Csaba Farkas

Singular Poisson equations on Finsler–Hadamard manifolds

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler–Laplace operator on Finsler–Hadamard manifolds having finite reversibility constant.

Anisotropic elliptic problems involving sublinear terms

In the present paper we prove a multiplicity result for an anisotropic sublinear elliptic problem with Dirichlet boundary condition, depending on a positive parameter λ. By variational arguments, we prove that for enough large values of λ, our anisotropic problem has at least two non-zero distinct solutions. In particular, we show that at least one of the solutions provides a Wulff-type symmetry.

Sobolev interpolation inequalities on Hadamard manifolds

We prove Sobolev-type interpolation inequalities on Hadamard manifolds and their optimality whenever the Cartan-Hadamard conjecture holds (e.g., in dimensions 2, 3 and 4). The existence of extremals leads to unexpected rigidity phenomena.

New Conditions for the Existence of Infinitely many Solutions for a Quasi-Linear Problem

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

A sublinear differential inclusion on strip-like domains

This paper deals with a sublinear differential inclusion problem (P_λ) depending on a parameter λ > 0 which is defined on a strip-like domain subject to the zero Dirichlet boundary condition. By variational methods, we prove that for large values of λ, problem (P_λ) has at least two non-zero axially symmetric weak solutions.