Quasilinear Schrödinger Equation involving Critical Hardy Potential and Choquard type Exponential nonlinearity

Abstract

In this article, we study the following quasilinear Schrödinger equation involving Hardy potential and Choquard type exponential nonlinearity with a parameter $\alpha$ \begin{equation*} \left\{ \begin{array}{l} - Δ_N w - Δ_N(|w|^{2α}) |w|^{2α- 2} w - λ\frac{|w|^{2αN-2}w}{\left( |x| \log\left(\frac{R}{|x|} \right) \right)^N} = \left(\int_Ω \frac{H(y,w(y))}{|x-y|^μ}dy\right) h(x,w(x))\; \mbox{in }\; Ω, w > 0 \mbox{ in } Ω\setminus \{ 0\}, \quad \quad w = 0 \mbox{ on } \partial Ω, \end{array} \right. \end{equation*} where $N\ge 2$, $\alpha >\frac{1}{2}$, $0\le \lambda <{\left(\frac{N-1}{N}\right)}^N$, $0<\mu <N$, $h:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $H(x,t)={\int }_0^{t}h(x,s)ds$ is the primitive of $h$. With the help of Mountain Pass Theorem and critical level which is obtained by the sequence of Moser functions, we establish the existence of a positive solution for a small range of $\lambda$. Moreover, we also investigate the existence of a positive solution for a non-homogeneous problem for every $0\le \lambda <{\left(\frac{N-1}{N}\right)}^N.$ To the best of our knowledge, the results obtained here are new even in case of $N$-Laplace equation with Hardy potential.