I am working on $p$-Laplace equation. that is $$-\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$$ I do not know when we say a PDE has singular solution and when, it has a degenerate solution?
why it has singular solution for $p<2$ and degenerate for $p>2$. Actually my problem is that I do not know what is the difference between degenerate and singular solution.
please help me.
I believe these words describe the structure of the equations at the critical points of $u$, where $|\nabla u|=0$. Recall that $p$-harmonic functions are $C^{1,\alpha}$ (this is a difficult result, see the beginninf of Chapter 4 in Notes on the p-Laplace equation by Peter Lindqvist). Let us think of $|\nabla u|^{p-2}$ as a coefficient $\sigma$ in linear PDE $\operatorname{div}(\sigma \nabla u)=0$. At the points where $\nabla u$ does not vanish, $\sigma$ is Hölder continuous, and the Schauder estimates (Chapter 6 of Gilbarg-Trudinger) lead to the following "bootstrap" proof of higher regularity:
$$\sigma\in C^{\alpha}\implies u\in C^{2,\alpha}\implies \sigma\in C^{1,\alpha} \implies u\in C^{3,\alpha}\implies \dots$$
and we conclude that $u\in C^\infty$. The above breaks down at the critical points of $u$, but the nature of the problem is different depending on $p$:
The aforementioned Chapter 4 of Lindqvist's notes is a good illustration of how the cases $p<2$ and $p>2$ must be treated differently.
Which case is worse? I guess it depends on who you ask. One can even dualize the problem to convert between the two cases, but this requires the consideration of differential forms instead of scalar functions. Here is a relevant quote from D'Onofrio-Iwaniec:
Hodge duality is an effective device, for it reduces the $p$-harmonic type systems with the inconvenient case $p<2$ to the $q$-harmonic type systems with the preferable case $q>2$. However, caution must be exercised because in dimension higher than $2$ the Hodge dual to a scalar $p$-harmonic type equation is a $q$-harmonic type system.