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Csaba Farkas

Publications and source records attributed to Csaba Farkas.

17 recordsLinked to original sources
arXiv2025

Higher-order Sobolev and Rellich inequalities via iterated Talenti's principle

In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality) and manifolds with non-positive sectional curvature (satisfying the Cartan-Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].

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arXiv2025

Critical double phase problems involving sandwich-type nonlinearities

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad \text{in } \Omega,\quad u= 0 \quad\text{on } \partial \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overline{\Omega})$, $\lambda$, $\theta$ are real parameters, $w$ is a suitable weight and $B\colon \overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $\lambda$ and $\theta$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$.

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arXiv2023

Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Krist\'aly [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

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arXiv2021

On a class of critical double phase problems

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on $\partial\Omega$, where $\Omega \subset \mathbb{R}^N$ is a bounded Lipschitz domain, $1<\vartheta 0$, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of $\lambda^*>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $\lambda\in (0,\lambda^*)$.

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arXiv2021

Singular Finsler double phase problems with nonlinear boundary condition

In this paper we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

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arXiv2020

An existence result for singular Finsler double phase problems

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general assumptions and based on variational tools we prove the existence of at least one nontrivial weak solution for such a problem. This is the first work dealing with a Finsler double phase operator even in the nonsingular case.

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arXiv2020

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings \`a la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

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arXiv2020

Critical elliptic equations on non-compact Finsler manifolds

In the present paper, we deal with a quasilinear elliptic equation involving a critical Sobolev exponent on non-compact Randers spaces. Under very general assumptions on the perturbation, we prove the existence of a non-trivial solution. The approach is based on the direct methods of the calculus of variations. One of the key steps is to prove that the energy functional associated with the problem is weakly lower semicontinuous on small balls of the Sobolev space, which is provided by a general inequality.

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arXiv2019

Topological rigidity of compact manifolds supporting Sobolev-type inequalities

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere $(\mathbb S^n,g_0)$, we prove that $(M,g)$ is topologically close to $(\mathbb S^n,g_0)$. Moreover, the Sobolev constants on $(M,g)$ are precisely the optimal constants on the sphere $(\mathbb S^n,g_0)$ if and only if $(M,g)$ is isometric to $(\mathbb S^n,g_0)$; in particular, the latter result answers a question of V.H. Nguyen.

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arXiv2019

On a critical Kirchhoff-type problem

In the present paper, we study a Kirchhoff type problem involving the critical Sobolev exponent. We give sufficient conditions for the sequentially weakly lower semicontinuity and the Palais Smale property of the energy functional associated to the problem.

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arXiv2018

Energy properties of critical Kirchhoff problems with applications

In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the parameters for the convexity and validity of the Palais-Smale condition of the same energy functional. We then apply these regularity properties in the study of some elliptic problems involving the critical Kirchhoff term.

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arXiv2018

Schrodinger-Maxwell systems on compact Riemannian manifolds

In this paper we are focusing to the following Schr\"odinger-Maxwell system $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$: \[ \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mathrm{in}\ M -\Delta_{g}\phi+\phi=qu^{2} & \mathrm{\mathrm{in}\ M} \end{cases} \] where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $\beta\in C^{\infty}(M)$ and $\Psi\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$ is established.

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arXiv2017

A characterization related to Schr\"odinger equations on Riemannian manifolds

In this paper we consider the following problem $$\begin{cases} -\Delta_{g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{in }M\\ u\geq0, & \mbox{in }M\\ u\to0, & \mbox{as }d_{g}(x_{0},x)\to\infty \end{cases}$$where $(M,g)$ is a $N$-dimensional ($N\geq3)$, non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, $\lambda$ is a real parameter, $V$ is a positive coercive potential, $\alpha$ is a bounded function and $f$ is a suitable nonlinearity. By using variational methods we prove a characterization result for existence of solutions for our problem.

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arXiv2016

Multipolar Hardy inequalities on Riemannian manifolds

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schr\"odinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

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arXiv2016

Schr\"odinger-Maxwell systems on Hadamard manifolds

In this paper we study nonlinear Schr\"odinger-Maxwell systems on $n-$dimensional Hadamard manifolds, $3\leq n\leq 5.$ The main difficulty resides in the lack of compactness of such manifolds which is recovered by exploring suitable isometric actions. By combining variational arguments, some existence, uniqueness and multiplicity of isometry-invariant weak solutions are established for the Schr\"odinger-Maxwell system depending on the behavior of the nonlinear term.

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arXiv2016

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.

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