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I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered $$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$$ After reading this I began to be interested in the higher dimension case so I found the paper ON THE CLASSIFICATION OF SOLUTIONS OF $-\Delta u=e^u$ ON $\mathbb{R}^N$ : STABILITY OUTSIDE A COMPACT SET AND APPLICATIONS, it states that let $3 \leq N \leq 9$. Equation $$ -\Delta u=e^u \quad \text { on } \mathbb{R}^N, \quad N \geq 2, $$ does not admit any $C^2$ solution stable outside a compact set of $\mathbb{R}^N$, so no finite Morse index solution.

I want to ask that:

1.If there are infinite Morse index solutions, what are these infinite Morse index solutions? If we consider the solutions as the critical point of the energy functional, what type of critical point are these infinite Morse index solutions?

2.If I want to learn the method of using Morse index to study elliptic PDE by directly reading papers, can you recommend me some appropriate ones?

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    $\begingroup$ A comment on your first question: note that it might be different considering critical points of the energy and $C^2$-solutions as the Dancer-Farina paper does. Even for $3\le N \le 9$, infinite index solutions could be singular/unbounded. For example $u(x):=-2\log|x|+C_N$ is a weak solution (for some dimensional $C_N$) to $-\Delta u = e^u$ for every $N\ge 3$, but it has infinite index. Indeed, if it would have finite index then by a result of Figalli-Zhang it would be bounded and smooth, and this is not the case. $\endgroup$ Sep 7, 2023 at 10:40

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NEW ANSWER:

Yes there exist infinite Morse index solutions in all dimensions $N\geq 3$. For example you can take the solution in the Li Chen paper $$ \phi(x,y) = \frac{\ln(32)}{(4+|(x,y)|^2)^2} $$ and trivially cross with $\mathbb{R}$ to define $$ \phi(x,y,z) = \frac{\ln(32)}{(4+|(x,y)|^2)^2} $$ Clearly this continues to solve the PDE on $\mathbb{R}^3$. Moreover, by the second link (Dancer--Farina) it cannot have finite Morse index. This works for all $N\geq 3$.

SOME COMMENTS:

This example might feel cheap, but in many situations infinite Morse index comes from having a periodic or quasi periodic structure. For example, the helicoid has infinite Morse index as a critical point of the area functional. The reason for this is that the helicoid has a discrete translation symmetry.

Thus, we see that either the Helicoid is stable or it has infinite Morse index. Proof: if the Helicoid is unstable, this means there is a compact piece that's unstable (stability is always considering compactly supported variations). Now, we can take this unstable piece and translate it sufficiently many times upwards to be disjoint from itself. This gives two $L^2$-orthogonal destabilizations. We can repeat this to get arbitrarily many. Thus the Morse index must be infinity.

Of course it could a priori happen that a solution with symmetries is stable (the helicoid is not, but this is just due to the specifics of the problem). For example, the flat plane is a stable minimal surface and it of course has lots of symmetries.

This is not to say that all infinite Morse index solutions have a discrete symmetry. In fact, one would expect that in general, infinite Morse index solutions can be very disordered. However, a lot of examples that we can cook up do have some symmetries (maybe this is more due to our limited knowledge than any deep truth). For an example of infinite index with only quasi-symmetries see e.g. the genus 1 helicoid.


EDIT: As pointed out by Willie Wong, the original answer considers the wrong question.

There are finite Morse index solutions for $N\geq 10$. See remark 1(i) in the second paper you reference.

[...] for every N ≥ 10 the equation (1.1) possesses a radial stable solution. The existence of such a solution is a consequence of the analysis performed in [12], as was remarked in [6].

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  • $\begingroup$ I am not sure how this addresses the question asked. The OP is asking (a) if there exists "infinite morse index" solutions and (b) what the quoted phrase would mean. You answered something about the existence of finite morse index solutions. $\endgroup$ Aug 30, 2023 at 15:03
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    $\begingroup$ @WillieWong, good point. I updated my answer to consider the question asked by OP. $\endgroup$ Aug 30, 2023 at 17:10

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