Existence of solutions for nonlinear elliptic degenerate equations

In this paper, we study the problem$\text{−div}\text{a(x,u,}\text{∇}\text{u)−div}\text{φ(u)+g(x,u)=f}\text{in}\text{Ω}$in the setting of the weighted sobolev space $\text{W}{}_0{}^{\mathrm{1,p}}\text{(Ω,ν)}$. The main novelty of our work is L^∞ estimates on the solutions, and the existence of a weak and renormalized solution.     

The question of interior blow-up points for an elliptic Neumann problem: the critical case

In contrast with the subcritical case, we prove that for any bounded domain $\text{Ω}$ in $\text{R}{}^3$, the Neumann elliptic problem with critical nonlinearity $\text{−Δu+μu=u}{}^5\text{,}\text{u>0}\text{in}\text{Ω;}\text{∂u/∂ν=0}\text{on}\text{∂Ω}$ has no solution blowing up at only interior points as μ goes to infinity.     

On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent

In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponent$\text{Δ}{}^2\text{u=u}{}^{\mathrm{n+4/(n-4)}}\text{in}\text{Ω,}\text{u>0}\text{in}\text{Ω,}\text{u=}\text{Δ}\text{u=0}\text{on}\text{∂Ω,}$where $\text{Ω}$ is a bounded domain of $\text{R}{}^{\mathrm{n}}\text{(n⩾5)}$ with a smooth boundary $\text{∂Ω}$.     

Une caractérisation des fonctions holomorphes injectives en analyse ultramétriqueA characterization of injective holomorphic functions in ultrametric analysis

We prove that a non constant holomorphic function f defined over an analytic subspace of $\text{C}{}_{\mathrm{p}}$ is injective if and only if

$\text{f(x)?f(y)}\text{x?y}{}^2\text{=}\text{f′(x)·f′(y)}\text{,}\text{for every distinct}\text{x}\text{and}\text{y.}$

This characterization proves the analogue, for holomorphic functions, of a conjecture of A. Escassut and M.C. Sarmant. On the other hand we give a counter-example to this conjecture, that concerns bi-analytic elements. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 441–446.     

Partial differential equations involving subcritical,critical and supercritical nonlinearities

We establish the existence of at least one nonnegative solution for the problem$\text{(}\text{P}\text{)}{}_{\mathrm{\lambda }}\text{−div(a(|}\text{∇}\text{u|)}\text{∇}\text{u)}\text{=}\text{λf(u)}\text{in}\text{Ω,}\text{u}\text{=}\text{0}\text{on}\text{∂Ω,}$where a and f satisfy conditions near zero. Here the novelty is that we do not need restrictions on the nonlinearities at infinity. Therefore, we can consider subcritical, critical and supercritical cases.