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 $\text{∂Ω}$.     

Über die nichtlineare diophantische Approximation von quadratischen,algebraischen Zahlen

In this paper I prove the following Theorem: Let α be a quadratic irrational number and let ? > 0, $\text{t =}\text{1}\text{1728}$. Then there exists an effectively computable, positive constant c depending on α and ? such that

$\text{6q}{}^2\text{α6 < c(}\text{log}\text{q)}{}^{\mathrm{t??}}\text{q}{}^{\mathrm{?2}}$

has no solutions in natural numbers q.     

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.     

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.