Nonradial solutions of semilinear elliptic equations on annuli (Q1335547)

The authors consider the semilinear elliptic Dirichlet problem \((*)\;- \Delta u= \lambda u+ g(u)\), \(u>0\) in \(\Omega_ a\), \(u|\partial \Omega_ a=0\). Here \(\Omega_ a= \{x\in \mathbb{R}^ n\): \(a<| x|< a+1\}\) is an annulus, \(\lambda\in \mathbb{R}\), \(g\in C^ 1( \mathbb{R}, \mathbb{R})\) satisfies \(g(0)= g'(0) =0\), \(0\leq g'(t)\leq C(1+ t^{p-1})\) for \(t>0\) and some \(p>2\), \(\mu g(t)\leq tg'(t)\) for \(t>0\) and some \(\mu>1\). The authors show the following results: i) If \(n=2\), \(\lambda<1\) the number of nonradial (nonequivalent) solutions to \((*)\) increases to \(\infty\), as \(a\to \infty\). ii) If \(n\geq 3\), \(\lambda\leq 0\) and additionally \(p< (n+2)/ (n-2)\), then for \(a\) sufficiently large, the Dirichlet problem \((*)\) has a nonradial solution. As the authors explain, their results are not new, but their proofs are relatively simple, cf. e.g. \textit{S. S. Lin} [Trans. Am. Math. Soc. 332, No. 2, 775-791 (1992; Zbl 0764.35009)] and \textit{T. Suzuki} [Generation of positive nonradial solutions for semilinear elliptic equations: a variational approach, Preprint].