Positive solutions of elliptic equations on Riemannian manifolds of a special type

A. P. Sazonov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2015), pp. 6-18

Résumé

In this paper we study the asymptotic behavior of positive solutions of elliptic equations $\Delta u+p(r)u^{\gamma}=0$ and $\rm{div}\left(\sigma(r)\nabla u\right)+p(r)u^\gamma=0$ on complete Riemannian manifolds. The conditions of existence and nonexistence of positive solutions of the equations studied on such manifolds. Let $M$ — complete Riemannian manifold can be represented as a union of $M=B\cup D$, where $B$ — a compact and $D$ isometric to the direct product of $[0;\infty)\times S$, where $S$ — compact Riemannian manifold with metric $$ds^2=h^2(r)dr^2+q^2(r)d\theta^2.$$ Where $h(r)$ and $q(r)$ — a positive, smooth on $[0;\infty)$ functions, and $d\theta$ — the standard Riemannian metric on the sphere $S$. The following assertions. Theorem 1. Let the manifold $M$ is such that $$\int_1^\infty\frac{h(t)dt}{q^{n-1}(t)}=\infty.$$ Then every non-negative solution (1) is identically zero. Theorem 2. Let the manifold $M$ is such that $$\int_1^\infty\frac{h(t)dt}{q^{n-1}(t)}=\infty$$ and let it go $$-\frac{\gamma+3}{\gamma+1}h(r)q^{n-1}(r)p(r)+ \frac{4(n-1)}{\gamma+1}q^{2n-3}(r)q'(r)p(r)\int_r^\infty\frac{h(t)dt}{q^{n-1}(t)}+$$ $$+\frac{2}{\gamma+1}q^{2n-2}(r)p'(r)\int_r^\infty\frac{h(t)dt}{q^{n-1}(t)}\leq0.$$ Then for every $\alpha>0$ the equation (1) is on $M$ a positive radially symmetric solution such that $u(0)=\alpha$. Theorem 3. Let the manifold $M$ is such that $$\int_1^\infty\frac{h(t)dt}{\sigma(r)q^{n-1}(t)}=\infty.$$ Then every non-negative solution (2) is identically zero. Theorem 4. Let the manifold $M$ is such that $$\int_1^\infty\frac{h(t)dt}{\sigma(r)q^{n-1}(t)}\infty.$$ and let it go $$-\frac{\gamma+3}{\gamma+1}h(r)q^{n-1}(r)p(r)+ \frac{4(n-1)}{\gamma+1}\sigma(r)q^{2n-3}(r)q'(r)p(r)\int_r^\infty\frac{h(t)dt}{\sigma(t)q^{n-1}(t)}+$$ $$+\frac{2}{\gamma+1}\sigma(r)q^{2n-2}(r)p'(r)\int_r^\infty\frac{h(t)dt}{\sigma(t)q^{n-1}(t)}+$$ $$+\frac{2}{\gamma+1}\sigma'(r)q^{2n-2}(r)p(r)\int_r^\infty\frac{h(t)dt}{\sigma(t)q^{n-1}(t)}\leq0.$$ Then for every $\alpha>0$ the equation (2) is on $M$ a positive radially symmetric solution such that $u(0)=\alpha$. In addition, the found conditions under which the equations (1) and (2) haven't a positive radially symmetric solutions.

Détail

Mots-clés : elliptic equations
Keywords: theorems of Liouville, model Riemannian manifolds, radially symmetric solutions, problem of Cauchy.