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On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form $$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha \delta_0\ .$$ Here $\delta_0$ is the Dirac distribution in $0$, $\alpha \in \mathbb{R}\setminus \{ 0 \}$, and $a\in C_b^\infty$, $0 < c \leq a\leq C < \infty$, so that the homogeneous Dirichlet problem is uniformly elliptic (and possibly monotone).

I'm interested in the asymptotic behavior of generalized solutions $u$ at $0$ in dimension $n>1$ (in any sense). For example, if a=1, it is easy to see that $u-\alpha E$ is smooth at 0, if $E$ is a fundamental solution of the Laplacian.

Questions:

  • Let $u \in W^{s,q}(U)$ be a generalized solution of the nonlinear problem. Does $u$ admit an asymptotic expansion near $0$?

  • More modestly, under which conditions is there a $c \in \mathbb{R}$ such that $u-c E$ is more regular than $u$?

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