Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains. II
[For part I see ibid., Ser. A 64, No. 1, 1--4 (1988; Zbl 0664.35067).] The main object is to show the existence of nonradial solutions of the nonlinear eigenvalue problem (P): \[ -\Delta u=\lambda e^ u\text{ in } \Omega \text{ and } u=0\text{ on } \partial \Omega, \] where \(\Omega\) is an annulus in \({\mathbb R}^ 2.\) After introducing \( K_{\infty}(\mu)=\left\{v\in H^ 1_ 0(\Omega)\mid v\text{ is radial and }\int_{\Omega}e^ v \,dx=\mu\right\}\) and \[ K_ k(\mu)=\left\{v\in H^ 1_ 0(\Omega)\mid v\text{ is invariant under the rotation of }2\pi/k\text{ in } {\mathbb R}^ 2\text{ and } \int_{\Omega}e^ v \,dx=\mu \right\}, \] variational problems \[ j_ k(\mu)=\inf \left \{\int_{\Omega}| \nabla v|^ 2 \,dx\mid v\in K_ k(\mu)\right\}\quad (k=1,2,3,...,\infty) \] are considered. The minimizers \(u_ k\in K_ k(\mu)\) exist and are proved to be solutions of (P). From the classification of critical values \(j_{\infty}(\mu)\) and \(j_ k(\mu)\) \((k=1,2,3,...)\) follows the fact that \(u_ k\) are nonradial for \(\mu\) large enough.
scientific article; zbMATH DE number 4117045
scientific article