Normalized solutions of quasilinear Schrödinger-Poisson system with critical nonlinear term in bounded domain

This work examines a quasilinear Schrödinger-Poisson system involving a critical nonlinearity, expressed as $$-\mathrm{\Delta }u+\phi u+\lambda u=\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u,x\in {\mathrm{\Omega }}_r,$$ $$-\mathrm{\Delta }\phi -{\epsilon }^4{\mathrm{\Delta }}_4\phi =u^2,x\in {\mathrm{\Omega }}_r,$$ $$u=\phi =0,x\in \mathrm{\partial }{\mathrm{\Omega }}_r$$ subject to the normalized condition $${\int }_{{\mathrm{\Omega }}_r}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{d}x=b^2\mathrm{.}$$ Here $\epsilon >0$, $q\in (2,8\mathrm{/}3)$, ${\mathrm{\Omega }}_r\subset {\mathbb{R}}^3$ is a bounded domain. By means of a truncation method combined with genus theory, we establish the existence of multiple families of normalized solutions. Due to the presence of a critical exponent in the nonlinear term, the associated energy functional fails to satisfy the usual compactness properties. To address this issue, we invoke the concentration-compactness principle. Furthermore, we derive the asymptotic result that the aforementioned system reduces to the classical Schrödinger-Poisson system (with $\epsilon =0$). Our findings extend several recent results concerning problems of this type.