Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation - Deng Yinbin; Liu Chenchen; Yang Xian (Advances in Nonlinear Analysis)
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,-Δ u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+∞ }{\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)drh_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}ru^{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 wkw_{k} which changes sign exactly kk times. Moreover, we also prove that wkw_{k} belongs to L2(R2)L^{2}\left({{\mathbb{R}}}^{2}) provided p>10p\gt 10.