论文标题
关于具有奇异或消失势势的quasilinearschrödinger方程的注释
A note on quasilinear Schrödinger equations with singular or vanishing radial potentials
论文作者
论文摘要
在本说明中，我们完成了一项先前的研究，在该研究中，我们获得了准椭圆方程的存在结果\ begin {equation*} -Δw+ v \ left（\ left | x \ right | \右） \ end {equation*}带有单数或消失的连续径向电势$ v（r）$，$ k（r）$。在我们的Previuos研究中，由于技术原因，我们假设$ k（r）$消失为$ r \ rightarrow 0 $，而在本文中，我们消除了此阻塞。为了面对问题，我们应用了变量的合适更改$ W = f（u）$，我们发现通过应用变异方法存在非负解决方案。我们的解决方案满足上述方程式的弱公式，但实际上它们是$ \ Mathbb {r}^{n} \ setMinus \ {0 \} $中的经典解决方案。非线性$ g $具有双重动力行为，其标准示例为$ g（t）= \ min \ {t^{q_1 -1 -1}，t^{q_2 -1} \} $（$ t> 0 $），在$ q_1 = q_2 $时恢复单点行为的常见情况。
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t>0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.
