Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary
Let \(D\subset \mathbb{R}^n\) be a bounded domain whose boundary consists of two parts \(\Gamma_0\) and \(\Gamma\), where \(\Gamma\) is relatively open with respect to \(\partial D\) and satisfies an interior and exterior sphere condition. The authors consider solutions \(u\) of the quasilinear equation \[ Lu(x)\equiv \sum^n_{i, j= 1} a_{ij}(x) u_{x_i x_j}(x)+ \sum^n_{i= 1} b_i(x) u_{x_i}(x)= g(x, u(x)), \] where \(L\) is a uniformly elliptic operator with \(a_{ij}\), \(b_i\in C^\alpha(\overline D)\), \(a_{ij}= a_{ji}\), and \(u\) satisfies \(u(x)\to \infty\) if \(x\to \Gamma\). Under some structure conditions on \(g\) the asymptotic behaviour of the solution \(u\) near \(\Gamma\) can be expressed in terms of \(\delta_\Gamma(x)\). Here \(\delta_\Gamma(x)\) is the distance from \(x\) to \(\Gamma\) in the metric \(ds^2= \sum^n_{i, j= 1} b_{ij}(x) dx_i dx_j\), where \((b_{ij}(x))\) is the inverse matrix of \((a_{ij}(x))\). For \(L= \Delta\) this result was established in a series of former papers. In the general case, the authors construct a lower and an upper comparison function for the solution \(u\) near \(\Gamma\) by solving appropriate Poisson equations. The asymptotic behaviour of \(u\) then follows from the result for \(L= \Delta\) applied to the comparison functions. In the last section, the previous result together with a blow up argument is used to study the asymptotic behaviour of the derivatives.
scientific article; zbMATH DE number 834233
scientific article