Interior singularity problem of some nonlinear elliptic equations
Let \(L\) be a uniformly elliptic operator on \(R^d\). For a bounded \(C^2\)-domain \(D\) containing \(0\), consider the interior singularity problem \(Lu=u^\alpha\) in \(D\setminus \{0\}\) with boundary condition \(u|_{\partial D}=\varphi\) for \(1<\alpha\leq 2\). The main purposes of this paper are to give the limiting bahavior of \(u\) near \(0\) and probabilistic representations of all nonnegative solutions. Let \(\xi=(\xi_t,\Pi_x)\) be a diffusion process with generator \(L\) and \(X=(X_\tau,Y_\tau,P_\mu)\) the \((L,\alpha)\)-superdiffusion. A nonnegative unbounded solution \(u\) is called a weak [resp. strong] singularity at \(0\) if \(\lambda=\lim\sup_{x\to 0} u(x)/G_D(x,0)<\infty\) [resp.~\(=\infty\)], where \(G_D(x,y)\) is the Green function of \(\xi\). The main result of this paper says that, for \(2\leq d\leq 2+2/(\alpha-1)\), \(u_\varphi(x)=-\log P_{\delta_x} \exp\langle -\varphi,X_{\tau_D}\rangle\) is the unique bounded solution. Furthermore \(u\) is a weak singularity at \(0\) iff \(u(x)=-\log P_{\delta_x}\exp(-\langle\varphi,X_{\tau_D}\rangle -\lambda y_D(0))\), where \(y_D\) is a density of \(Y_{\tau_D}\). Representations of minimal and maximal strong singularity at \(0\) are also given.
scientific article; zbMATH DE number 1906965