论文标题
带有凹形和凸非线性的QuasilinearSchrödinger方程
Quasilinear Schrödinger equations with concave and convex nonlinearities
论文作者
论文摘要
在本文中，我们考虑以下QuasilIrinearSchrödinger方程\ begin {align*}-ΔU-uδ（u ^{2}）= k（x）= k（x）\ left \ welet \ vert u \ vert u \ right \ vert ^{q-2} {q-2} d^{1,2}（\ mathbb {r}^{n}）\ text {，} \ end {align*}其中$ 1 <q <q <2 <s <s <+\ infty $。与文献中的大多数结果不同，此处的指数$ s $被允许为超临界$ s> 2 \ cdot2^{\ ast} $。通过利用非线性转换$ f $的几何特性和克拉克定理的变体，我们在小于$ d^{1,2}的空间中获得了一系列具有负能量的解决方案（\ Mathbb {r}^{n}）$。还获得了负能级下的非负溶液。
In this paper, we consider the following quasilinear Schrödinger equation \begin{align*} -Δu-uΔ(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,} \end{align*} where $1<q<2<s<+\infty$. Unlike most results in the literature, the exponent $s$ here is allowed to be supercritical $s>2\cdot2^{\ast}$. By taking advantage of geometric properties of a nonlinear transformation $f$ and a variant of Clark's theorem, we get a sequence of solutions with negative energy in a space smaller than $D^{1,2}(\mathbb{R}^{N})$. Nonnegative solution at negative energy level is also obtained.
