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Aleks Jevnikar

Publications and source records attributed to Aleks Jevnikar.

At least 19 recordsLinked to original sources
arXiv2026

Qualitative bifurcation diagram for Grad-Shafranov type equations

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities. In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary. The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.

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arXiv2026

Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-\Delta)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^\mu}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental-solution term may occur at the singular point. For non-removable blow-up singularities satisfying a fundamental-order upper bound and a weighted source condition, we derive the corresponding Riesz decomposition and retain its nonnegative singular term in an off-center Kelvin moving-spheres argument. In the range determined by two nonnegative Kelvin weights, this yields radial symmetry and strict radial monotonicity. We also construct and classify positive radial homogeneous singular solutions in the corresponding convergence regime. If a nonzero positive radial homogeneous scaling limit is independently known to exist and to solve the limiting equation, then its coefficient is uniquely determined.

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arXiv2026

Sharp spectral estimates for free boundary problems arising in plasma physics

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say $\sigma_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\geq 2$, where we prove that in fact $\sigma_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

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arXiv2026

Existence results for a non-relativistic Chern-Simons model with purely mutual interaction

We are concerned with a skew-symmetric singular Liouville system arising in non-relativistic Chern-Simons theory. Based on its variational structure, we establish existence and multiplicity results. Since the energy functional is indefinite, standard variational approaches do not apply directly. We overcome this difficulty by introducing a suitable constrained problem and implementing a Morse-theoretical argument

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arXiv2025

The Rabinowitz continuum of subcritical Gelfand problems and free boundary-type equations arising in plasma physics

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual $L^\infty$ norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.

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arXiv2025

Critical sinh-Gordon flow with non-negative weight functions

The aim of this article is twofold: one one side we introduce and study the properties of a critical sinh-Gordon type flow \begin{equation*} {\frac{\partial}{\partial t}}e^u=\Delta_gu+8\pi\left({\frac{h_1e^u}{\int_{\Sigma}h_1e^udV_g}}-1\right)-\rho_2\left({\frac{h_2e^{-u}}{\int_{\Sigma}h_2e^{-u}dV_g}}-1\right), \end{equation*} where $\rho_2<8\pi$, $h_1,h_2$ are non-negative weight functions and $\Sigma$ is a closed Riemannian surface. Secondly, under suitable geometric conditions, we prove the convergence of the flow to a solution of the critical sinh-Gordon equation, extending the result of Zhou (2008) to the case of non-negative weights. The argument is based on a careful blow-up analysis. Some remarks about a Toda flow are also given.

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arXiv2025

Nonlocal problems with Hardy-Littlewood-Sobolev critical exponent and Hardy potential

We are concerned with a Brezis-Nirenberg type problem for a critical Choquard equation, in the sense of Hardy-Littlewood-Sobolev inequality, and with the Hardy potential in a smooth bounded domain. By exploiting variational methods we obtain existence results, which extend to different perturbation terms. Some estimates of independent interest about a nonlocal minimization problem are also derived.

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arXiv2025

Symmetry and monotonicity of singular solutions for the Hartree equation

In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-\Delta u\!=\Big( \int_{\mathbb{R}^N\setminus \Gamma}\frac{F(u(y))}{|x-y|^\mu}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminus\Gamma,$$ where $F$ is the primitive of $f$ and $\Gamma$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties.

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arXiv2025

A Lane-Emden system of free boundary type: existence, uniqueness and monotonicity of solutions

We consider a Hamiltonian system of free boundary type, showing first uniform bounds and existence of solutions and of the free boundary. Then, for any smooth and bounded domain, we prove uniqueness of positive solutions in a suitable interval and show that the associated energies and boundary values have a monotonic behavior. Some consequences are discussed about the parametrization of the unbounded Rabinowitz continuum for a class of superlinear strongly coupled elliptic systems.

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arXiv2025

Classification of singular limits for free boundary and singularly perturbed elliptic problems: the Dancer-Yan spikes revisited

We classify the singular limits relative to a free boundary problem arising in plasma physics in dimension $d=2$, under suitable natural integral bounds. It turns out that one of the asymptotic behaviors allowed corresponds to the Dancer-Yan spikes (J. London Math. Soc. ({\bf 78}) 2008, 639--662). Interestingly enough, roughly speaking and unlike the higher dimensional case, it is not true that any solution in the limit is a Dancer-Yan spike. Indeed, the spiking structure is more rich and we succeed in a detailed description of the singular behavior by a careful analysis, from local to global, of the tiny difference between the maximum value of the spikes and their ``vanishing level'' defining the free boundary.

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arXiv2024

Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities

On a $2m$-dimensional closed manifold we investigate the existence of prescribed $Q$-curvature metrics with conical singularities. We present here a general existence and multiplicity result in the supercritical regime. To this end, we first carry out a blow-up analysis of a $2m$th-order PDE associated to the problem and then apply a variational argument of min-max type. For $m>1$, this seems to be the first existence result for supercritical conic manifolds different from the sphere.

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arXiv2024

Sharp estimates, uniqueness and spikes condensation for superlinear free boundary problems arising in plasma physics

We are concerned with Grad-Shafranov type equations, describing in dimension $N=2$ the equilibrium configurations of a plasma in a Tokamak. We obtain a sharp superlinear generalization of the result of Temam (1977) about the linear case, implying the first general uniqueness result ever for superlinear free boundary problems arising in plasma physics. Previous general uniqueness results of Beresticky-Brezis (1980) were concerned with globally Lipschitz nonlinearities. In dimension $N\geq 3$ the uniqueness result is new but not sharp, motivating the local analysis of a spikes condensation-quantization phenomenon for superlinear and subcritical singularly perturbed Grad-Shafranov type free boundary problems, implying among other things a converse of the results about spikes condensation in Flucher-Wei (1998) and Wei (2001). Interestingly enough, in terms of the "physical" global variables, we come up with a concentration-quantization-compactness result sharing the typical features of critical problems (Yamabe $N\geq 3$, Liouville $N=2$) but in a subcritical setting, the singular behavior being induced by a sort of infinite mass limit, in the same spirit of Brezis-Merle (1991).

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arXiv2023

On the global bifurcation diagram of the equation $-Δu=μ|x|^{2α}e^u$ in dimension two

The aim of this note is to present the first qualitative global bifurcation diagram of the equation $-Δu=μ|x|^{2α}e^u$. To this end, we introduce the notion of domains of first/second kind for singular mean field equations and base our approach on a suitable spectral analysis. In particular, we treat also non-radial solutions and non-symmetric domains and show that the shape of the branch of solutions still resembles the well-known one of the model regular radial case on the disk. Some work is devoted also to the asymptotic profile for $μ\to-\infty$.

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arXiv2023

Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems

In the present paper we derive Liouville type results and existence of periodic solutions for $χ^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates for this class of problems. In this study, we have to face new difficulties due to the non-homogenous nonlinearities. To overcome this issue, we carry out delicate integral estimates for this class of nonlinearities and modify the usual scaling and blow up arguments. This seems to be the first result for parabolic systems with non-homogeneous nonlinearities.

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arXiv2023

On the first eigenvalue of Liouville-type problems

The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenfunction. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case.

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arXiv2023

Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains

In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan matrices and eigenvalue analysis of linearized Liouville-type problems. This seems to be the first result for this class of problems and it covers all the Lie algebras of any rank.

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