Symmetry, degeneracy, and universality in semilinear elliptic equations. Infinitesimal symmetry-breaking.

*(English)*Zbl 0702.35016We consider the bifurcation of radial solutions of semilinear elliptic equations on n-balls,
\[
(*)\quad \Delta u(x)+f(u(x))=0,\quad x\in D^ n;\quad \alpha u(x)-\beta du(x)/dn=0,\quad x\in \partial D^ n,
\]
to asymmetric solutions. We show that for some fairly broad classes of nonlinear functions f, “infinitesimal” symmetry-breaking occurs, in the sense that there exist infinitely many degenerate radial solutions of (*), the kernel of whose linearized operators contain asymmetric elements. In fact if we write an element w in the kernel in its spherical harmonic decomposition, \(w=\sum_{N\geq 0}a_ N(r)\phi_ N(\theta)\), where \(\phi_ N\) lies in the N-th eigenspace of the Laplacian on \(S^{n-1}\), then each summand is in the kernel, and these kernels contain asymmetric elements \(a_ N\phi_ N\), \(a_ N\not\equiv 0\), of arbitrarily high modes; i.e., N sufficiently large. This means that on each such degenerate solution, there exists the possibility of symmetry breaking in the sense of bifurcation of a radial solution into an asymmetric one. [Indeed, in Invent. Math. 100, 63-95 (1990), we prove that actual symmetry breaking must occur.] For these functions f, we show that there is an integer \(N_ 0\) with the property that if \(N\geq N_ 0\), there are k-distinct radial solutions for which \(a_ N\not\equiv 0\), (here k represents the nodal-class of radial solutions). For a certain subclass of f’s, we show that \(N_ 0\) is actually independent of f.

Reviewer: J.Smoller

##### MSC:

35B32 | Bifurcations in context of PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

##### Keywords:

infinitesimal symmetry-breaking; radial solutions; semilinear elliptic equations; asymmetric solutions; kernel; linearized operators; eigenspace; Laplacian
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\textit{J. Smoller} and \textit{A. Wasserman}, J. Funct. Anal. 89, No. 2, 364--409 (1990; Zbl 0702.35016)

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