Radial positive solutions for a nonpositone problem in an annulus (Q2877602)

Let \(\Omega=\{x\in \mathbb{R}^n: R<|x|<\hat{R}\}\) be an annulus in \(\mathbb{R}^N\) (\(N>2\)), let \(f:[0,,+\infty)\rightarrow \mathbb{R}\) be a continuously differentiable function, with \(f(0)<0\) and \(\lim_{t\rightarrow +\infty}f(t)/t=+\infty\), and let \(\lambda\) be a positive parameter. The authors study the existence of positive radial solutions to the nonpositone superlinear elliptic problem NEWLINE\[NEWLINE(P_\lambda)\quad \begin{aligned} -\Delta u=\lambda f(u) \,\, & \mathrm{ in}\,\, \Omega, \,\, \\ u=0 \,\,& \mathrm{on}\,\, \partial\Omega, \end{aligned}NEWLINE\]NEWLINE \noindent in the case in which the nonlinearity \(f\) has more then one zero.NEWLINENEWLINEMore precisely, under the following further conditions on the nonlinearity \(f\):NEWLINENEWLINE- there exist \(\beta_1,\beta_2,\beta_3\in (0,+\infty)\), with \(\beta_1<\beta_2<\beta_3\), such that \(f(\beta_i)=0\), \(f'(\beta_i)\neq 0\) for all \(i=1,2,3\), and \(f'(t)\geq 0\) for all \(t\in [\beta_3,+\infty)\);NEWLINENEWLINE- \(\inf_{t\in [0,+\infty)}\biggl(\int_0^tf(s)ds-\frac{N-2}{2N}f(t)t\biggr)>-\infty\),NEWLINENEWLINE\noindent the authors establish the existence of at least a positive radial solution for \(\lambda\) small enough.NEWLINENEWLINE\noindent To prove this result, the authors consider the auxiliary Cauchy problemNEWLINENEWLINE\(-u''-\frac{N-1}{r}u'=\lambda f(u)\) \ \ in \((R,+\infty)\), \ \ \(u(R)=0\), \(u'(R)=d\).NEWLINENEWLINEThey first show that this problem admits a unique positive solution \(u(\lambda,d,\cdot)\) for all \(\lambda,d>0\). Then, they prove that, for \(\lambda\) small enough, there exists \(\delta^*>0\) such that \(u(x):=u(\lambda,d^*,|x|)\) is a solution of problem \((P_\lambda)\).