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Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. (English) Zbl 0952.35054

J. Differ. Equations 163, No. 2, 429-474 (2000); addendum 172, No. 2, 445-447 (2001).
The author considers the problem \[ \begin{cases} \Delta u+ hu +f(u)=0 \quad &\text{in } \;\Omega, \\ u=0\quad &\text{on} \partial \;\Omega, \\ u>0\quad &\text{in} \;\Omega, \end{cases} \] where \(\Omega=\{x\in {\mathbb R}^N \colon |R-1|<|x|<R+1 \}\) and \(f\) and \(h\) satisfy suitable assumptions. It is shown that when \(R\to \infty,\) a critical orbital set produces a solution of this problem whose energy is concentrated around a scaled critical orbital set.
In the addendum the proof of Lemma 4.5 is corrected.

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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