arXiv subjects
Publications and source records attributed to Zakaria Boucheche.
We investigate a borderline between existence and non-existence of positive solution for a nonlinear elliptic equation involving a critical Sobolev exponent in three-dimensional ball. The method is relied on a suitable choice of the functions used to test two natural ingredients for the associated variational problem.
This paper concerns a nonlinear elliptic equation involving a critical Sobolev growth and a lower-order term. Under a Lions's condition, we prove the existence of at least one positive solution. Our approach consists in constructing a relatively compact Palais--Smale sequence for the associated variational problem.
We are concerned with a semi-linear elliptic equation on a smooth bounded domain $Ω$ of $\mathbb{R}^n,\,n\geq 5,$ which involves a critical nonlinearity and a linear term of the form $K(x)u^{(n+2)/(n-2)}$ and $μu,$ respectively. By using a test function procedure, we give an existence criterion involving the parameter $μ$ and the function $K(x).$ For a particular case of $Ω,\,K(x)$ and $n,$ we prove its optimality through a Pohozaev type identity.
In this paper, we consider the following nonlinear elliptic equation with Dirichlet boundary condition: $-Δu=K(x)u^{\frac{n+2}{n-2}},\, u>0$ in $Ω,\, u=0$ on $\partialΩ$, where $Ω$ is a smooth bounded domain in $\mathbb{R}^n,$ $n\geqslant 4,$ and $K$ is a $\mathcal{C}^1$-positive function in $\barΩ$. Under the assumption that the order of flatness at each critical point of $K$ is $β\in ]\,n-2,\,n[,$ we give precise estimates on the looses of the compactness, and we prove an existence result through an Euler-Hopf type formula.