On the existence and smoothness of radially symmetric solutions of a BVP for a class of nonlinear, non-Lipschitz perturbations of the Laplace equation

J. Hegedûs, Bolyai Institute, Szeged, Hungary

E. J. Qualitative Theory of Diff. Equ., No. 18. (2002), pp. 1-28.

Abstract: The existence of radially symmetric solutions $u(x;a)$ to the Dirichlet problems
$$\Delta u(x)+f(|x|,u(x),|\nabla u(x)|)=0\qquad x\in B,\ u|_\Gamma=a\in{\R}\ (\Gamma:=\partial B)$$
is proved, where $B$ is the unit ball in ${\R}^n$ centered at the origin $(n\ge2)$, $a$ is arbitrary $(a>a_0\ge-\infty);f$ is positive, continuous and bounded. It is shown that these solutions belong to $C^2(\ov{B})$. Moreover, in the case $f\in C^1$ a sufficient condition (near necessary) for the smoothness property $u(x;a)\in C^3(\ov{B})\quad\forall a>a_0$ is also obtained.

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