85 research outputs found

    The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution

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    Let L ∆ be the logarithmic Laplacian operator with Fourier symbol 2 ln |ζ|, we study the expression of the diffusion kernel which is associated to the equation ∂ t u + L ∆ u = 0 in (0, N 2) × R N , u(0, •) = 0 in R N \ {0}. We apply our results to give a classification of the solutions of ∂ t u + L ∆ u = 0 in (0, T) × R N u(0, •) = f in R N and obtain an expression of the fundamental solution of the associated stationary equation in R N , and of the fundamental solution in a bounded domain, i.e. L ∆ u = kδ 0 in D (Ω) such that u = 0 in R N \ Ω

    Counting function and lower bound for Dirichlet eigenvalues of the m-order logarithmic Laplacian

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    Our aim in this article is to obtain the limit of counting function for the Dirichlet eigenvalues involving the m-order logarithmic Laplacian in a bounded Lipschitz domain and to derive also the lower bound.Comment: 1

    On Schr\"odinger equations involving the regional fractional Laplacian in a ball with the zero boundary condition

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    Our purpose in this article to show the existence of positive classical solutions of (Δ)B1su+u=h1up+ϵh2in B1,u=0on B1 ( - \Delta )_{B_1}^s u +u=h_1 u^p+\epsilon h_2 \quad {\rm in} \ \, B_1,\qquad u = 0 \quad {\rm on}\ \partial B_1 for ϵ>0\epsilon>0 small enough, where (Δ)B1s( - \Delta )_{B_1}^s is the regional fractional Laplacian, p>1p>1, hih_i with i=1,2i=1,2 are H\"older continuous and satisfy some additional conditions. Our existence is based on the solution of (Δ)B1su+u=1in B1,u=0on B1. ( - \Delta )_{B_1}^s u +u=1 \quad {\rm in} \ \, B_1,\qquad u = 0 \quad {\rm on}\ \partial B_1. Comment: 2

    Singularities of fractional Emden's equations via Caffarelli-Silvestre extension

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    We study the isolated singularities of functions satisfying (E) (−∆) s v±|v| p−1 v = 0 in Ω\{0}, v = 0 in R N \Ω, where 0 1 and Ω is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + × R N. We emphasize the obtention of a priori estimates, analyse the set of self-similar solutions via energy methods to characterize the singularities

    On fractional harmonic functions

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    Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined in a principle value sense at infinity. Secondly, we study the fractional harmonic functions in half space with singularities on the boundary and the related distributional identities.Comment: 20 page

    Boundary singularities of semilinear elliptic equations with Leray-Hardy potential

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    International audienceWe study existence and uniqueness of solutions of (E 1) −∆u + µ |x| ^{-2} u + g(u) = ν in Ω, u = λ on ∂Ω, where Ω ⊂ R N + is a bounded smooth domain such that 0 ∈ ∂Ω, µ ≥ − N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and ν and λ two Radon measures respectively in Ω and on ∂Ω. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem (E 1)

    Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model

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    International audienceThis article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged O(3) sigma model, (E) −∆u + A 0 (k j=1 |x − p j | 2nj) −a e u (1 + e u) 1+a = 4π k j=1 n j δ pj − 4π l j=1 m j δ qj in R 2 , where {δ pj } k j=1 (resp. {δ qj } l j=1) are Dirac masses concentrated at the points {p j } k j=1 , (resp. {q j } l j=1), n j and m j are positive integers and a ≥ 0. We set N = k j=1 n j and M = l j=1 m j. In previous works [10, 31], some qualitative properties of solutions of (E) with a = 0 have been established. Our aim in this article is to study the more general case where a > 0. The additional difficulty of this case comes from the fact that the nonlinearity is no longer monotone and we cannot construct directly supersolutions and subsolutions anymore. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in gauged O(3) sigma model. Without the gravitational term, i.e. a = 0, problem (E) has a layer's structure of solutions {u β } β∈(−2(N −M), −2] , where u β is the unique non-topological solution such that u β = β ln |x| + O(1) for −2(N − M) 0, the set of solutions to problem (E) has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that u tends to −∞ at infinity, of non-topological solutions in type II, which tend to ∞ at infinity. The existence of these types of solutions depends on the values of the parameters N, M, β and on the gravitational interaction associated to a

    Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

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    18 pages, 34 ref.International audienceWe provide bounds for the sequence of eigenvalues {λ i (Ω)} i of the Dirichlet problem L ∆ u = λu in Ω, u = 0 in R N \ Ω, where L ∆ is the logarithmic Laplacian operator with Fourier transform symbol 2 ln |ζ|. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough, hence the principle eigenvalue is no longer always positive. We also give asymptotic estimates of the sum of the first k eigenvalues. To study the principle eigenvalue, we construct lower and upper bounds by a Li-Yau type method and calculate the Rayleigh quotient for some particular functions respectively. Our results point out the role of the volume of the domain in the bound of the principle eigenvalue. For the asymptotic of sum of eigenvalues, lower and upper bounds are built by a duality argument and by Kröger's method respectively. Finally, we obtain the limit of eigenvalues and prove that the limit is independent of the volume of the domain

    Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model

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    This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged O(3) sigma model, (E) −∆u + A 0 (k j=1 |x − p j | 2nj) −a e u (1 + e u) 1+a = 4π k j=1 n j δ pj − 4π l j=1 m j δ qj in R 2 , where {δ pj } k j=1 (resp. {δ qj } l j=1) are Dirac masses concentrated at the points {p j } k j=1 , (resp. {q j } l j=1), n j and m j are positive integers and a ≥ 0. We set N = k j=1 n j and M = l j=1 m j. In previous works [10, 31], some qualitative properties of solutions of (E) with a = 0 have been established. Our aim in this article is to study the more general case where a > 0. The additional difficulty of this case comes from the fact that the nonlinearity is no longer monotone and we cannot construct directly supersolutions and subsolutions anymore. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in gauged O(3) sigma model. Without the gravitational term, i.e. a = 0, problem (E) has a layer's structure of solutions {u β } β∈(−2(N −M), −2] , where u β is the unique non-topological solution such that u β = β ln |x| + O(1) for −2(N − M) 0, the set of solutions to problem (E) has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that u tends to −∞ at infinity, of non-topological solutions in type II, which tend to ∞ at infinity. The existence of these types of solutions depends on the values of the parameters N, M, β and on the gravitational interaction associated to a
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