Existence and qualitative theorems for nonnegative solutions of a semilinear elliptic equation (Q810278)

The authors study existence and qualitative properties for positive solutions of the Dirichlet problem: \[ (1)\quad \Delta u+f(u)=0\text{ in } B_{{\mathbb{R}}},\quad u=0\text{ on } \partial B_{{\mathbb{R}}}, \] where \(B_{{\mathbb{R}}}\) is the ball centered at the origin and radius R. The function f is assumed to be continuous on \([0,\infty)\) and \(satisfying\) A\({}_ 1)\lim_{s\to 0} \sup f(s)/s=-m<0,\) A\({}_ 2)\) there exists a unique \(\zeta_ 0\in (0,\infty)\) such that \(F(\zeta_ 0)=0\), \(F(\zeta)<0\) for \(\zeta \in (0,\zeta_ 0)\) and \(f(\zeta_ 0)>0\), where \(F(\zeta)=\int^{\zeta}_{0}f(s)ds,\) A\({}_ 3)\) if \(\alpha =\sup \{\zeta <\zeta_ 0:\) \(f(\zeta)=0\}\) and \(\beta =\inf \{\zeta >\zeta_ 0:\) \(f(\zeta)=0\}\) let \(0<\alpha <\beta <\infty,\) A\({}_ 4)\) f is Lipschitz continuous in a neighborhood of \(\beta\). By using a variational approach the authors obtain the following results: I) There exists an \(R_ 0>0\) such that, for any \(R>R_ 0\) problem (1) admits a positive radially symmetric solution u satisfying \(\zeta_ 0<u(0)<\beta\), \(u_ r(r)<0\) on (0,R). II) Let \(R=\infty\) and define \(u(\infty)=\lim_{r\to \infty}u(r)\). For some \(\mu \in (\zeta_ 0,\beta)\) there exists a nonnegative radially symmetric solution u of (1) satisfying \(u(0)=\mu\). If \(R_ 1=\inf \{r>0:\) \(u(r)=0\}<\infty\) then \(u_ r(r)<0\) on \((0,R_ 1)\) and \(u=0\) on \((R_ 1,\infty)\).