Boundary blow-up solutions for a class of elliptic equations on a bounded domain
This paper deals with existence of solutions of the quasilinear elliptic equation \[ \Delta u=f(x,u,\nabla u) \] in \(\Omega\) satisfying the boundary blow-up condition \(u(x)\to\infty\) as \({x\to\partial\Omega}\), where \(\Omega\subset\mathbb R^ N\) is a bounded domain with smooth boundary \(\partial\Omega\). Our main result applies to the existence of blow-up solutions of a class of equations which appear in Stochastic Control Theory, namely \[ \Delta u=a(x)g(u)+\lambda| \nabla u| ^ \sigma+\Psi(x) \] in \(\Omega\), where \(a(x)\), \(g(u)\), \(\Psi(x)\) are suitable functions and \(\lambda\), \(\sigma\) are nonnegative constants. Our approach employs an approximation procedure combined with both a non-monotone iteration variant of the method of lower and upper solutions and fixed point arguments.
scientific article; zbMATH DE number 5083659
scientific article