Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition
This paper is devoted to the study of positive solutions of the following semilinear elliptic problem with a nonlinear boundary condition \[ \begin{cases} -\Delta u=\lambda\left(m(x)u-u^2\right) &\text{ in }\Omega,\\ \frac{\partial u}{\partial \mathbf{n}}=\lambda b(x)v^2 &\text{ on }\partial\Omega, \end{cases} \tag{1} \] where \(\Omega\subset \mathbb R^N\), \(N=2, 3\), is a bounded domain with a smooth boundary \(\partial\Omega\). \(m\in C^{\theta}(\Omega)\) and \(b\in C^{\theta+1}(\Omega)\) are Hölder continuous functions with exponent \(0 <\theta<1\), and \(m\) is positive somewhere in \(\Omega\), that is, it may change sign, while \(b>0\) on \(\partial\Omega\). The author separately considers the cases that \(\int_{\Omega}mdx<0\), \(\int_{\partial\Omega}b\phi_1^3ds\geq\int_{\Omega}\phi_1^3dx\) and \(\int_{\Omega}mdx\geq0\). In the case of \(\int_{\Omega}mdx<0\) and \(\int_{\partial\Omega}b\phi_1^3ds\geq\int_{\Omega}\phi_1^3dx\), the positive solution that bifurcates subcritically at \((\lambda_1, 0)\) is unique and unstable for any \(\lambda\) less than but close to \(\lambda_1\). Additionally, if \(\int_{\partial\Omega}bds\leq |\Omega|\), the positive solution \((\lambda, u_\lambda)\) satisfies \(\lim_{\lambda\downarrow0}\min_{x\in \overline{\Omega}}u_\lambda(x)\to\infty\) and if \(\int_{\partial\Omega}bds> |\Omega|\), \(u_\lambda\to C(m,b)\) in \(C^{2+\theta}(\overline{\Omega})\) as \(\lambda\downarrow0\), \(C(m,b)\in \mathbb R\) dependents on functions \(m,b\). Then in the case of \(\int_{\Omega}mdx\geq0\), if \(\int_{\partial\Omega}bds< |\Omega|\), the first solution bifurcating from \(\{(0,c):c\geq 0 \text{ is a constant}\}\) is unstable and it tends to \(\infty\) in \(C(\overline{\Omega})\) as \(\lambda\downarrow0\), the second one is unstable for \(\lambda\) sufficiently small and its minimum value in \(\overline{\Omega}\) tends to infinity as \(\lambda\downarrow0\) and if \(\int_{\partial\Omega}bds\geq |\Omega|\), problem (1) has no positive solution for \(0<\lambda<\overline{\lambda}\). The proofs depend on local bifurcation theory, Lyapunov-Schmidt reduction and the result of a priori upper bounds for positive solutions.
scientific article; zbMATH DE number 5988422
scientific article