Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation

Abstract

This study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: −Δu+hu2(∣x∣)∣x∣2+∫∣x∣+∞hu(s)su2(s)dsu=∣u∣p−2u,x∈R2,-\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(s)}{s}{u}^{2}\left(s){\rm{d}}s\right)u={| u| }^{p-2}u,\hspace{1.0em}x\in {{\mathbb{R}}}^{2}, where p>6p\gt 6 and hu(s)=12∫0sru2(r)dr{h}_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}r{u}^{2}\left(r){\rm{d}}r. By variational methods, we prove that for any integer k≥0k\ge 0, the above equation has a nodal solution wk{w}_{k} which changes sign exactly kk times. Moreover, we also prove that wk{w}_{k} belongs to L2(R2){L}^{2}\left({{\mathbb{R}}}^{2}) provided p>10p\gt 10.

Keywords

• gauged schrödinger equation
• nodal solution
• zero mass
• variational methods
• 35j20
• 35q55
• 35r09