On the existence of positive solutions for a class of semilinear elliptic equations
The authors are concerned with the existence and uniqueness of large, i.e., unbounded, solutions to semi-linear elliptic equations. The first step is to establish conditions for the existence of positive solutions to the one-dimensional nonlinear boundary value problem \[ (A(x)y^{\prime }(x))^{\prime }=A(x)p(x)f(y(x)) \quad\text{for }0\leq x<\infty, \] where \(f\) is Lipschitz continuous. Then they apply their result to partial differential equations in the special case when the solution is radial, to prove the existence of a large solution, since in this case the semi-linear partial differential equation \(\Delta u=q(x)F(u)\) reduces to a nonlinear ordinary differential equation. Otherwise, when \(q\) is not radial but positive, an average function of \(q\) is used to prove the existence of a bounded positive solution to \(\Delta u=q(x)F(u)\) over \(\mathbb{R}^{n}\).
nonlinear ordinary differential equation
scientific article; zbMATH DE number 1880006