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.     

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.     

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 $\text{∂Ω}$.     

Viscoélastodynamique monodimensionnelle avec conditions de SignoriniOne-dimensional viscoelastodynamics with Signorini boundary conditions

Let α be a positive number. The one-dimensional viscoelastic problem

$\text{u}{}_{\mathrm{tt}}\text{?u}{}_{\mathrm{xx}}\text{?αu}{}_{\mathrm{xxt}}\text{=f,}\text{x∈(?∞,0],}\text{t∈[0,+∞),}$

with unilateral boundary conditions

$\text{u(0,·)?0,}\text{(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{)(0,·)?0,}\text{(u(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{))(0,·)=0,}$

can be reduced to the following variational inequality:

$\text{λ}{}_1\text{1w=g+b,}\text{w?0,}\text{b?0,}\text{〈w,b〉=0.}$

Here $\text{λ}\text{?}{}_1\text{(ω)}$ is the causal determination of $\text{i}\text{ω}\text{1+}\text{i}\text{αω}$. We show that the energy losses are purely viscous; this result is a consequence of the relation $\text{〈}\text{w}\text{?}\text{,b〉=0}$; since a priori, b is a measure and $\text{w}\text{?}$ is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988.     

Random fixed points of K-set- and pseudo-contractive random maps

Let $\text{(Ω,Σ)}$ be a measurable space, X and Y separable Banach spaces, and C a weakly compact subset of X. Let $\text{f}\text{:}\text{Ω×C→Y}$ and $\text{T}\text{:}\text{Ω×C→Y}$ be continuous random operators. Then the deterministic solvability of the equation$\text{f(ω,x)−T(ω,x)=0}\text{(ω∈Ω,x∈C)}$implies the stochastic solvability of it provided that (f−T)(ω,.) is demiclosed at zero and T(ω,C) is bounded for each $\text{ω∈Ω}$. As applications, random fixed points of various types of pseudo-contractive and k-set-contractive random operators are obtained.     

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.