79 research outputs found
Boundary blow-up solutions in the unit ball : asymptotics, uniqueness and symmetry
We calculate the full asymptotic expansion of boundary blow-up so- lutions,
for any nonlinearity f . Our approach enables us to state sharp qualitative
results regarding uniqueness and ra- dial symmetry of solutions, as well as a
characterization of nonlinearities for which the blow-up rate is universal. At
last, we study in more detail the standard nonlinearities f (u) = u^p, p >
On the fractional Lane-Emden equation
We classify solutions of finite Morse index of the fractional Lane- Emden
equatio
Uniqueness of large solutions
Given a nondecreasing nonlinearity , we prove uniqueness of large
solutions in the following two cases: the domain is the ball or the domain has
nonnegative mean curvature and the nonlinearity is asymptotically convex
Regularity of radial extremal solutions for some non local semilinear equations
We investigate stable solutions of elliptic equations of the type
\begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad
{\mbox{ in }} \\ u&= 0 \qquad{\mbox{ on ,}}\end{aligned}\right . \end{equation*} where , ,
and is any smooth positive superlinear function. The
operator stands for the fractional Laplacian, a
pseudo-differential operator of order . According to the value of
, we study the existence and regularity of weak solutions
On a nonlocal equation arising in population dynamics
We study a one-dimensional nonlocal variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modelled by a convolution operator. We prove that as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits
The Lane–Emden equation in strips
International audienceWe study the Lane-Emden equation in strips
- …