition guaranteed by the topological properties of the nonlinear operators.     

Multiple positive solutions of non-local initial value problems for first order differential systems

The paper gives a new and natural method for the existence of multiple positive solutions for first order differential systems with non-local initial value conditions involving linear functionals. The case of higher order differential equations is also considered. The results are accompanied by numerical examples confirming the theory and proving for practice the importance of the bounds in solution localization.     

Positive solutions for discontinuous problems with applications to $$\phi $$-Laplacian equations

We establish existence and localization of positive solutions for general discontinuous problems for which a Harnack-type inequality holds. In this way, a wide range of ordinary differential problems such as higher order boundary value problems or \(\phi \)-Laplacian equations can be treated. In particular, we study the Dirichlet–Neumann problem involving the \(\phi \)-Laplacian. Our results rely on Bohnenblust–Karlin fixed point theorem which is applied to a multivalued operator defined in a product space.     

Stationary Solutions of Fokker-Planck Equations with Nonlinear Reaction Terms in Bounded Domains

Using an operator approach, we discuss stationary solutions to Fokker-Planck equations and systems with nonlinear reaction terms. The existence of solutions is obtained by using Banach, Schauder and Schaefer fixed point theorems, and for systems by means of Perov’s fixed point theorem. Using the Ekeland variational principle, it is proved that the unique solution of the problem minimizes the energy functional, and in case of a system that it is the Nash equilibrium of the energy functionals associated to the component equations.

     

Critical point theorems in cones and multiple positive solutions of elliptic problems

We obtain critical point variants of the compression fixed point theorem in cones of Krasnoselskii. Critical points are localized in a set defined by means of two norms. In applications to semilinear elliptic boundary value problems this makes possible the use of local Moser-Harnack inequalities for the estimations from below. Multiple solutions are found for problems with oscillating nonlinearity.     

Affine pavings of Hessenberg varieties for semisimple groups

In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of $\mathfrak{gl }_n(\mathbb C )$ are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.