Four positive solutions of semilinear elliptic equations in exterior domains (Q884497)

The authors consider the semilinear elliptic equation \[ \begin{gathered} -\Delta_x u+ u= |u|^{p-2} u+ h(z)\quad\text{in }\Omega,\\ u\in H^1_0(\Omega),\end{gathered}\tag{1} \] where \(\Omega= \mathbb{R}^N\setminus\overline{\mathbb{D}}\), \(\mathbb{D}\) is a \(C^{1,1}\) bounded domain in \(\mathbb{R}^N\), \(2< p< 2^*\), \(2^*= {2N\over N-2}\) for \(N\geq 3\). Assuming that \(h\) is nonnegative and \(\| h\|_{L^2}> 0\), they prove that if \(\| h\|_{L^2}\) is sufficiently small, then there exist at least four positive solutions of (1).