Global Structure of the Solution Set for a Semilinear Elliptic Problem Related to the Liouville Equation on an Annulus

Abstract

A semilinear elliptic problem related to the Liouville equation on a two-dimensional annulus is studied. The problem appears as the limiting problem of the Liouville equation as the inside radius of the annulus tends to 0, and is derived by the method of matched asymptotic expansions. Our concern is the solution set of the problem in the bifurcation diagram. We find explicit solutions including non-radially symmetric solutions and determine the connected component containing the solutions. As a consequence, we provide a suggestive evidence for the global structure of the solution set of the Liouville equation.

Appendix

In this appendix, we prove the following lemma which is used in Sect. 4.

Lemma 1

Let $\tilde{\varPhi}\in X$ and suppose that there are positive constants C ₀, μ ₀ such that the inequality $|\varDelta_{(s,\theta)} \tilde{\varPhi}(s,\theta)| \le C_{0} e^{-\mu_{0} |s|}$ holds for all (s,θ)∈ℝ×S ¹. Then there exist positive constants C, μ, s ₀ and real numbers c ₊, c ₋, d ₊, d ₋ such that

(31)

(32)

for all ±s≥s ₀ and θ∈S ¹.

Proof

Let $\tilde{\varphi}_{0} (s)/2 + \sum_{j=1}^{\infty} ( \tilde{\varphi}_{j} (s) \cos j\theta+ \tilde{\psi}_{j} (s) \sin j\theta)$ be the Fourier series of $\tilde{\varPhi}$ with respect to θ and put $F(s,\theta):=\varDelta_{(s,\theta)} \tilde{\varPhi}(s,\theta)$. Then $\tilde{\varphi}_{j}$ satisfies

$$ (\tilde{\varphi}_j)_{ss} -j^2 \tilde{\varphi}_j = f_j(s), \quad s \in {\mathbb{R}}, $$

where $f_{j}(s)=\pi^{-1}\int_{0}^{2\pi} F(s,\tau) \cos j\tau d\tau$. In particular, we have $(\tilde{\varphi}_{0})_{ss} (s)=O(e^{-\mu_{0} |s|})$ as |s|→∞. From this it follows easily that for some constants c ₊, c ₋, d ₊, d ₋, $\varphi_{0} (s)/2 -c_{\pm} s -d_{\pm} =O(e^{-\mu_{0} |s|})$, $(\varphi_{0})_{s} (s)/2 -c_{\pm}=O(e^{-\mu_{0} |s|})$ as s→±∞. We estimate $\tilde{\varphi}_{j}$ for j≥1. Since $\tilde{\varPhi}\in X$, $\tilde{\varphi}_{j}$ does not have exponential growth. Hence $\tilde{\varphi}_{j}$ is given by

$$ \tilde{\varphi}_j(s) = -\frac{1}{2j} \int_{-\infty}^\infty e^{-j|s-t|} f_j(t)\,dt. $$

From the assumption, we have

Therefore we see that for any fixed 0<μ<min{μ ₀,1}, there exist a constant C>0 independent of s and j such that $|\tilde{\varphi}_{j}(s)| \le C j^{-2} e^{-\mu|s|}$. Since the same estimate holds also for $\tilde{\psi}_{j}$, we have $|\tilde{\varPhi}(s,\theta) -\tilde{\varphi}_{0} (s)/2| \le C ( \sum_{j=1}^{\infty}j^{-2} ) e^{-\mu|s|}$. Thus (31) follows.

We verify (32). Since

for j≥1, we have

Here $z=|s-t|+\sqrt{-1}(\theta-\tau)$. In a similar way,

These estimates yield

where 0<μ<min{μ ₀,1}. The last integral of the above inequality is finite, and consequently (32) is verified. □