Positive solutions of diffusive logistic equations (Q5935721)

The existence and uniqueness of positive solutions in the space \(C^{2+\theta}(\overline{D}),\) \(0<\theta <1,\) is studied for a class of degenerate elliptic boundary value problems: NEWLINE\[NEWLINE \begin{cases} -\Delta u=\lambda (m(x)-h(x)u)u\;\;& \text{in }D, \cr Bu:=a(x')\frac{\partial u}{\partial {\mathbf n}}+b(x')u=0\;\;& \text{on }\partial D,\end{cases} NEWLINE\]NEWLINE where \(D\) is a bounded domain of Euclidean space \({\mathbb R}^{N},\) \(N\geq 2,\) with smooth boundary \(\partial D.\) The boundary condition \(B\) is degenerate in the following sense: NEWLINE\[NEWLINE a(x')+b(x')>0 \text{ on }\partial D,\text{ and } b(x')\neq 0\text{ on }\partial D. NEWLINE\]NEWLINE The main result of the paper describes the changes that occur in the structure of the positive solutions when the parameter \(\lambda \) varies near the first eigenvalue of the corresponding linearized eigenvalue problem: NEWLINE\[NEWLINE \begin{cases} -\Delta \varphi =\lambda m(x)\varphi & \text{in }D, \cr B\varphi =0 & \text{ on }\partial D.\end{cases} NEWLINE\]NEWLINE It generalizes a result of \textit{J. M. Fraile, P. Koch Medina, J. López-Gómez} and \textit{S. Merino} [J. Differ. Equations 127, No.~1, 295-319, Art. No.0071 (1996; Zbl 0860.35085)].