Clustered layers for the Schrödinger-Maxwell system on a ball
The existence of multiple layered solutions for the singular perturbed Schrödinger-Maxwell system with Dirichlet boundary conditions \[ \begin{cases} -\varepsilon^2\Delta v+v+\omega \phi v-\varepsilon ^{\frac{p-1}{2}}v^p=0,\,\,\,\text{in}\,\,\,B_1,\\ -\Delta \phi=4\pi \omega v^2,\,\,\,\text{in}\,\,\,B_1,\\ v,\,\phi>0\,\,\,\text{in}\,\,\,B_1\,\,\,\,\,\text{and}\,\,\,\,\,v=\phi=0,\,\,\,\text{on}\,\,\,\partial B_1,\end{cases}\eqno(1) \] is investigated in this paper. Here \(B_1\) is the unit ball centered at \(0\) in \(\mathbb{R}^3\), \(\omega>0\) and \(p\in (7/3,5]\) are constants, and \(\varepsilon>0\) is a small parameter. Using the localized energy method, which is a combination of the Lyapunov-Schmidt reduction procedure and the variational principle, for a sufficiently large integer \(N\), the authors construct a positive radially symmetric solution \((v_{\varepsilon},\phi_{\varepsilon})\) of the problem (1) such that \(v_{\varepsilon}\) concentrates on \(N\) spheres \(\{| x| =s_i^{\varepsilon},\,\,\,i=1,\ldots, N\}\) and these spheres converge to a shere \(\{| x| =r_N\}\) as \(\varepsilon\) tends to zero.
scientific article; zbMATH DE number 5141179
scientific article