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We study properties of nonnegative functions satisfying (E)$\;-\Delta u+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1, Comment: 83 pages, 5 figures

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-\Delta u+u^p-M|\nabla u|^q=\mu$ in a domain $\Omega\subset\BBR^N$ where $\mu$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $\Omega\setminus K$ where $K$ is a compact subset of $\Omega$ can be extended as a solution of the same equation in $\Omega$, Comment: 26 pages

We study properties of positive functions satisfying (E) --$��$u + u p -- M |$\nabla$u| q = 0 is a domain $��$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the boundary except at one point. This analysis depends on the existence of separable solutions in R N +. We consruct various types of positive solutions with an isolated singularity on the boundary. We also study conditions for the removability of compact boundary sets and the Dirichlet problem associated to (E) with a measure for boundary data., 44 pages, 26 ref

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; \Delta u - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v =&\; (-\Delta_{\R^2})^{-1} u := \frac 1{2\pi} \int_{R^2} \, \log \frac 1{|x-z|}\,u(z,t)\, dz, \\ & \qquad\ u(\cdot ,0) = u_0 \geq 0\quad\hbox{in } R^2. \end{aligned} \right. \end{equation} We consider the {\em critical mass case} $\int_{R^2} u_0(x)\, dx = 8\pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8\pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of \equ{ks0} is globally defined and blows-up in infinite time. As $t\to+\infty $ it has the approximate profile $$ u(x,t) \approx \frac 1{\lambda^2} \ch{U}\left (\frac {x-\xi(t)}{\lambda(t)} \right ), \quad \ch{U}(y)= \frac{8}{(1+|y|^2)^2}, $$ where $\lambda(t) \approx \frac c{\sqrt{\log t}}, \ \xi(t)\to q $ for some $c>0$ and $q\in \R^2$. This result answers affirmatively the nonradial stability conjecture raised in \cite{g}., Comment: 95 pages; final version; comments are welcome

International audience; We study local and global properties of positive solutions of $-{\Delta}u=u^p]{\left |{\nabla u}\right |}^q$ in a domain ${\Omega}$ of ${\mathbb R}^N$, in the range $1

International audience; This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in W1,p(S1) with p ≥ 2. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a p-Laplacian type operator. It is remarkable that the carré du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever p≠2.

We study local and global properties of positive solutions of $$-\Delta u=u^p+M\left| \nabla u\right| ^q$$ in a domain $$\Omega $$ of $$\mathbb {R}^N$$ , in the range $$\min \{p,q\}>1$$ and $$M\in \mathbb {R}$$ . We prove a priori estimates and existence or non-existence of ground states for the same equation.

We study global properties of positive radial solutions of \begin{document}$ -\Delta u = u^p+M\left |{\nabla u}\right |^{\frac{2p}{p+1}} $\end{document} in \begin{document}$ \mathbb R^N $\end{document} where \begin{document}$ p>1 $\end{document} and \begin{document}$ M $\end{document} is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at \begin{document}$ 0 $\end{document} according to the values of \begin{document}$ M $\end{document} and \begin{document}$ p $\end{document} .

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $\mu$ is a Dirac mass, we prove the existence of several critical exponents p., Comment: Advanced Nonlinear Studies, Walter de Gruyter GmbH, A Para{\^i}tre

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., to appear J. Funct. Anal

We study the boundary behaviour of the solutions of (E) $-��_p u+|\nabla u|^q=0$ in a domain $��\subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partial��$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed., Calculus of Variations and Partial Differential Equations DOI 10.1007/s00526-015-0911-5

We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

In this article we study quasilinear systems of two types, in a domain $\Omega$ of $R^N$ : with absorption terms, or mixed terms: \begin{eqnarray} (A): \mathcal{A}_{p} u=v^{\delta}, \mathcal{A}_{q}v=u^{\mu},\\ (M): \mathcal{A}_{p} u=v^{\delta}, -\mathcal{A}_{q}v=u^{\mu}, \end{eqnarray} where $\delta$, $\mu>0$ and $1 0$; the model case is $\mathcal{A}_p = \Delta_p$, $\mathcal{A}_q = \Delta_q.$ Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type: \begin{eqnarray} u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leq Cd(x,\partial\Omega)^{-\frac{q(p-1)+p\mu}{D}}. \end{eqnarray} Concerning system $(M)$, we show that $v$ always satisfies Harnack inequality. In the case $\Omega=B(0,1)\backslash \{0\}$, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.

We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.

Let (M^n,g) be a n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Ricc_g and sectional curvature Sec_g. Assume Ricc_g\geq (1-n)B^2, and either p>2 and Sec_g(x)=o(dist^2(x,a)) when dist^2(x,a)\to\infty for a\in M, or 1p-1> 0, 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_{n,p}B\dist (x,a)} for any (a,x)\in M\times M., Comment: 6 pages

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain, $H$ a Caratheodory function defined in $\Omega \times \mathbb{R\times R}^{N},$ and $\mu $ a bounded Radon measure in $\Omega .$ We study the problem% \begin{equation*} -\Delta_{p}u+H(x,u,\nabla u)=\mu \quad \text{in}\Omega,\qquad u=0\quad \text{on}\partial \Omega, \end{equation*} where $\Delta_{p}$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm \left\| \nabla u\right\| ^{q}$ ($q>0$). We obtain an existence result under subcritical growth assumptions on $H,$ we give necessary conditions of existence in terms of capacity properties, and we prove removability results of eventual singularities. In the supercritical case, when $\mu \geqq 0$ and $H$ is an absorption term, i.e. $% H\geqq 0,$ we give two sufficient conditions for existence of a nonnegative solution., Comment: To appear in Contemporary Mathematics

We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$). We prove that such solutions exist, the spherical part $\omega$ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N --1 and that the exponents $\beta$ = $\beta$ + > 0 and $\beta$ = $\beta$ -- < 0 are uniquely determined if the domain is smooth.

In this work we study the nonnegative solutions of the elliptic system \[ \Delta u=|x|^{a}v^{\delta},\qquad\Delta v=|x|^{b}u^{\mu}% \] in the superlinear case $\mu\delta>1,$ which blow up near the boundary of a domain of $\mathbb{R}^{N},$ or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension $N$. We also show the existence of infinitely many solutions blowing up at $0.$ Furthermore, we show that there exists a global positive solution in $\mathbb{R}^{N}\backslash\left\{ 0\right\} ,$ large at $0,$ and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation \[ \Delta^{2}u=\left\vert x\right\vert ^{b}\left\vert u\right\vert ^{\mu}. \] Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, introduced in [4], combined with the nonradial upper estimates of [5].

In this article we study the positive solutions of the parabolic semilinear system of competitive type in Ω × (0, T), where Ω is a domain of ℝN, and p,q > 0, pq ≠ 1, Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case pq > 1 of the form u(x, t) ≦ Ct-(p+1)/(pq-1), v(x, t) ≦ Ct-(q+1)/(pq-1) in ω × (0, T1), for any domain ω ⊂⊂ Ω, T1 ∈ (0, T), and C = C(N, p, q, T1, ω). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Ω. Finally we prove that the punctual singularities at time 0 are removable when p, q ≧ 1 + 2/N.