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Sign-changing solutions for a parameter-dependent quasilinear equation. (English) Zbl 1480.35222

Summary: We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:
\[ \begin{cases} \Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \quad\text{in }\Omega,\\ u = 0\quad\text{on }\partial\Omega, \end{cases} \] 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.

MSC:

35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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