Symmetry-Breaking at Positive Solutions of Elliptic Equations

Abstract

We consider a Dirichlet boundary value problem

$$ \begin{matrix}\Delta \text w(\text x)+ \text f(\text w(\text x),\lambda)= 0 \;\;, &\text x \in \Omega\;,\\ \text w(\text x)= 0 \;\;, & \text x \in \partial \Omega\;,\end{matrix} $$

(1)

where Ω = {xεℝⁿ | |×|<1 } is the unit ball in ℝⁿ (n≥2), and f: ℝ×ℝ→ℝ is smooth. If (w₀,λ₀) is a solution of (1) such that W₀(x) > 0 for all × ε Ω, then a theorem of GIDAS, NI and NIRENBERG (1979) implies that w₀ is spherically symmetric. In this paper we give some results on the structure of the solution set of (1) near (w₀,λ₀) in the case where w₀ has a neighborhood (in an appropriate space) containing nonpositive functions. In particular we give sufficient conditions for symmetry-breaking at (w₀,λ₀); by definition this means that each neighborhood of (w₀,λ₀) contains solutions of (1) which are not spherically symmetric (and hence also nonpositive). Our results complement and improve the work of SMOLLER and WASSERMAN (1984, 1986), who considered the case f(w,λ) = $\lambda \tilde{\text f}(\text w).$ Other related work was published by CERAMI (1986), POSPIECH (1986) and BUDD (1986). Full details and proofs are given elsewhere (VANDERBAUWHEDE, 1986).