Existence and asymptotic behaviour of large solutions of semilinear elliptic equations (Q1611955)

The authors investigate nonlinear differential equations of the form \[ Lu=u\cdot \varphi(\cdot,u) \quad\text{and}\quad \Delta u=p(x) f(u), \] where \(L\) is the generator of a submarkovian resolvent on a domain \(\Omega\) in \(\mathbb{R}^n\) and \(\varphi,p,f\) are given functions satisfying various conditions. Existence and uniqueness results are proved by analyzing the equation \[ u+V\bigl(u \varphi(\cdot,u) \bigr)=s_1- s_2, \] where \(V=(-L)^{-1}\) is the potential kernel and \(s_1,s_2\) are surmedian functions. Moreover, the existence of positive solutions of \[ \Delta u-u\varphi (\cdot,u)= 0\text{ on } \Omega,\quad u=g\text{ on }\partial\Omega \] is shown for nonnegative continuous \(g\) and suitable \(\varphi\). In further sections, large solutions of \(\Delta u=p(x)f(u)\) are studied and several examples are given.