On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions
We use an implicit probabilistic representation and Schauder's fixed point theorem to obtain sufficient conditions for the existence of positive continuous solutions to the following semilinear elliptic equation with Dirichlet boundary conditions: \[ {\textstyle {1\over 2}} \Delta u+F(x,u) =-g(x) \qquad \text{for } x\in D, \] \[ u(x) =\varphi(x) \qquad \text{for } x\in \partial D, \qquad \lim_{{\textstyle {{| x|\to\infty}\atop {x\in D}}}} u(x)= \alpha \quad \text{when \(D\) is unbounded}, \] where \(D\) is a regular domain in \(\mathbb{R}^ d\) (\(d\geq 3\)), \(\varphi\) is a nonnegative continuous function on \(\partial D\), \(\alpha\) is a nonnegative constant such that when \(\partial D\) is unbounded, \(\lim_{| x|\to\infty,x\in\partial D} \varphi(x) =\alpha\), \(g\) is a nonnegative Green-tight function on \(D\), and \(F\) is a real-valued Borel measurable function defined on \(D\times (0,b)\) for some \(b\in(0,\infty]\) such that \(F(x,\cdot)\) is continuous on \((0,b)\) for each \(x\in D\) and \(-U(x)u\leq F(x,u)\leq V(x)f(u)\) for all \((x,u)\in D\times (0,b)\), where \(U\) and \(V\) are nonnegative Green-tight functions on \(D\) and \(f\) is a nonnegative Borel measurable function defined on \((0,b)\). Special cases of equations of this type arise in connection with superprocesses and problems in geometry concerning conformal deformation of metrics.
existence of positive continuous solutions
scientific article; zbMATH DE number 562495
scientific article