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A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth
- Source :
- Journal of Differential Equations, 2022
- Publication Year :
- 2022
-
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 $\eta<2$.<br />Comment: 37 pages
- Subjects :
- Mathematics - Analysis of PDEs
Subjects
Details
- Database :
- arXiv
- Journal :
- Journal of Differential Equations, 2022
- Publication Type :
- Report
- Accession number :
- edsarx.2203.14028
- Document Type :
- Working Paper