论文标题
不平衡的$（P，2）$ - 分数问题随着关键增长
Unbalanced $(p,2)$-fractional problems with critical growth
论文作者
论文摘要
我们研究了以下双重非本地问题的非负解决方案的存在，多样性和规律性结果：$$（p_ \ la）\ left \ {\ {\ begin {array} {lr} {lr} \ ds \ quad（-δ）^{s_1} u++\ ba（-Δ）^{s_2} _ {p} u = \ la a（x）| u |^{q-2} u+++ \ left（\ int _ {\ om} \ frac {| u（y）|^r} {| x-y |^μ} 〜dy \ right）| u |^{r-2} u \ quad \ quad \ quad \ text {in} \; \ om， \ Quad \ Quad \ Quad \ Quad U = 0 \ Quad \ text {in} \ Quad \ Quad \ mb r^n \ setMinus \ om，\ end \ end {array} \ right。 $$ $ \ om \ subset \ mb r^n $是一个有界域，带有$ c^2 $ boundard $ \ pa \ om $，$ 0 <s_2 <s_1 <s_1 <1 $，$ n> 2 s_1 $，$ 1 <q <q <q <p <p <2 $，$ 1 <r \ leq 2^{**} $ 2^{*}_μ= \ frac {2n-μ} {n-2s_1} $，$ \ la，\ ba> 0 $和$ a \ in l^{\ frac {d} {d-q} {d-q}}}（\ om）$ $ q <d <2^{*} _ {s_1}：= \ frac {2n} {n-2s_1} $，是一个更改功能的符号。我们证明$（p_ \ la）$的每个非负弱解决方案都是有限的。此外，我们使用Nehari歧管方法获得了一些存在和多重性结果。
We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-Δ)^{s_1}u+\ba (-Δ)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^r}{|x-y|^μ}~dy\right)|u|^{r-2} u \quad \text{in}\; \Om, \quad \quad\quad \quad u =0\quad \text{in} \quad \mb R^n\setminus \Om, \end{array} \right. $$ where $\Om\subset\mb R^n$ is a bounded domain with $C^2$ boundary $\pa\Om$, $0<s_2 < s_1<1$, $n> 2 s_1$, $1< q<p< 2$, $1<r \leq 2^{*}_μ$ with $2^{*}_μ=\frac{2n-μ}{n-2s_1}$, $\la,\ba>0$ and $a\in L^{\frac{d}{d-q}}(\Om)$, for some $q<d<2^{*}_{s_1}:=\frac{2n}{n-2s_1}$, is a sign changing function. We prove that each nonnegative weak solution of $(P_\la)$ is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.
