I'm dealing with an elliptic PDE, it depends on the dimension of the domain, for some situations, I highly doubt that there is no $C^2$ solution, so I searched for some papers related to this equation and found that there are papers proving no finite Morse index solution, I want to ask that:
1.What is the difference between no classical solution and no finite Morse index solution?
2.If there are infinite Morse index solutions, what this infinite Morse index solutions look like?
In fact, I'm reading this 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 $\mathbb{R}^N$.
When $N=2$, there is a celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, after reading this I began to be interested in the higher dimension case so I found the paper I mentioned I'm reading.