Quasilinear Schrödinger Equations with a Singular Operator and Critical or Supercritical Growth

We consider the following singular quasilinear Schrödinger equations involving critical exponent \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u-\frac{\alpha }{2}\Delta (|u|^{\alpha })|u|^{\alpha -2}u=\theta |u|^{k-2}u+|u|^{2^{*}-2}u+\lambda f(u), x\in \Omega ,\\ \hspace{1.65in}u=\,0, x\in \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle -\mathrm{\Delta }u-\frac{\alpha }{2}{\mathrm{\Delta }(|u|}^{\alpha }{)|u|}^{\alpha -2}u={\theta |u|}^{k-2}u+{|u|}^{2^{\ast }-2}u+\lambda f(u),x\in \mathrm{\Omega },}\hfill \\ u=0,x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(0<\alpha <1\) $0<\alpha <1$ . By using the variational methods, we first prove that for small values of \(\lambda \) $\lambda$ and \(\theta \) $\theta$ , the above problem has infinitely many distinct solutions with negative energy. Besides, we point out that odd assumption on f is required; the problem has at least one nontrivial solution. Finally, a new modified technique is used to consider the existence of infinitely many solutions for far more general equations.