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

2024/11/28 by Sarika Goyal, Malhotra, Shammi, Goyal, Sarika +2
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2411.19321

Abstract

In this article, we study the following quasilinear Schrödinger equation involving Hardy potential and Choquard type exponential nonlinearity with a parameter α \ - Δ_N w - Δ_N(|w|^2α) |w|^{2α- 2} w - λ\frac|w|^2αN-2w( |x| log((R)/(|x|) ) )^N = (∫_Ω (H(y,w(y)))/(|x-y|^μ)dy) h(x,w(x)) in Ω, w · gt; 0 in Ω∖ \ 0\, w = 0 on ∂ Ω, . where N≥ 2, α>\frac12, 0≤ λ< ((N-1)/(N))^N, 0 < μ< N, h : \mathbb R^N × \mathbb R → \mathbb R is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and H(x,t)= ∫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 λ. Moreover, we also investigate the existence of a positive solution for a non-homogeneous problem for every 0≤ λ<((N-1)/(N))^N. To the best of our knowledge, the results obtained here are new even in case of N-Laplace equation with Hardy potential.

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