Existence of positive radial solutions for semilinear elliptic equations in the annulus (Q1106404): Difference between revisions

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.