arXiv subjects

Hoang-Hung Vo

Publications and source records attributed to Hoang-Hung Vo.

At least 19 recordsLinked to original sources

arXiv2026

Principal eigenvalues of nonlocal operators with advection: sharp scaling limits and spectral phase transitions

We consider generalized principal eigenvalues of one-dimensional nonlocal dispersal operators with a drift of fixed sign. A principal difficulty in this non-self-adjoint setting is the absence of a variational characterization of the principal value. In the symmetric drift-free problem, the critical nonlocal-to-local limit can be treated through a quadratic variational structure and Sobolev-seminorm approximation as Berestycki-Coville-Vo \cite{BCV}. The drift destroys this structure and, on a bounded interval, introduces at the same time a one-sided inflow--outflow boundary geometry. Replacing the missing variational argument by estimates which remain stable under singular rescaling is therefore a central technical issue. Our approach is direct : we work with a single generalized principal value and prove the maximum principle, simplicity in the positive cone and the scaling limits from the equation itself, without passing through auxiliary generalized principal eigenvalues analogous to $λ_1'$ and $λ_1''$ in Berestycki--Rossi \cite{BR}. The replacement mechanisms are directional Harnack inequalities and coefficient barriers, direct--adjoint identities, logarithmic Collatz--Wielandt transforms, Fourier coercivity of zero extensions and scale-dependent localization. At the critical diffusive scale, Fourier compactness and an exact energy--transport identity recover the missing inflow Dirichlet condition and identify the local Dirichlet limit without a Rayleigh quotient. For variable coefficients we determine the complete small-range phase diagram on bounded intervals and on the line, and we obtain sharp large-range three-term asymptotics by a rank-one reduction and a corner Laplace analysis in the advective travel-time variable. In the homogeneous whole-line problem the critical correction is of order $σ^2$.

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arXiv2026

Harnack Theory and Rigidity for Singular and Degenerate Fully Nonlinear Elliptic Equations with Hamiltonians

We study regularity, Harnack inequalities, Liouville rigidity, and principal eigenvalues for viscosity solutions of singular or degenerate fully nonlinear elliptic equations $Φ(x,|\nabla u|)F(D^2u)-H(x,\nabla u)+c(x)|u|^{i(Φ)}u=h(x)$ in a bounded domain $Ω$, where $Φ$ describes the singular or degenerate dependence on the gradient and $H$ is a Hamiltonian. We first prove global $C^{1,γ}$ regularity for the Dirichlet problem without lower-order terms. The argument combines a global $L^\infty$ estimate from the Alexandroff--Bakelman--Pucci inequality, boundary barriers yielding a global Lipschitz bound, and a compactness-based iterative approximation scheme. We next establish an additive Harnack inequality for nonnegative viscosity solutions by sliding from below a cusp function of the form $-|x|^{1/2}$. Under an additional homogeneity assumption, this yields the classical Harnack inequality and Liouville-type theorems in $\mathbb{R}^n$. Finally, under suitable homogeneity and comparison assumptions, we develop a generalized Dirichlet principal-eigenvalue theory for the full operator. We prove the existence of principal eigenfunctions and characterize the associated eigenvalues through maximum and minimum principles. These results provide a unified framework for global regularity, Harnack estimates, Liouville rigidity, and principal eigenvalues for a broad class of singular and degenerate fully nonlinear equations with Hamiltonian terms.

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arXiv2026

Long-time dynamics and threshold Phenomena for a free-boundary SIS Model with asymmetric kernels in advective periodic environments

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of $S$ and $I$ via an $ω$-limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.

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arXiv2026

Spectral Theory of Fractional Cooperative Systems and Threshold Dynamics in Epidemic Models

Spectral analysis has long been recognized as a fundamental tool for studying the existence, uniqueness, and qualitative behavior of solutions to semilinear elliptic and parabolic equations, as well as their long-time dynamics. In modern mathematics, fractional Laplacians are widely used to model nonlocal or long-range diffusion processes arising in biology, including anomalous movement, long-distance dispersal, and Levy-flight migration of organisms, cells, and epidemics. In this paper, we employ the spectral fractional Laplacian introduced by Caffarelli and Stinga (2016) to develop the eigentheory for a cooperative system describing an infectious epidemic process and to analyze its long-term behavior. Using Fredholm theory and related analytical techniques, building in part on ideas of Lam and Lou (2016), we establish a sharp criterion ensuring the existence and simplicity of the principal eigenvalue, together with variational characterizations and consequences for the validity of maximum principles. We further derive the asymptotic behavior of the principal eigenvalue with respect to diffusion coefficients, fractional orders, and domain scaling, complementing recent developments by Zhao and Ruan (2023) and Feng, Li, Ruan, and Xin (2024). As an application of this spectral framework, we prove the existence, uniqueness, and threshold-type long-time dynamics of solutions to an endemic reaction-diffusion system with fractional diffusion, providing a perspective that differs from earlier approaches such as Hsu and Yang (2013). Our results contribute to the growing interaction between spectral theory and nonlocal analysis, in line with recent advances in the area.

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arXiv2026

Principal spectral theory and asymptotic analysis for time-periodic cooperative systems with temporally nonlocal dispersal

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.

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arXiv2025

Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon Nonlinearities

In this paper we study Liouville-type properties for a class of degenerate elliptic equations driven by the fractional infinity Laplacian with nonlinear lower-order terms, \[ Δ_\infty^βu - c\,H(u,\nabla u) - λ\, f(|x|,u)=0 \qquad \text{in }\mathbb{R}^n, \] where $β\in[0,2]$, $Δ_\infty^β$ denotes the fractional infinity Laplace operator, and the nonlinearities $H$ and $f$ represent Hamiltonian and Hardy--Hénon type effects, respectively. We extend the Liouville theory for the classical and normalized infinity Laplacian by establishing a new weighted comparison principle together with sharp local Lipschitz estimates for viscosity solutions. Our Liouville theorems are derived from precise growth conditions for bounded nonnegative solutions when $f$ exhibits power-type behavior, i.e.\ $f\sim u^γ$. We also treat the exponential case $f\sim e^u$, for which the equation becomes strongly supercritical: under suitable assumptions on the growth of $u$ at spatial infinity, only partial Liouville-type conclusions can be obtained. The analysis relies on radial reduction, barrier constructions, and refined comparison arguments. Altogether, the results provide a unified framework linking regularity, comparison principles, and Liouville-type phenomena for degenerate elliptic equations involving fractional infinity Laplacians and nonlinear lower-order effects.

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arXiv2025

A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general Hénon-type superquadratic and gradient growth

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^γ\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric, translation-invariant, uniformly elliptic integro--differential operator of order $2s\in(0,2)$, and $H,G$ satisfy general structural and growth conditions. A unified analytical framework is developed to identify the precise critical balance $γ+p=2s$, which separates the supercritical, critical, and subcritical situations. In the supercritical case $γ+p>2s$, the diffusion dominates the nonlinear term and every globally defined solution with subcritical growth must be constant; in the critical case $γ+p=2s$, all bounded positive solutions are constant, showing that the nonlocal diffusion prevents the formation of nontrivial equilibria; in the subcritical case $γ+p<2s$, we are able construct a unique, positive, radially symmetric, and monotone entire solution with explicit algebraic decay \[ u(x)\sim (1+|x|^2)^{-β}, \qquad β=\frac{2s+γ-p}{1-p}>0. \] The proofs rely on new nonlocal analytical techniques, including quantitative cutoff estimates for general integro--differential kernels, a fractional Bernstein-type transform providing pointwise gradient control, and moving plane and sliding methods formulated in integral form to establish symmetry and uniqueness. The current investigation provides an equivalent and unifying contribution to Liouville properties and related existence results, comparable to the recent deep studies of Chen--Dai--Qin~\cite{Chen2023} and Biswas--Quaas--Topp~\cite{Biswas2025} on this direction.

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arXiv2025

The eigentheory for nonlocal cooperative-advective system and its role in the study of free boundary system for directional epidemic models

In this paper, we propose and analyze a nonlocal cooperative reaction--diffusion system with free boundaries and drift terms, motivated by directional epidemic spread. Lacking a variational structure but requiring sharper regularity of solutions, the model poses substantial analytical challenges compared with previous works~\cite{Du,Berestycki2016a,Berestycki2016b,Cao2019,NguyenVo2022,Tang2024a,Tang2024b}. We first establish the well-posedness of the local problem and the global existence and uniqueness of classical solutions in $C^1$ space. We then study the associated nonlocal eigenvalue problem, proving the existence, simplicity, qualitative properties, and asymptotic behavior of the principal eigenvalue. The analysis employs Fredholm theory, the Crandall--Rabinowitz bifurcation theorem, and Hadamard-type derivative formulas to describe its parameter dependence and connection with the basic reproduction number~$R_0$. Building on this spectral characterization, we show that the system admits a \emph{sharp vanishing--spreading dichotomy} in its long-term dynamics. When $R_0\le1$, all solutions vanish; for $R_0>1$, the outcome depends on the initial domain size~$h_0$ and the free-boundary expansion rate~$μ$. There exists a critical habitat length~$\mathcal L^\ast$ such that if $h_0<\mathcal L^\ast$, a threshold $\widehatμ>0$ separates vanishing ($μ\in(0,\widehatμ]$) from spreading ($μ>\widehatμ$). In the spreading regime, solutions converge to the unique positive steady state, while in the vanishing regime they decay uniformly to zero. These results provide a rigorous framework for the threshold dynamics of cooperative--advective nonlocal systems and offer mathematical insight for further studies in epidemic modeling, ecological invasion, and population dynamics.

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arXiv2025

Spatio-temporal dynamics of an age-structured reaction-diffusion system of epidemic type subjected by Neumann boundary condition

This paper is concerned with the spatio-temporal dynamics of an age-structured reaction-diffusion system of KPP-epidemic type (SIS), subject to Neumann boundary conditions and incorporating $L^1$ blow-up type death rate. We first establish the existence of time dependent solutions using age-structured semigroup theory. Afterward, the basic reproduction number $\mathcal{R}_0$ is derived by linearizing the system around the disease-free equilibrium state. In the case $\mathcal{R}_0<1$, the existence, uniqueness and stability of disease-free equilibrium are shown by using $ω$-limit set approach of Langlais \cite{langlais_large_1988}, combined with the technique developed in recent works of Zhao et al. \cite{zhao_spatiotemporal_2023} and Ducrot et al. \cite{ducrot_age-structured_2024}. We highlight that the absence of a general comparison principle for the age-structured SIS-model and non-separable variable mortality rate prevent the direct application of the semi-flow technique developed in \cite{ducrot_age-structured_2024} to study the long time dynamics.

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arXiv2022

On the characterization of Bow-up and Global Radial solution for free boundary system with nonlinear inhomogeneous gradient and source term

This paper concerns the characterization of blowup and global radial solutions of a two-free boundaries system read by \begin{align}\label{bs_pr} \tag{1.1} \left\{\begin{array}{rl} u_t(t,r)= Δu(t,r) - λ(t,x)|\nabla u(t,r)|^α + a(t,x)v^{p}(t,r),& t>0,\ 0 0,\ 0 < r < g(t), \end{array}\right. \end{align} where $r = |x|,\ x \in \R^N$, $p, α>1 $ are given constants and $λ(t,x), a(t,x)$ satisfy suitable prescribed growth conditions. First, we show the well-posedness of the local solution to (\ref{bs_pr}). Second, we succeed to classify the blowup and global phenomena by establishing some relations between $α$, $p$ and growth rate of the coefficients, in which proving a comparison principle based on the Stampacchia truncation method plays the central role. In particular, if $1<α< p$ and $ (u(0,r);v(0,r)) = A (ϕ(r); φ(r))$ is an initial data of (\ref{bs_pr}), we find two certain positive thresholds $A^*_G$ and $A^*_B$ such that the global fast solution exists for $0 < A < A_G^{*}$ and the global slow solution exists for a suitable value of $A$ such that $A^*_G \leq A < A^*_{B}$ while blow-up solutions hold for $A \geq A^{*}_{B}$. On the other hand, if $α\geq p$ incorporating with a suitable comparison on $β, p$ and $α$, then there exist global solutions with nonnegative initial data of exponential decay. Our approach is being far different from the celebrated works \cite{MF,PS}, where the authors can only handled the equations with constant coefficients. To our knowledge, this is the first work revealing the influence of the inhomogeneous coefficients to the blow-up and global phenomena to the cooperative system with nonlinear gradient and different free boundaries.

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arXiv2022

Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $\mathbb{R}^N$ and applications

Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a contribution in the development of the theory of \textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.

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arXiv2021

Liouville theorems for infinity Laplacian with gradient and KPP type equation

In this paper, we prove new Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ where $γ\in [0, 2]$ and $Δ^γ_\infty$ is a $(3-γ)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-γ}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-γ}$ decreasing in $s$, we further obtain the uniqueness by improving sliding method for infinity Laplacian operator with nonlinear gradient. Otherwise, if $\limsup_{|x|\to\infty}\sup_{[δ_1,δ_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop novel techniques to overcome the difficulties stemming from the degeneracy of infinity Laplacian and nonlinearity of the gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopf's lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further investigate some deeper qualitative properties of the principal eigenvalue of the corresponding nonlinear operator $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + c(x)u^{3-γ},$$ with Dirichlet boundary condition in smooth bounded domains, which may be of independent interest. The results obtained here could be considered as sharp extension of the Liouville type results obtained in [1, 2, 11, 24, 48, 52].

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arXiv2021

Dynamics for a two phases free boundaries system in an epidemiological model with nonlocal dispersals

The present paper is devoted to the investigation of the long time dynamics for a double free boundary system with nonlocal diffusions, which models the infectious diseases transmitted via digestive system such as fecal-oral diseases, cholera, hand-foot and mouth, etc...We start by proving the existence and uniqueness of the Cauchy problem, which is not a trivial step due to presence of couple dispersals and new types of nonlinear reaction terms. Next, we provide simple conditions on comparing the basic reproduction numbers $\CR_0$ and $\CR^*$ with some certain numbers to characterize the global dynamics, as $t\to\infty$. We further obtain the sharp criteria for the spreading and vanishing in term of the initial data. This is also called the vanishing-spreading phenomena. The couple dispersals yield significant obstacle that we cannot employ the approach of Zhao, Zhang, Li, Du \cite{WWZ1} and Du-Ni \cite{DN}. To overcome this, we must prove the existence and the variational characterization for the principal eigenvalue of a linear system with nonlocal dispersals, then use it to obtain the right limits as the dispersal rates and domain tend to zero or infinity. The maximum principle and sliding method for the nonlocal operator are ingeniously employed to achieve the desired results.

math.DS↗

arXiv2020

Principal eigenvalue and positive solutions for Fractional $P-Q$ Laplace operator in quantum field theory

This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-Δ_p\right)^αu + \left(-Δ_q\right)^βu \,= λ\left[a(x) \left|u\right|^{p-2}u + b(x) \left|u\right|^{q-2}u \right]\quad\quad\textrm{in $Ø$},$$ where $Ø$ is a smooth bounded domain in $\R^N$ extended by zero outside. When $Ø=\R^N$ and $b\equiv0$, we further show that there exists a continuous family of the eigenvalue if $1<q<p<q^*_β=\frac{Nq}{N-qβ}$ and $0\leq a\in L^{\left(\frac{q_β^*}{s}\right)'}\left(\R^N\right)\bigcap L^{\infty}\left(\R^N\right)$ with $s$ satisfies $\dfrac{p-t}{p_α^*}+ \dfrac{p\left(1-t\right)}{s} =1$, for some $t\in \left(0, \sqrt{\dfrac{p-q}{p}}\right).$ Our approach replies strongly on variational analysis, in which the Mountain pass theorem plays the key role. The main difficulty in this study is that how to establish the Palais-Smale conditions. In particular, in $\R^N$, due to the lack of spatial compactness and the embedding $W^{α, p}\left(\R^N\right) \hookrightarrow W^{β, q}\left(\R^N\right)$, we must employ the concentration-compactness principle of P.L. Lions \cite{PLL} to overcome the difficulty.

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arXiv2019

Principal Spectral Theory of time-periodic nonlocal dispersal operators of Neumann type

In this communication, we prove some important limits of the principal eigenvalue for nonlocal operator of Neumann type with respect to the parameters, which are significant in the understanding of dynamics of biological populations. We obtained a complete picture about limits of the principal eigenvalue in term of the large and small dispersal rate and dispersal range classified by "Ecological Stable Strategy" of persistence. This solves some open problems remainning in the series of work [3, 28, 29], in which we have to overcome the new difficulties comparing to [3, 28, 29] since principal eigenvalue of nonlocal Neumann operator is not monotone with respect to the domain. The maximum principle for this type of operator is also achieved in this paper.

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arXiv2017

On the generalized principal eigenvalue of quasilinear operators

The notions of generalized principal eigenvalue for linear second order elliptic operators in general domains introduced by Berestycki et al. \cite{BNV,BR0,BR3} have become a very useful and important tool in analysis of partial differential equations. In this paper, we extend these notions for quasilinear operator of the form $$\CK_V[u]:=-Δ_p u +Vu^{p-1},\quad\quad u \geq0.$$ This operator is a natural generalization of self-adjoint linear operators. If $Ø$ is a smooth bounded domain, we already proved in \cite{NV} that the generalized principal eigenvalue coincides with the (classical) first eigenvalue of $\CK_V$. Here we investigate the relation between three types of the generalized principal eigenvalue for quasilinear operator on general smooth domain (possibly unbounded), which plays an important role in the investigation of their asymptotic properties. These results form the basis for the study of the simplicity of the generalized principal eigenvalues, the maximum principle and the spectrum of $\CK_V$. We further discuss applications of the notions by providing some examples.

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arXiv2017

Nonlocal dispersal equations in time-periodic media: principal spectral theory, bifurcation and asymptotic behaviors

This paper is devoted to the investigation of the following nonlocal dispersal equation $$ u_{t}(t,x)=\frac{D}{σ^m}\left[\int_ΩJ_σ(x-y)u(t,y)dy-u(t,x)\right]+f(t,x,u(t,x)), \quad t>0,\quad x\in\overlineΩ, $$ where $Ω\subset\mathbb{R}^{N}$ is a bounded and connected domain with smooth boundary, $m\in[0,2)$, $D>0$ is the dispersal rate, $σ>0$ characterizes the dispersal range, $J_σ=\frac{1}{σ^{N}} J\left(\frac{\cdot}σ\right)$ is the scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of generalized KPP type. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at $u\equiv0$. We obtain an easily verifiable and general condition for the existence of the principal eigenvalue as well as important sup-inf characterizations for the principal eigenvalue. We next study the influence of the principal eigenvalue on the global dynamics and confirm the criticality of the principal eigenvalue being zero. It is then followed by the study of the effects of the dispersal rate $D$ and the dispersal range characterized by $σ$ on the principal eigenvalue and the positive time-periodic solution, and prove various asymptotic behaviors of the principal eigenvalue and the positive time-periodic solution when $D,σ\to0^{+}$ or $\infty$. Finally, we establish the maximum principle for time-periodic nonlocal operator.

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arXiv2016

On the definition and the properties of the principal eigenvalue of some nonlocal operators

In this article we study some spectral properties of the linear operator $\mathcal{L}\_Ω+a$ defined on the space $C(\barΩ)$ by :$$ \mathcal{L}\_Ω[φ] +aφ:=\int\_ΩK(x,y)φ(y)\,dy+a(x)φ(x)$$ where $Ω\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue $λ\_p(\mathcal{L}\_Ω+a)$ defined by $$λ\_p(\mathcal{L}\_Ω+a):= \sup\{λ\in \mathbb{R} \,|\, \exists φ\in C(\bar Ω), φ\textgreater{}0, \textit{such that}\, \mathcal{L}\_Ω[φ] +aφ+λφ\le 0 \, \text{in}\;Ω\}. $$ We establish some new properties of this generalised principal eigenvalue $λ\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $λ\_p(\mathcal{L}\_Ω+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $λ\_{p} \left(\mathcal{L}\_{σ,m,Ω} -\frac{1}{σ^m}+a\right)$ where $\mathcal{L}\_{σ,m,Ω}$ is defined by $\displaystyle{\mathcal{L}\_{σ,m,Ω}[φ]:=\frac{1}{σ^{2+N}}\int\_ΩJ\left(\frac{x-y}σ\right)φ(y)\, dy}$. In particular, we prove that $$\lim\_{σ\to 0}λ\_p\left(\mathcal{L}\_{σ,2,Ω}-\frac{1}{σ^{2}}+a\right)=λ\_1\left(\frac{D\_2(J)}{2N}Δ+a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $λ\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $φ\_{p,σ}$.

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