Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity

Abstract

In this article, we study the following quasilinear Schrödinger equation: −Δu+V(∣x∣)u−κuΔ(u2)+qh2(∣x∣)∣x∣2(1+κu2)u+q∫∣x∣+∞h(s)s(2+κu2(s))u2(s)dsu=(Iα∗∣u∣p)∣u∣p−2u,x∈R2,-\Delta u+V\left(| x| )u-\kappa u\Delta \left({u}^{2})+q\frac{{h}^{2}\left(| x| )}{| x\hspace{-0.25em}{| }^{2}}\left(1+\kappa {u}^{2})u+q\left(\underset{| x| }{\overset{+\infty }{\int }}\frac{h\left(s)}{s}\left(2+\kappa {u}^{2}\left(s)){u}^{2}\left(s)\hspace{0.1em}\text{d}\hspace{0.1em}s\right)u=\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where κ\kappa , q>0q\gt 0, p>8p\gt 8, Iα{I}_{\alpha } is a Riesz potential, α∈(0,2)\alpha \in \left(0,2) and V∈C(R2,R)V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}). By using Jeanjean’s monotone trick, it can be explored that the aforementioned equation has a ground state solution under appropriate assumptions.

Keywords

• chern-simons-schrödinger equations
• choquard type nonlinearity
• ground state solution
• monotone trick
• 35j60
• 35j20