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On the stability of radial solutions of semilinear elliptic equations in all of \(\mathbb R^n\). (English. Abridged French version) Zbl 1081.35029

Summary: We establish that every nonconstant bounded radial solution \(u\) of \(-\Delta u=f(u)\) in all of \(\mathbb R^n\) is unstable if \(n \leqslant 10\). The result applies to every \(C^1\) nonlinearity \(f\) satisfying a generic nondegeneracy condition. In particular, it applies to every analytic and every power-like nonlinearity. We also give an example of a nonconstant bounded radial solution \(u\) which is stable for every \(n \geqslant 11\), and where \(f\) is a polynomial.

MSC:

35J60 Nonlinear elliptic equations
35B35 Stability in context of PDEs
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