论文标题
双曲线空间中一类非均匀椭圆方程的阳性溶液多样性
Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space
论文作者
论文摘要
该论文涉及对该类型问题的积极解决方案 \ begin {equation*} -Δ_ {\ Mathbb {b}^n} u-λu= a（x）| u |^{p -1} \; u \， + \，f \ \，\ \ \; \; \; \ text {in} u \ in H^{1} {（\ Mathbb {b}^{n}）}， \ end {equation*} 其中$ \ mathbb {b}^n $表示夸张的空间，$ 1 <p <2^* - 1：= \ frac {n+2} {n-2} {n-2} $，$ \;λ<\;λ<\ frac {（n-1） \ not \ equiv 0 $）是一个非负功能。 l^\ infty（\ mathbb {b}^n）$中的潜在$ a \被认为是严格的阳性，使得$ \ lim_ {d（x，0）\ rightArrow \ rightarrow \ infty} a（x）\ rightArrow 1，$ d（x）$ d（x，x，x，0）$表示地板距离。首先，在假设$ a（x）\ leq 1 $的假设下证明了三种积极解决方案。然后考虑了$ a（x）\ geq 1 $的情况，并证明了两个积极解决方案的存在。在这两种情况下，都假定$μ（\ {x：a（x）\ neq 1 \}）> 0。存在的证据结合了变异参数，涉及双曲气泡的关键能量估计。
The paper is concerned with positive solutions to problems of the type \begin{equation*} -Δ_{\mathbb{B}^N} u - λu = a(x) |u|^{p-1}\;u \, + \, f \, \;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \end{equation*} where $\mathbb{B}^N$ denotes the hyperbolic space, $1<p<2^*-1:=\frac{N+2}{N-2}$, $\;λ< \frac{(N-1)^2}{4}$, and $f \in H^{-1}(\mathbb{B}^N)$ ($f \not\equiv 0$) is a non-negative functional. The potential $a\in L^\infty(\mathbb{B}^N)$ is assumed to be strictly positive, such that $\lim_{d(x, 0) \rightarrow \infty} a(x) \rightarrow 1,$ where $d(x, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that $a(x) \leq 1$. Then the case $a(x) \geq 1$ is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that $μ( \{ x : a(x) \neq 1\}) > 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.
