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

Abstract

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.