On Regions of Existence and Nonexistence of solutions for a System of pp-qq-Laplacians

We give a new region of existence of solutions to the superhomogeneous Dirichlet problem \quad \begin{array}{l} -\Delta_{p} u= v^\delta\quad v>0\quad {in}\quad B,\cr -\Delta_{q} v = u^{\mu}\quad u>0\quad {in}\quad B, \cr u=v=0 \quad {on}\quad \partial B, \end{array}\leqno{(S_R)} where $B$ is the ball of radius $R>0$ centered at the origin in \RR^N. Here $\delta, \mu >0$ and $\Delta_{m} u={\rm div}(|\nabla u|^{m-2}\nabla u)$ is the $m-$Laplacian operator for $m>1$.Comment: 17 pages, accepted in Asymptotic Analysi

On the invertibility of mappings arising in 2D grid generation problems

In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube

A note on the moving hyperplane method

We give more precision on the regularity of the domain that is needed to have the monotonicity and symmetry results recently proved by Damascelli and Pacella, result concerning p-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method.Comment: 4 pages, 2 figure

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in \RR^N with smooth boundary. We distinguish the cases where either $f(u)=-\lambda|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=\lambda|u|^{p-2}u-|u|^{r-2}u$, with $p$, $q>1$, $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the first case we show the existence of infinitely many weak solutions for any $\lambda>0$. In the second case we prove the existence of a nontrivial weak solution if $\lambda$ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces

Entire solutions of quasilinear elliptic systems on Carnot Groups

We prove general a priori estimates of solutions of a class of quasilinear elliptic system on Carnot groups. As a consequence, we obtain several non existence theorems. The results are new even in the Euclidean setting.Comment: 21 pages submitte