Existence of positive radial solutions for semilinear elliptic equations in the annulus
This paper studies the existence of positive radial solutions of the Dirichlet problem for semilinear elliptic equations in an annular region in R n, i.e., in \(\Omega =\{x\in R\) n: \(0<\lambda <| x| <R\}\). The equation has the form \(\Delta u(x)+f(u(x))=0,\) \(x\in \Omega\), and the boundary condition is \(u(x)=0\), \(x\in \partial D\). The author shows that for such regions, certain growth conditions at infinity on f that are needed for the existence of positive radial solutions of \(\Omega\) is a ball are not needed here. Since these radial solutions satisfy the second order ordinary differential equation \[ u''(r)+((n+1)/r)u'(r)+f(u(r))=0 \] where \(r=| x|\), the paper contains a very detailed study of positive solutions u(r), \(x<r<R\), satisfying the boundary conditions \(u(\lambda)=u(R)=0\). A basic condition on f is that there exists an \(A\geq 0\) such that \(f(u)>0\) for \(u>A\) and F(u)\(\leq 0\) for \(0\leq u<A\), where \(F(u)=\int^{x}_{0}f(s)ds\) and that there exist constants \(b>0\), \(d_ 1>0\), \(d_ 2>0\), and \(k>-1\) such that \(d_ 1u\) \(k\leq f(u)\leq \delta_ 2u\) k for \(u\geq b\). The case \(R=+\infty\), the ``exterior'' problem, is also dealt with.
scientific article; zbMATH DE number 4061765