Symmetry breaking in a bounded symmetry domain
The authors deal with the semilinear equation \[ -\Delta u+u= |u|^{p-2}u\text{ in }\Omega\quad u\in H_0^1(\Omega),\tag{1} \] where \(\Omega\) is a domain in \(\mathbb{R}^N\), \(N\geq 2\), \(2<p<2^*\) with \(2^*= \infty\) if \(N=2\), \(2^*=\frac{2N}{N-2}\) for \(N\geq 3\). The authors prove that the equation (1) possesses three positive solutions in \[ \Omega_t=A^r_{-t,t}\setminus\overline{B^N(x,0);r_1)} \] for \(t\geq t_0\), where \(t_0>0\) is a some positive number and \[ B^N(z_0,s):=\bigl \{z\in \mathbb{R}^N\mid|z-z_0|<s\bigr\}, \] \[ A^r_{s,t}:=\bigl\{(x,y)\in A^r \mid s<y<t \bigr\},\;A^r=\bigl\{(x,y)\in\mathbb{R}^N\mid |x|<r\bigr\},\;r_1< r_0. \]
scientific article; zbMATH DE number 2113835