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.     

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.     

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.     

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.     

Spectral properties of a differential system

The spectrum of the differential operators associated with the differential system$\text{u′=}\text{r}\text{D}\text{D}\text{p}\text{u,}\text{D}\text{y=−y″+qy}\text{on}\text{L}{}^2\text{([0,∞))}$is determined. For this the coefficients are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix.