A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

In this paper, we aim to investigate the following class of singularly perturbed elliptic problem {-epsilon(2 )delta u + |x|(eta)u = |x|(eta)f (u) in A,u=0 on & part;A, where epsilon > 0, eta is an element of R, A = {x is an element of R-2N : 0 < a < |x| < b}, N >= 2 and f is a nonlinearity of C-1 class with supercritical growth. By a reduction argument, we show that there exists a nodal solution u(epsilon)( )with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension N - 1 as epsilon -> 0. In particular, we establish different concentration phenomena of four peaks when the parameter eta > 2, eta = 2 and eta < 2. (c) 2022 Elsevier Inc. All rights reserved.