Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

Abstract

We consider the boundary value problem \(\Delta u + \left| x \right|^{2\alpha } \left| u \right|^{p - 1} u = 0, - 1 < \alpha \ne 0\), in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are located alternately near the origin and the other m points \(\widetilde{q_l } = (\lambda \cos \frac{{2\pi (l - 1)}} {m},\lambda \sin \frac{{2\pi (l - 1)}} {m}),l = 2, \cdots ,m + 1 \), such that as p goes to +∞, $$p\left| x \right|^{2\alpha } \left| {u_p } \right|^{p - 1} u_p \rightharpoonup 8\pi e(1 + \alpha )\delta _0 + \sum\limits_{l = 2}^{m + 1} {8\pi e( - 1)^{l - 1} \delta _{\widetilde{q_l }} } $$ , where λ ∈ (0, 1), m is an odd number with (1+α)(m+2)−1 > 0, or m is an even number. The same techniques lead also to a more general result on general domains.