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

We consider the following singular quasilinear Schrödinger equations involving critical exponent

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u-\frac{\alpha }{2}\Delta (|u|^{\alpha })|u|^{\alpha -2}u=\theta |u|^{k-2}u+|u|^{2^{*}-2}u+\lambda f(u), x\in \Omega ,\\ \hspace{1.65in}u=\,0, x\in \partial \Omega , \end{array} \right. \end{aligned}$$

where \(0<\alpha <1\). By using the variational methods, we first prove that for small values of \(\lambda \) and \(\theta \), 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.