Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli
The asymptotic behaviour of positive solutions to the semilinear elliptic Dirichlet problem \(\Delta u+ f(u)= 0\) in \(\Omega\), \(u= 0\) on \(\partial \Omega\), is studied, where \(\Omega= \Omega_a= \{x\in \mathbb{R}^N: a< |x|< a+ 1\}\) are expanding annuli in \(\mathbb{R}^N\) as \(a\to + \infty\), \(N\geq 2\), and \(f\) satisfies the following conditions: (H-0) \(f\in C^1(\mathbb{R}^1)\) and \(f(u)> 0\) for large \(u\), (H-1) \(f(0)= 0\) and \(f'(0)\leq 0\), (H-2) there exists \(\sigma> 0\) such that \(uf'(u)\geq (1+ \sigma) f(u)\) for all \(u\geq 0\), (H-3) for large \(u\), \[ f(u)\leq \begin{cases} Cu^p\quad & \text{for some } p< (N+ 2)/(N- 2)\text{ and }C> 0\text{ if }N\geq 3,\\ \exp A(u)\quad & \text{with } A(u)= o(u^2)\text{ as }u\to \infty\text{ if } N= 2.\end{cases} \] A variational formulation of the problem is given using Nehari-type functionals. Then a priori bounds are obtained for the solution. The limiting behaviour (as \(a\to \infty\)) is investigated in the radial case. The case of least-energy solutions with partial symmetry is discussed. An Appendix contains some basic properties of Bessel functions, necessary in conducting the analysis. This is a very technical paper.
scientific article; zbMATH DE number 795273