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.     

Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations

We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x)→n_∞(x/|x|) as |x|→∞. We prove that the Sommerfeld condition at infinity still holds true under the weaker form

$\text{1}\text{R}\text{∫}\text{|x|?R}\text{?u?}\text{i}\text{n}{}_{\mathrm{\infty }}{}^{\mathrm{1/2}}\text{x}\text{|x|}\text{u}\text{x}\text{|x|}{}^2\text{d}\text{x→0,}\text{as}\text{R→∞.}$

Our approach consists in proving this estimate in the framework of the limiting absorbtion principle. We use Morrey–Campanato type of estimates and a new inequality on the energy decay, namely

$\text{∫}\text{R}{}^{\mathrm{d}}\text{?}\text{?ω}\text{n}{}_{\mathrm{\infty }}\text{(ω)}{}^2\text{|u|}{}^2\text{|x|}\text{d}\text{x?C,}\text{ω=}\text{x}\text{|x|}\text{.}$

It is a striking feature that the index n_∞ appears in this formula and not the phase gradient, in apparent contradiction with existing literature. To cite this article: B. Perthame, L. Vega, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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.