For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\partial C_S$, of the form $u\left(x\right)={\left|x\right|}^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Nella prima parte di questa Nota si dimostrano dei risultati di simmetria unidimensionale e radiale per le soluzioni di $\mathrm{\Delta }u+f\left(u\right)=0$ in ${\mathbb{R}}^N$. Questi risultati sono legati a due congetture (De Giorgi, 1978 e Gibbons, 1994) riguardanti la classificazione delle soluzioni dell’equazione $\mathrm{\Delta }u+u\left(1-u^2\right)=0$ in ${\mathbb{R}}^N$. Si dimostra, in particolare, la seguente generalizzazione della congettura di Gibbons: se $N>1$ e se l’insieme degli zeri di $u$ è limitato nella direzione $\nu$, allora $u\left(x\right)=u_0\left(\nu \cdot x\right)$, ovvero, $u$ è unidimensionale. Nella seconda parte si considerano...

Étant donné $L$ un opérateur différentiel d’ordre $m$ sur un ouvert $\Omega$ de ${\mathbf{R}}^N$, $K$ un compact de $\Omega$, $\gamma \&gt;1$ et ${\gamma }^{\text{'}}=\gamma /(\gamma -1)$, nous montrons que toute solution de “$Lu+u^{\gamma }=0$ sur $\Omega \setminus K,u\ge 0$” est solution de “$Lu+u^{\gamma }=0$ sur $\Omega$” dès que la $W^{m,{\gamma }^{\text{'}}}$-capacité de $K$ est nulle. Cette condition s’avère nécessaire quand $L$ est un opérateur elliptique d’ordre 2. Dans ce cas, nous montrons aussi que $``Lu+u|u{|}^{\gamma -1}=\mu ,u{|}_{\partial \Omega }=0^{\text{'}\text{'}}$ où $\mu$ est une mesure de Radon bornée sur $\Omega$, a une solution si et seulement si $\mu$ ne charge pas les ensembles de $W^{2,{\gamma }^{\text{'}}}$-capacité nulle.

In this paper we prove that some classes of semilinear elliptic problems, formulated in very general terms by using the theory of maximal monotone graphs, admit a finite propagation speed. More concretely we show that if the data of these problems have compact supports, then the same happens to their solutions. These same thechniques will also be applied to some evolution problems. The first results in this direction are due to H. Brézis and to O. Oleinik & A. S. Kalashnikov & C. Yuilin...

In this paper we study the solutions of Toda systems on Riemann surface in the critical case, proving a sufficient condition for existence.