85 research outputs found

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

Let L âˆ† be the logarithmic Laplacian operator with Fourier symbol 2 ln |Î¶|, we study the expression of the diffusion kernel which is associated to the equation âˆ‚ t u + L âˆ† u = 0 in (0, N 2) Ã— R N , u(0, â€¢) = 0 in R N \ {0}. We apply our results to give a classification of the solutions of âˆ‚ t u + L âˆ† u = 0 in (0, T) Ã— R N u(0, â€¢) = 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 âˆ† u = kÎ´ 0 in D (â„¦) such that u = 0 in R N \ â„¦

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

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

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

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

We study the isolated singularities of functions satisfying (E) (âˆ’âˆ†) s vÂ±|v| pâˆ’1 v = 0 in â„¦\{0}, v = 0 in R N \â„¦, where 0 1 and â„¦ is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + Ã— 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

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

International audienceWe study existence and uniqueness of solutions of (E 1) âˆ’âˆ†u + Âµ |x| ^{-2} u + g(u) = Î½ in â„¦, u = Î» on âˆ‚â„¦, where â„¦ âŠ‚ R N + is a bounded smooth domain such that 0 âˆˆ âˆ‚â„¦, Âµ â‰¥ âˆ’ N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and Î½ and Î» two Radon measures respectively in â„¦ and on âˆ‚â„¦. 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)