Logistic Dirichlet problems with discontinuous coefficients.
The author studies existence of positive solutions to semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. Precisely, let \(\Omega\subset {\mathbb R}^N,\) \(N\geq3,\) be a bounded and \(C^{1,1}\)-smooth domain and \[ {\mathcal L}:= -\sum_{i,j=1}^N a^{ij}(x){{\partial^2}\over{\partial x_i\partial x_j}} +\sum_{i=1}^N b^{i}(x){{\partial}\over{\partial x_i}} +c(x) \] a uniformly elliptic operator with \(a^{ij}\in VMO\cap L^\infty,\) \(b^i,\;c\in L^\infty,\) \(c(x)\geq0\) for almost all \(x\in\Omega.\) The paper deals with the problem \[ \begin{cases} {\mathcal L}u=\lambda\big(m(x)u-h(x)u^2\big) &\text{in}\;\Omega,\\ u=0 &\text{on}\;\partial\Omega, \end{cases} \] where \(\lambda>0\) is a parameter, \(m\in C(\overline\Omega)\) may change its sign in \(\Omega\) whereas \(h\in C(\overline\Omega)\) and \(h(x)>0\) in \(\overline\Omega.\) A complete study is carried out of the changes which occur in the global structure of positive solutions to the above Dirichlet problem as the parameter \(\lambda\) varies from the principal eigenvalue \(\lambda_1(m)\) of the linearized problem \[ \begin{cases} {\mathcal L}u=\lambda m(x)u &\text{in}\;\Omega,\\ u=0 &\text{on}\;\partial\Omega. \end{cases} \]
scientific article; zbMATH DE number 2012734
scientific article