Hilbert's projective metric and nonlinear elliptic problems
The paper deals with eigenvalue problems for a class of positive nonlinear operators. Precisely, let \(K\) be a closed (not necessarily solid) cone in a real Banach space \(X,\) and let \(d\) be the Hilbert projective metric in \(K^+=K\setminus\{0\}\). For given \(f\in K^+\) and \(r>0,\) let \(K_f=\{x\in K^+:\;d(x,f)<\infty\}\) and \(E_{f,r}=\{x\in K_f:\;\|x\|=r\}\). The authors consider the eigenvalue problem \[ Tx=\lambda x,\quad x\in E_{f,r},\;\lambda>0, \] and the associated fixed point equation \(Tx=x\) for compact cone mappings \(T:\;K\to K\) which are of the form \(T=T_1+T_p,\) where \(T_j\) is increasing \(j\)-concave for \(j=1,p\) and \(0<p<1.\) Assuming \(T_1f\in K_f\) and \(T_pf\in K_f\) for some \(f\in K^+,\) and \(T(K^+)\subseteq K_f,\) it is proved that for each \(r>0\) there exists a unique pair \((x_r,\lambda_r)\in E_{f,r}\times(0,\infty)\) such that \(Tx_r=\lambda_rx_r.\) Moreover, the authors prove that \(T\) has a unique fixed point in \(K^+\) under the additional assumption that \(tf-T_1f\in K\) for some \(t<1.\) The results are applied to the study of nonlinear elliptic problems of the type \[ \begin{cases} \Delta u+a(|x|)\varphi(u)=0\quad \text{in}\;\Omega,\\ u=0\quad \text{on}\;\partial\Omega. \end{cases} \]
scientific article; zbMATH DE number 5586792
scientific article