Separable solutions of some quasilinear equations with source reaction

We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.Comment: 34 page

Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations

We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.Comment: 17 page

Boundary singularities of positive solutions of some nonlinear elliptic equations

We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and existence or non existence and uniqueness results for solutions of some nonlinear elliptic equations on the half unit sphere.Comment: 6 page

Local and global properties of solutions of quasilinear Hamilton-Jacobi equations

We study some properties of the solutions of (E) \;-\Gd_p u+|\nabla u|^q=0 in a domain \Gw \sbs \BBR^N, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the positive solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result in expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete non compact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.Comment: to appear J. Funct. Ana

Quasilinear Lane-Emden equations with absorption and measure data

We study the existence of solutions to the equation -\Gd_pu+g(x,u)=\mu when $g(x,.)$ is a nondecreasing function and \gm a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We study particularly the cases where g(x,u)=\abs x^{\beta}\abs u^{q-1}u and g(x,u)=\abs x^{\tau}\rm{sgn}(u)(e^{\tau\abs u^\lambda}-1). The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz-Bessel capacities.Comment: 28 page

Isolated Boundary Singularities of Semilinear Elliptic Equations

Given a smooth domain \Omega\subset\RR^N such that $0 \in \partial\Omega$ and given a nonnegative smooth function $\zeta$ on $\partial\Omega$, we study the behavior near 0 of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u = \zeta$ on $\partial\Omega\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then u(x)\leq C \abs{x}^{-\frac{2}{q-1}} and we compute the limit of \abs{x}^{\frac{2}{q-1}} u(x) as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains

Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

International audienceWe study global properties of positive radial solutions of −∆u = up +M |∇u|p+1 in RN wherep > 1 and M is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at 0 according to the values of M and p.o x y (A) x t 0 y t 0 y t > 0 (D) x t 0 L C P

Entire large solutions for semilinear elliptic equations

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $\rho$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.Comment: Journal of Differential Equations 2012, 28 page

Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights.Comment: 19 page

Quasilinear Elliptic Hamilton-Jacobi Equations on Complete Manifolds

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 445-449.Let $(M^n,g)$ be a $n$-dimensional complete and non-compact Riemannian manifold with Ricci tensor $Ricc_g$ and sectional curvature $Sec_g$. Assume $Ricc_g\geq (1-n)B^2$ and $scal_g(x)=o(dist^2(x,a))$ for some $a\in M$ if $p>2$. Then for $q>p-1\geq 1$, any $C^1$ solution of (E) -\Gd_pu+\abs{\nabla u}^q=0 on $M$ satisfies \abs{\nabla u(x)}\leq c_{n,p,q}B^{\frac{1}{q+1-p}} for some constant $c_{n,p,q}>0$. As a consequence there exists $c_{n,p}>0$ such that any positive $p$-harmonic function $v$ on $M$ satisfies v(a)e^{-c_{n,p}B\dist (x,a)}\leq v(x)\leq v(a)e^{c_{d,p}B\dist (x,a)} for any $x\in M$