Stability and Instability of Solutions of Semilinear Problems with Dirichlet Boundary Condition on Surfaces of Revolution

We consider the equation ∆u + f (u) = 0 on a surface of revolution with Dirichlet boundary conditions. We obtain conditions on f , the geometry of the surface and the maximum value of a positive solution in order to ensure its stability or instability. Applications are given for our main results.

In this paper, we extend some of these results for certain classes of surfaces of revolution in R 3 . We consider the positive solutions of where S ⊂ R 3 is a surface of revolution with metric g, ∆ g stands for the Laplace-Beltrami operator in S and f ∈ C 1 (R + ). Since S satisfies certain conditions we proceeded as in [13] to prove as the stability or instability can often be determined by the maximum value of u(x). Basically, we studied the sign of h(u) − h(α) in (0, α) where h(u) = 2 u 0 f (t)dt − u f (u) and α is the maximum value of positive solution u(x) in S. Under some conditions, we conclude that if h(u) − h(α) > 0 (h(u) − h(α) < 0) then the solution is stable (unstable). It is common to consider the function h in such matters, we cite [10,[12][13][14] and references therein.

M. Sônego
Recently it has been considered by some authors the question of stability in problems on surfaces of revolution or, in a more general setting, on compact Riemannian manifold. For example, see [1,6,7,16,18] for problems on surfaces without boundary or with Neumann boundary conditions and [2] for a problem with Robin boundary conditions. This work seems to be the first to consider the problem with Dirichlet boundary conditions.
In the final section, we explained how our results can be applied to obtain the multiplicity of solutions, in addition we present two simple examples. More specifically, we introduced a positive parameter λ in (1.1) (i.e. ∆ g u(x) + λ f (u(x)) = 0) and, if u(x) is a positive solution, then u L ∞ (S) uniquely identifies the solution pair (λ, u(x)) [5]. Hence, the solution set of (1.1) can be depicted by planar curves in (λ, u L ∞ (S) ) plane and our stability and instability results indicate the turning points of this curve. For more detail on this subject, see [12,14] for instance.
This paper is divided as follows. In Section 2 we recall some material from differential geometry and stability of solution. Moreover we prove two essential propositions to our approach. In Section 3 we present a result of instability for a class of surfaces of revolution that has only one pole. In Section 4 we consider S a cylindrical surface to obtain conditions for stability and instability while Section 5 is devoted to applications.

Preliminaries
We begin with some definitions and known results from differential geometry which will be used in the following sections.

Surface of revolution
Consider M = (M, g) a 2-dimensional Riemannian manifold with a metric given in local coordinates x = (x 1 , x 2 ) given by (using Einstein summation convention) Given a smooth vector field X on M, the divergence operator of X is defined as and the Riemannian gradient, denoted by ∇ g , of a sufficiently smooth real function φ defined on M, as the vector field We will see how the operator ∆ g can be expressed for the particular case where M is a surface of revolution. Let C be the curve of R 3 parametrized by where ψ, χ ∈ C 2 (I), ψ > 0 in (0, l) and (ψ ) 2 + (χ ) 2 = 1 in I. Moreover, and Let M be the surface of revolution parametrized by Setting x 1 = s, x 2 = θ then a surface of revolution in R 3 with the above parametrization is a 2-dimensional Riemannian manifold with metric By (2.1) and (2.2) M has no boundary and we always assume that M and the Riemannian metric g on it are smooth (see [3], for instance). The area element on M is dσ = ψdθds and the gradient of u with respect to the metric g is given by Hence,

Stability analysis
Consider a solution u(x) of (1.1) with S ⊂ M a surface of revolution with boundary. The eigenvalue problem for the corresponding linearized equation is It is well know that if the principal eigenvalue µ 1 is positive then u(x) is stable and if µ 1 is negative then u(x) is unstable. In the case µ 1 = 0, u(x) is sometimes called neutrally stable. This is so called linear stability and, roughly speaking, means that solutions of the corresponding parabolic equation, with the initial data near u will tend to u, as t → ∞. Some properties on the principal eigenpair (µ 1 , φ 1 ) of (2.5) have fundamental role in this work. Namely, µ 1 is a simple eigenvalue (i.e. the eigenspace corresponding to µ 1 is onedimensional); φ 1 can be assumed positive in S and S φ 2 1 dσ = 1. We outline the proof of the first one below. The others we omitted since classical argument of linearized stability can be applied to the present situation (e.g., see [9]) Let us start with a simple observation concerning solutions of (1.1). This result was observed in [1,6,18] for Neumann boundary condition, in [2] for Robin boundary conditions, and for convenience of the reader we will prove it in our case. Proposition 2.1. Every solution u(x) of problem (1.1), which depends on the angular variable θ, is unstable.
Proof. We have that u satisfies the equation Now, if we differentiate this equation with respect to θ we see that u θ is an eigenfunction of (2.5) with corresponding eigenvalue µ = 0. Since u θ must change sign it cannot be the eigenfunction corresponding to the lowest eigenvalue. Hence µ 1 < 0.

Proposition 2.2.
If φ 1 is an eigenfunction corresponding to the principal eigenvalue µ 1 of problem Proof. We first observe that for any θ 0 > 0, φ 1 (s, θ + θ 0 ) is also an eigenfunction corresponding to µ 1 . Moreover we have that φ 1 is 2π-periodic in θ and It is well known that µ 1 is a simple eigenvalue. We outline the proof for the reader's convenience. We suppose that φ 2 (= φ 1 (s, θ + θ 0 ), for instance) is also an eigenfunction corresponding to µ 1 and then φ 1 and φ 2 satisfy the equation We can assume φ 1 > 0, φ 2 > 0 and is not difficult to see that Using Green's theorem it follows that To use Green's theorem we define (φ 2 /φ 1 ) and each component of ∇ g (φ 2 /φ 1 ), as well as its derivatives, on ∂S, using a limit process so as to make it functions of H 1 (S). Therefore, we prove that φ 2 differs from φ 1 by a multiplicative constant and our claim follows.

A result of instability
Consider S = D ⊂ M a surface of revolution with boundary such that D has one of the poles but not the other. For example, consider ψ(1) > 0 and ∂D = C 1 where 0 < 1 < l. Then C 1 is parametrized in the local coordinates (s, θ) By (2.4), the problem (1.1) on D reduces to u ss + ψ s ψ u s + 1 and the eigenvalue problem for the corresponding linearized equation is In order to state our main result, we define where F(u) = u 0 f (t)dt. We use the notation v (s) instead of v s (s) when it is convenient.
Proof. As noted above, we can consider the positive solutions of (3.2). By Proposition 2.1, the problem (3.2) reduces to It should be remarked that the boundary condition u s (0) = 0 follows from a simple computation taking into consideration the hypothesis on ψ and χ. Now note that u (s) ≤ 0, for all s ∈ [0, 1).
Indeed, assuming otherwise, u(s) would have a point of local minimum in (0, 1), at which the left hand side of (3.5) is positive (see (h 3 )), a contradiction. Thus, we can conclude that u(0) is the maximum value of solution, i.e., u L ∞ (D) = u(0).

Now, from the equations (1.1) and (2.5) we have
On the other hand, in view of (h 5 ) and (3.7), which is a contradiction.

Remark 3.2.
(i) It is not difficult to get D with ψ satisfying (h 1 ) and (h 2 ). A simple example is ψ(s) = (2/π) sin(πs/2), χ(s) = (2/π) cos(πs/2) with s ∈ (0, 1). In this case D is the north hemisphere of a sphere of radius 2/π. Obviously, surfaces that have the south pole and not the north pole can also be obtained. However, a careful analysis of the proof above shows that symmetry conditions on ψ are required if S has no poles. Such conditions reduce S to a cylindrical surface and this is the subject of the next section.
(ii) For the hypothesis (h 3 ) and (h 4 ), f (u) = e u 1+ u , > 0, is an important example since it is related to perturbed Gelfand problem.

Stability and instability on cylindrical surfaces
In this section we consider the problem (1.1) with S = C, where C ⊂ M is a cylindrical surface parametrized by Here ψ(s) = a in [0, 1] and a > 0 is a constant. By (2.4), the problem (4.1) reduces to (i) If h(α) < h(u), for all u < α, (4.4) then the positive solution of (4.1), with u L ∞ (C) = α, is unstable.
To prove (ii) assume (4.10) implies that p(s) is increasing in (0, 1/2) and decreasing in (1/2, 1). Similarly to item (i) we have a contradiction, The theorem is proved. It is easy to see that the symmetry of C makes it possible.

Applications
In this section consider S a surface of revolution which can be either D or C. Let u be a positive solution of (1.1) with a positive parameter λ introduced. Moreover, suppose that u is independent of θ, i.e., u = u(s) is solution of (ψu ) + ψλ f (u) = 0, s ∈ (0, 1) If S = D we have that u(0) is the maximum value of u and, as mentioned in the Introduction, α = u(0) uniquely identifies the solution pair (λ, u). Hence the solution set of (5.1) can be depicted by planar curves in (λ, α) plane. The same is true if S = C, with α = u(1/2) the maximum value of u.
The behavior of the solution curves has been extensively studied for many different types of equations, for instance, see [11,12,14,19] or the recent work [15] and references therein. In [13] a simple and general method was presented which we apply here to surfaces of revolution.
For the next result, we write u = u(s, α) and λ = λ(α). Recall that α = u(0) (or α = u(1/2)) uniquely identify the pair (λ, u). The result below appears in [13] (Proposition 1) and the same proof, which is based on Sturm comparison theorem, can be used to our case (the presence of the function ψ adds no significant additional difficulty).
Proposition 5.1 allows to apply the Theorems 3.1 and 4.1 in order to obtain the solution curve behavior in the (λ, α) plane and, consequently, multiplicity results. In particular, some problems presented in [11][12][13][14]19] can be extended to surfaces of revolution considered here. For example, the Theorem 3.1 can be used to extend Theorem 5.3 of [13] where the perturbed Gelfand problem was considered on a 2-dimensional unit ball. In short, there is a interval (λ 1 , λ 2 ) so that for any λ ∈ (λ 1 , λ 2 ) the problem ∆ g u(x) + λe u 1+ u = 0 for x ∈ D (with > 0 small, see Remark 3.2) and u(x) = 0 when x ∈ ∂D, has at least three positive solutions that are independent of θ. In fact, in a future paper we consider only this case in order to prove that the solution curve is S-shaped.
For the reader's convenience, we detail another simple case. Take ψ(s) = 1, χ(s) = s for s ∈ [0, 1] and f (u) = au − u sin(u) with a > 1. Then S = C (i.e. S is a cylindrical surface) and we consider the problem where λ > 0 is a parameter (the same problem on a interval was considered in [19] when a = 2). The solutions that are independent of θ satisfies u + λu(a − sin(u)) = 0, s ∈ (0, 1) First, we note that the positive solutions of (5.3) lie in a bounded in λ strip. We follow the steps of [11,Lemma 3] to conclude that if (5.3) has a positive solution, then where λ 1 is the principal eigenvalue of −u on the interval (0, 1), with Dirichlet boundary conditions. We have that λ 1 = π 2 and φ 1 = sin(πs) is the corresponding eigenfunction.
There is a curve of positive independent of θ solutions of (5.2) (i.e. positive solutions of (5.3)) in the (λ, u(1/2)) plane, which bifurcates from the trivial one at λ 1 /2 ( [4]). Finally, by Proposition 5.1 and (5.4), we can conclude that this curve has infinitely many turns. This occurs because there are infinitely many changes of stability to u(1/2) increasing.