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*R*

^{n}.

This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schr*ö*dinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer $k ≥ 0$.

$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $ |

*N*≥ 3; 2 <

*p*< 2

^{*}= $\frac{{2N}}{{N -2}}$ and

*V*(

*x*) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [

$ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$ |

$λ$ |

$V(x)$ |

$Ω: = int(V^{-1}(0))$ |

$k$ |

$Ω_1,Ω_2,···,Ω_k$ |

$f(u)$ |

$λ→ +∞$ |

$-\Delta u + u - \mu \frac{u}{|x|^2}=|u|^{2^*-2}u + f(u),$

$u \in H^1_r (\R ^N),(1)$

where $2^$*$=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_r (\R ^N) \rightarrow L^{2^}$*$ (\R ^N)$, $\mu \in [0, \ (\frac {N-2}{2})^2)$ and $f: \R \rightarrow\R $ is a function satisfying some conditions. The main results obtained in this paper are that there exists a nontrivial solution of equation (1) provided $N\ge 4$ and $\mu \in [0, \ (\frac {N-2}{2})^2-1] $ and there exists at least a pair of nontrivial solutions $u^+_k$, $u^-_k$ of problem (1) for each k $\in N \cup \{0\}$ such that both $u^+_k$ and $u^-_k$ possess exactly k nodes provided $N\ge 6$ and $\mu \in [0, \ (\frac {N-2}{2})^2-4]$.

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