Clustered boundary layer sign-changing solutions for a supercritical problem (Q2843989)

Let \(\Omega\subset{\mathbb R}^2\) be a smooth bounded domain and assume that \(a:\overline\Omega\rightarrow {\mathbb R}\) is a smooth function such that \(0<a_1\leq a(x)\leq a_2<+\infty\) for all \(x\in\Omega\). Let \(\Delta_a\) denote the differential operator \(\Delta_au=\Delta u+\nabla\log a\cdot\nabla u\) for all \(u\in H^1_0(\Omega)\). This paper deals with the anisotropic Lane-Emden-Fowler equation \(\Delta_au+|u|^{p-1}u=0\) in \(\Omega\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The main purpose of this paper is to construct solutions of this nonlinear boundary value problem with positive and negative bubbles which accumulate to a certain on \(\partial\Omega\) as \(p\) goes to \(+\infty\). The proofs combine refined elliptic estimates and asymptotic analysis tools.