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.

Bibliographie

[1] S.\;S. Vikharev, “Some Liouville Theorems for Stationary Ginzburg–Landau Equation on Quasimodel Riemannian Manifolds”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:3 (2015), 127–135

[2] A.\;G. Losev, “Some Liouville Theorems on Riemannian Manifolds of Special Type”, Soviet Mathematics, 1991, no. 12, 15–24 | MR | Zbl

[3] A.\;G. Losev, “The Schrödinger Equation in the Curved Riemannian Products”, Trudy po analizu i geometrii, Izd-vo in-ta matematiki Publ., Novosibirsk, 2000, 350–369 | MR | Zbl

[4] A.\;G. Losev, A.\;P. Sazonov, “On the Asymptotic Behavior of Solutions of Semilinear Equations on Model Riemannian Manifolds”, Science Journal of Volgograd State University. Mathematics. Physics, 2013, no. 2 (19), 36–56 | DOI

[5] A.\;G. Losev, Yu.\;S. Fedorenko, “Positive Solutions of Quasilinear Elliptic Inequalities on Noncompact Riemannian Manifolds”, Mathematical Notes, 81:6 (2007), 867–878 | DOI | MR | Zbl

[6] A. Grigor'yan, “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”, Bull. Amer. Math. Soc., 36:2 (1999), 135–249 | DOI | MR | Zbl

[7] T. Kusan, M. Naito, “Positive entire solutions of superlinear elliptic equations”, Hirosima Math. J., 16:2 (1986), 361–366 | MR | Zbl

[8] M. Murata, “Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall parrurbations”, Rev. Mat. Iberoamericana, 27:3 (2011), 885–907 | DOI | MR | Zbl

[9] M. Murata, T. Tsuchida, “Uniqueness of $L^1$-harmonic functions on rotationally symmetric Riemannian manifolds”, Kodai Math. J., 37:1 (2014), 1–15 | DOI | MR | Zbl