论文标题
关于具有关键非线性的非局部方程的阳性解决方案的多样性
On multiplicity of positive solutions for nonlocal equations with critical nonlinearity
论文作者
论文摘要
本文介绍了具有关键非线性的以下非局部方程的积极解决方案的存在和多样性：\ begin {equation} \ tag {$ \ Mathcal E $} （-Δ）^s u = a（x）| u |^{2^*_ s-2} u+f（x）\; \; \ text {in} \; \; \ mathbb {r}^n}，\ quad u \ in \ dot {h}^s（\ mathbb {r}^{n}），\ end {qore {qoreation}其中$ s \ in（0,1）$，$ n> 2s $，$ 2_S $，$ 2_S^*：= \ frac {2n} l^\ infty（\ mathbb {r}^{n}）$和$ f $是$ \ dot {h}^s $的双空间中的非负函数。我们证明了能量为负的积极解决方案的存在。此外，在另外假设的情况下，$ a $是连续的函数，$ a（x）\ geq 1 $ in $ \ m athbb {r}^{n} $，$ a（x）\ to 1 $ as as $ | | x | x | x | \ to \ to \ infty $ and $ \ \ | f \ | f \ | _____ {\ dot {\ dot {足够小（但$ f \ not \ equiv 0 $），我们至少建立了至少两个正面解决方案（$ \ natercal e $）。
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\geq 1$ in $\mathbb{R}^{N}$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{\dot{H}^s(\mathbb{R}^{N})'}$ is small enough (but $f\not\equiv 0$), we establish existence of at least two positive solutions to ($\mathcal E$).
