×

zbMATH — the first resource for mathematics

Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli. (English) Zbl 0839.35039
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.

MSC:
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX Cite
Full Text: DOI