论文标题
具有单数或消失电势的一类Quasilinearschrödinger方程的存在结果
Existence results for a class of quasilinear Schrödinger equations with singular or vanishing potentials
论文作者
论文摘要
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 \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \ right）= k（| x |）g（w）\ quad \ text {in} \ mathbb {r}^{n}，\]，其中$ n \ geq3 $。为了研究此问题，我们应用了变量的更改$ W = f（u）$，已经由几位作者使用，并通过应用变异方法找到了非负解决方案的存在结果。我们结果的主要特征是，它们不需要在原点和无穷大的电位$ v $和$ k $之间的兼容性，并且它们本质上依赖于$ v $和$ k $的相对增长的功率类型估计，而不是分别分别的潜力。我们的解决方案满足了上述方程式的薄弱公式，但是我们能够证明它们实际上是$ \ Mathbb {r}^{n} {n} \ backslash \ {0 \} $中的经典解决方案。要应用变分方法，我们必须研究合适的功能空间嵌入到lebesgue空间之和$ l_ {k}^{q_ {1}}}+l_ {k}^Q_ {q_ {2}} $的紧凑性中（$ = l_ {k}^{q}+l_ {k}^{q} $）作为特定情况。 非线性$ 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} \} $，在$ q_1 = q_2 $时恢复了单个功率行为的常见情况。
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 $q_1 = q_2$.
