We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:
$ \begin{equation*} \left\{ \begin{aligned} &\Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \ \ \ \text{in}\,\,\Omega,\\ &u = 0\quad\text{on}\,\,\partial\Omega, \end{aligned} \right. \end{equation*} $
where $ \Omega\subset\mathbb{R}^N(N\geq3) $ is a bounded domain with smooth boundary, $ \lambda>0,\, r\in(2,4) $. We prove as $ \lambda $ becomes large the existence of more and more sign-changing solutions of both positive and negative energies.
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