Positive solutions with prescribed patterns in some simple semilinear equations
The authors show that given a bounded domain \(\Omega\) in \(\mathbb{R}^N\) \((N\geq 2)\) with smooth boundary \(\partial \Omega\) and an arbitrary set of finitely many disjoint closed subdomains \(D_1,\dots, D_m\) of \(\Omega\) one can find a simple continuous function \(f(x,u,\varepsilon) = f_\varepsilon(x,u)\) (sublinear in nature) such that the boundary value problem \[ -\Delta u=f_\varepsilon(x,u),\quad x\in\Omega,\;u=0,\;x\in\partial\Omega, \] has a unique positive solution \(u_\varepsilon\) and for \(\varepsilon\) small, \(u_\varepsilon\) has peaks concentrating exactly on \(D_1\cup\cdots \cup D_m\). Moreover, any positive solution (regardless of the initial value) of the parabolic problem \[ u_t-\Delta u= f_\varepsilon(x,u),\quad x\in\Omega,\;u = 0,\;x\in\partial\Omega,\;t > 0 \] satisfies \(\lim_{t\to\infty}u(x,t)= u_\varepsilon(x)\), uniformly for \(x\in\overline{\Omega}\).
scientific article; zbMATH DE number 1839917