Shooting from singularity to singularity and a semilinear Laplace–Beltrami equation

Abstract

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or \(\mathbb {R}^N\).

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Correspondence to Alfonso Castro.

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This work was partially supported by a grant from the Simons Foundation (# 245966 to Alfonso Castro). I. Ventura was partially supported by the NSF under Grant DMS-0946431.

Appendix A: Computation of the Laplace–Beltrami operator on surfaces of revolution in geodesic metric

Appendix A: Computation of the Laplace–Beltrami operator on surfaces of revolution in geodesic metric

In order to prove that the solutions to (1.5) are the rotationally symmetric solutions to (1.3) first we compute \(\partial ^2_{x_N}u\) for functions that are constant along normal directions to M. More precisely, let \(\hat{s} \in (0,a)\). Since rotationally symmetric functions satisfy \(u(\theta , x_N) = u(0, |\theta |, z)\) we simply write \(u(0,x_{N-1}, x_N) = u(x_{N-1}, x_N)\). Let \((0, \ldots , 0, G(t), z(t)):= (G(t), z(t))\) denote a local parametrization of

$$\begin{aligned}M \cap \{(x_1, \ldots , x_{N-1},z); x_1= \cdots = x_{N-2}=0\}\end{aligned}$$

around \((0, \ldots , 0, G(\hat{s}) , z(\hat{s}))\) such that \((G'(s))^2 + (z'(s))^2 = 1\). By the \(\epsilon \)-neighborhood theorem, see [9], there exists \(\delta >0\) and differentiable functions \(s:(-\delta , \delta ) \rightarrow \mathbb {R}\) and \(t:(-\delta , \delta ) \rightarrow \mathbb {R}\) such that

$$\begin{aligned} (G(\hat{s}), z(\hat{s}) + h) = (G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h))), \end{aligned}$$
(A.1)

for all \( h \in (-\delta , \delta )\), \(s(0) = \hat{s}\) and \( t(0) = 0\). Differentiating in (A.1) with respect to h we have

$$\begin{aligned} \begin{aligned} 0&= G'(s(h))s'(h) - z'(s(h))t'(h) - z''(s(h))s'(h) t(h) \\ 1&= z'(s(h))s'(h) + G'(s(h))t'(h) + t(h)G''(s(h))s'(h). \end{aligned} \end{aligned}$$
(A.2)

Taking \(h=0\) and using that \((G'(s))^2 + (z'(s))^2 = 1\) yield \(t'(0) = G'(\hat{s})\) and \(s'(0) =z'(\hat{s})\). Differentiating again gives

$$\begin{aligned} \begin{aligned} 0&= G''(\hat{s})(z'(\hat{s}))^2 + G'(\hat{s})s''(0) - z'(\hat{s}) t''(0) - 2z''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \ \hbox {and } \\ 0&= z''(\hat{s}) (z'(\hat{s}))^2 + z'(\hat{s}) s''(0) + G'(\hat{s})t''(0) + 2G''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \end{aligned} \end{aligned}$$
(A.3)

which in turn implies

$$\begin{aligned} \begin{aligned} s''(0)&= -G''(\hat{s})(z'(\hat{s}))^2 G'(\hat{s})+ 2z'(\hat{s})z''(\hat{s})(G'(\hat{s}))^2 - z''(\hat{s})(z'(\hat{s}))^3 \\&\quad - 2G'(\hat{s})G''(\hat{s})(z'(\hat{s}))^2 \\&= -G''(\hat{s}) G'(\hat{s})\left[ (z'(\hat{s}))^2 + 2(G'(\hat{s}))^2 -(z'(\hat{s}))^2 + 2(z'(\hat{s}))^2 \right] \\&= -2G''(\hat{s}) G'(\hat{s}), \ \ \hbox {and} \\ t''(0)&= -z''(\hat{s})((G'(\hat{s}))^2G'(\hat{s}) - G''(\hat{s})(z'(\hat{s}))^2 G'(\hat{s}). \end{aligned} \end{aligned}$$
(A.4)

Assuming now that u is constant along normal lines in an \(\epsilon \)-neighborhood of M and defining \(v(s) = u((G(s),z(s)) + t\eta (s))\) with \(\eta (s) = (-z' (s), G'(s))\) we have

$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_N^2}&(G(\hat{s}), z(\hat{s})) = \lim _{h \rightarrow 0}\left( \frac{u((0,G(\hat{s}), z(\hat{s})+h))}{h^2} \right. \\&\quad - \left. \frac{2u(0,G(\hat{s}) , z(\hat{s}))-u((0,G(\hat{s}),z(\hat{s})-h) )}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{u(0,G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h))) -2u(G(\hat{s}) , z(\hat{s}))}{h^2} \right. \\&\quad + \left. \frac{ u(0,G(s(-h)) - t(-h) z'(r(h)), z(s(-h)) + t(-h) G'(s(-h)))}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{v(s(h)) - 2v(s(0)) + v(s(-h))}{h^2} \right) . \end{aligned} \end{aligned}$$
(A.5)

Thus, by L’Hopital rule,

$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_N^2}(G(\hat{s}),y(\hat{s}))&= \lim _{h \rightarrow 0}\left( \frac{v'(s(h)) s'(h) -v'(s(-h))s'(-h)}{2h} \right) \\&= \frac{1}{2} \lim _{h \rightarrow 0}\left( v''(s(h)) (s'(h))^2 + v'(s(h))s''(h) \right. \\&\quad \left. +v''(s(-h))(s'(-h))^2 + v'(s(-h))s''(-h)\right) \\&= (z'(\hat{s}))^2 v''(\hat{s}) -2G''(\hat{s})G'(\hat{s})v'(\hat{s}). \end{aligned} \end{aligned}$$
(A.6)

Now we proceed to the computation of \(\frac{\partial ^2u}{\partial x_{N-1}^2}\). Using again the \(\epsilon \)-neighborhood theorem, for h small enough there exist differentiable functions s(h) and t(h) such that

$$\begin{aligned} (G(\hat{s}) + h, z(\hat{s})) = (G(s(h)), z(s(h)) + t(h) (-z'(s(h)), G'(s(h))). \end{aligned}$$
(A.7)

Differentiating with respect to h,

$$\begin{aligned} \begin{aligned} 1&= G'(s(h))s'(h) - z'(s(h))t'(h) - z''(s(h))s'(h) t(h) \\ 0&= z'(s(h))s'(h) + G'(s(h))t'(h) + t(h)G''(s(h))s'(h). \end{aligned} \end{aligned}$$
(A.8)

Also, \(t(0) = 0\) and \(s(0) = \hat{s}\). Hence \(1 = G'(\hat{s}) s'(0) - t'(0) z'(\hat{s})\) and \( 0= z'(\hat{s}) s'(0) + t'(0) G'(\hat{s})\). Thus

$$\begin{aligned} s'(0)=G'(\hat{s}), \ \ \hbox { and } \ \ t'(0) = -z'(\hat{s}). \end{aligned}$$
(A.9)

Differentiating in (A.8) with respect to h, replacing h by 0 and using (A.9) we have

$$\begin{aligned} \begin{aligned} 0&= G''(\hat{s})(G'(\hat{s}))^2 + G'(\hat{s})s''(0) - z'(\hat{s}) t''(0) + 2z''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \ \hbox {and } \\ 0&= z''(\hat{s}) (G'(\hat{s}))^2 + z'(\hat{s}) s''(0) + G'(\hat{s})t''(0) - 2G''(\hat{s}) G'(\hat{s}) z'(\hat{s}). \end{aligned} \end{aligned}$$
(A.10)

Using that \(z'(s)z''(s) = - G'(s) G''(s)\) for all s we have

$$\begin{aligned} \begin{aligned} s''(0)&= 2G'(\hat{s}) G''(\hat{s}) \ \ \hbox {and} \\ t''(0)&= \frac{(G'(\hat{s}))^2G''(\hat{s})}{ z'(\hat{s})}. \end{aligned} \end{aligned}$$
(A.11)

Since u is constant along normal lines in an \(\epsilon \)-neighborhood of M, defining \(v(s) = u(0,G(s),z(s)) + t\eta (s))\) with \(\eta (s) = ( 0,-z' (s), G'(s))\) we have

$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_{N-1}^2}&(0, G(\hat{s}), y(\hat{s})) = \lim _{h \rightarrow 0}\left( \frac{u(0,G(\hat{s})+h, z(\hat{s}) )}{h^2} \right. \\&\quad - \left. \frac{2u(0,G(\hat{s}) , z(\hat{s}))-u(0, G(\hat{s}) - h, z(\hat{s}) )}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{u(0, G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h)))- 2u(0,G(\hat{s}) , z(\hat{s})) }{h^2} \right. \\&\quad + \left. \frac{ u(0,G(s(-h)) - t(-h) z'(r(h)), z(s(-h)) + t(-h) G'(s(-h)))}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{v(s(h)) - 2v(s(0)) + v(s(-h))}{h^2} \right) . \end{aligned} \end{aligned}$$
(A.12)

Thus, by L’Hopital rule,

$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_{N-1}^2}(G(\hat{s}),y(\hat{s}))&= \lim _{h \rightarrow 0}\left( \frac{v'(s(h)) s'(h) -v'(s(-h))s'(-h)}{2h} \right) \\&= \frac{1}{2} \lim _{h \rightarrow 0}\left( v''(s(h)) (s'(h))^2 + v'(s(h))s''(h) \right. \\&\quad \left. +v''(s(-h))(s'(-h))^2 + v'(s(-h))s''(-h)\right) \\&= (G'(\hat{s}))^2 v''(\hat{s}) +2G''(\hat{s})G'(\hat{s})v'(\hat{s}). \end{aligned} \end{aligned}$$
(A.13)

From (A.9), (A.6), and (A.12),

$$\begin{aligned} \begin{aligned} \Delta _M(u)&= \frac{\partial ^2u}{\partial x_N^2}( G(s), z(s)) + \frac{\partial ^2u}{\partial x_{N-1}^2}( G(s), z(s)) + \frac{(N-2)}{G(s)} \frac{\partial u}{\partial x_{N-1}}(G(s), z(s)) \\&= v''(s) + \frac{(N-2)G'(s)}{G(s)} v'(s), \end{aligned} \end{aligned}$$
(A.14)

where we have used that the Laplacian operator \(\Delta \) in \(\mathbb {R}^{N-1}\) for radial functions is given by \(\partial ^2_r+ ((N-2)/r)\partial _r\). This proves that rotational symmetric solutions to (1.3) are given by solutions to (1.5) subject to (1.6).

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Castro, A., Ventura, I. Shooting from singularity to singularity and a semilinear Laplace–Beltrami equation. Boll Unione Mat Ital 12, 557–567 (2019). https://doi.org/10.1007/s40574-019-00191-y

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Keywords

  • Laplace–Beltrami operator
  • Pohozaev identity
  • Semilinear equation
  • Rotationally symmetric solution
  • Superlinear nonlinearity
  • Subcritical nonlinearity

Mathematics Subject Classification

  • 35J25
  • 58J05