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

TL;DR

This paper constructs a new class of nodal solutions for a singularly perturbed elliptic equation with supercritical growth, showing they concentrate on orthogonal spheres and exhibit different behaviors based on a parameter.

Contribution

It introduces a novel type of nodal solutions with multiple peaks concentrating on orthogonal spheres, expanding understanding of solution structures in supercritical elliptic problems.

Findings

Existence of nodal solutions with four peaks

Concentration on orthogonal spheres as perturbation parameter tends to zero

Different peak behaviors depending on the parameter ta>2, ta=2, or ta<2

Abstract

In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^\eta u =|x|^\eta f(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $\eta\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension $N-1$ as $\e\rightarrow0$. In particular, we establish different concentration phenomena of four peaks when the parameter $\eta>2$, $\eta=2$ and $\eta<2$.