In this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ where $N\geq 3$ , $1< p< N$ , $-\infty <\alpha <\frac{N-p}{p}$ , $\alpha \leq e\leq \alpha +1$ , $d=1+\alpha -e$ , $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$ (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.

中文翻译：

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具有一般势的简并拟线性Schrödinger方程的无穷多个解

在本文中，我们研究以下拟线性Schrödinger方程：$$-\ operatorname {div} \ bigl（a（x，\ nabla u）\ bigr）+ V（x）\ vert x \ vert ^ {-\ alpha p ^ {*}} \ vert u \ vert ^ {p-2} u = K（x）\ vert x \ vert ^ {-\ alpha p ^ {*}} f（x，u）\ quad \ text {in } \ mathbb {R} ^ {N}，$$其中$ N \ geq 3 $，$ 1 <p <N $，$-\ infty <\ alpha <\ frac {Np} {p} $，$ \ alpha \ leq e \ leq \ alpha + 1 $，$ d = 1 + \ alpha -e $，$ p ^ {*}：= p ^ {*}（\ alpha，e）= \ frac {Np} {N-dp } $（临界Hardy–Sobolev指数），V和K为非负电势，该函数a满足合适的假设，并且f为超线性，比Ambrosetti-Rabinowitz型条件弱。通过使用变分方法，我们得到了拟线性Schrödinger方程具有无限多个非平凡解。