We consider the anisotropic quasilinear elliptic Dirichlet problem $$\left\{\begin{array}{cc}{\displaystyle -\sum _{i=1}^N{D}^i{a}_i(x,u,\nabla u)+{\left|u\right|}^{s\left(x\right)-1}u=f+\lambda \frac{{\left|u\right|}^{p_0\left(x\right)-2}u}{{\left|x\right|}^{p_0\left(x\right)}}}\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded subset of ${ℝ}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^1\left(\Omega \right)$ and $\lambda$ is a positive constant.

We study the following singular elliptic equation with critical exponent ⎧$-\Delta u=Q\left(x\right)u^{2*-1}+\lambda u^{-\gamma }$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

Soit $\mathbf{F}$ le faisceau des sursolutions variationnelles d’un opérateur différentiel elliptique du second ordre à coefficients ${\mathbf{L}}_{\mathrm{loc}}^{\infty }$. Soit $\widehat{\mathbf{F}}$ le faisceau des régularitées essentielles inférieures des éléments de $\mathbf{F}$. On démontre que $\widehat{\mathbf{F}}$ est contenu dans un seul préfaisceau ${\mathbf{F}}^{*}$ maximal de cônes convexes de fonctions s.c.i. $\>-\infty$ vérifiant le principe du minimum sur une base d’ouverts suffisamment petits. On démontre que ${\mathbf{F}}^{*}$ possède toutes les bonnes propriétés d’une théorie locale du potentiel.

We prove the existence of classical solutions to certain fully non-linear second order elliptic equations with large zeroth order coefficient. The principal tool is an a priori estimate asserting that the $C^{2,\alpha }$-norm of the solution cannot lie in a certain interval of the positive real axis.

Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in ${ℝ}^N$.

We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.