## 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
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