We consider a class of generalized quasilinear Schrödinger equations $$ -\operatorname{div}\bigl(l^{2}(u)\nabla u\bigr)+l(u)l'(u) \vert \nabla u \vert ^{2}+V(x)u= f(u),\quad x\in \mathbb{R}^{N}, $$ where $l(t): \mathbb{R}\to\mathbb{R}^{+}$ is a nondecreasing function with respect to $|t|$, the potential function V is allowed to be sign-changing so that the Schrödinger operator $-\Delta+V$ possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory.

中文翻译：

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具有变号势的广义拟线性Schrödinger方程解的多重性与存在性。

我们考虑一类广义拟线性Schrödinger方程$$-\ operatorname {div} \ bigl（l ^ {2}（u）\ nabla u \ bigr）+ l（u）l'（u）\ vert \ nabla u \ vert ^ {2} + V（x）u = f（u），\ quad x \ in \ mathbb {R} ^ {N}，$$其中$ l（t）：\ mathbb {R} \ to \ mathbb {R} ^ {+} $是相对于$ | t | $的非递减函数，势函数V可以进行符号转换，以便Schrödinger运算符$-\ Delta + V $具有有限维负数空间。我们通过对称的山口定理和莫尔斯理论获得了该问题的存在性和多重性结果。