arXiv 2202.018722022-02-03math.APmath.FA

一类具奇异或零位势的拟线性薛定谔方程解的存在性

Existence results for a class of quasilinear Schrödinger equations with singular or vanishing potentials

Marino Badiale, Michela Guida, Sergio Rolando

备注：: To appear on Nonlinear Analysis. arXiv admin note: text overlap with arXiv:1912.07537

摘要：

给定两个连续函数$V\left(r \right)\geq 0$和$K\left(r\right)> 0$（$r>0$），它们在零点和无穷远处都可能是奇异的或为零，研究拟线性椭圆方程\[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \]，其中$N\geq3$ .为了研究这个问题，我们申请的变量$w=f(u)$，已经使用了几个作者，并找到存在性结果的非负解的应用变分方法。我们的研究结果的主要特点是，他们不需要任何兼容性之间的潜力$V$和$K$的行为在原点和无穷大，他们基本上依赖于电源类型估计的相对增长的$V$和$K$，而不是单独的潜力。我们的解决方案满足上述方程的弱公式，但我们能够证明，他们实际上是经典的解决方案在$\mathbb{R}^{N} \backslash \{ 0\}$。为了应用变分方法，我们必须研究一个合适的函数空间嵌入到勒贝格空间$L_{K}^{q_{1}}+L_{K}^{q_{2}}$的和中的紧性，从而作为一种特殊情况嵌入到$L_{K}^{q}$（$=L_{K}^{q}+L_{K}^{q}$）中。非线性$g$具有双幂行为，其标准示例是$g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$，恢复了$q_1 = q_2$时单幂行为的通常情况。

英文摘要：

Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of variables $w=f(u)$, already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and that they essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. Our solutions satisfy a weak formulations of the above equation, but we are able to prove that they are in fact classical solutions in $\mathbb{R}^{N} \backslash \{ 0\}$. To apply variational methods, we have to study the compactness of the embedding of a suitable function space into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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