We consider the generalized quasilinear Schrödinger equations (P) $-\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\right(u\left)\mathrm{\nabla }u\right)+g\left(u\right)g^{\prime }\left(u\right)|\mathrm{\nabla }u{|}^2+V\left(x\right)u=\lambda f(x,u),x\in {\mathcal{R}}^N,$(P) where $N\ge 3$, $g:\mathcal{R}\to {\mathcal{R}}^{+}$ is an even differentiable function, $f:{\mathcal{R}}^N\times \mathcal{R}\to \mathcal{R}$ satisfies the Carathéodory condition, the potential $V\left(x\right):{\mathcal{R}}^N\to (0,\mathrm{\infty })$ is continuous and λ is a parameter. The intention of the article is to determine the precise positive interval of λ when the problem possesses at least two nontrivial solutions. The main tool is the abstract critical point which is based on the Ekeland variational principle. Moreover, the existence of one solution and infinite solutions has been studied by the mountain pass theorem and the symmetric mountain pass theorem.

我们考虑广义拟线性薛定谔方程(P)$-\mathrm{d}\mathrm{一世}\mathrm{v}\left(G^2\right(你\left)\mathrm{\nabla }你\right)+G\left(你\right)G^{\text{'}}\left(你\right)|\mathrm{\nabla }你{|}^2+五\left(X\right)你=\lambda F(X,你),X\in {\mathcal{R}}^{ñ},$(P)哪里$ñ\ge 3$,$G：\mathcal{R}\to {\mathcal{R}}^{+}$是一个偶可微函数，$F：{\mathcal{R}}^{ñ}\times \mathcal{R}\to \mathcal{R}$满足 Carathéodory 条件，潜力$五\left(X\right)：{\mathcal{R}}^{ñ}\to (0,\mathrm{\infty })$是连续的，λ是一个参数。本文的目的是确定当问题具有至少两个非平凡解时λ的精确正区间。主要工具是基于 Ekeland 变分原理的抽象临界点。此外，通过山口定理和对称山口定理研究了单解和无穷解的存在性。