论文标题
nehari歧管（。
Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities
论文作者
论文摘要
在本文中，我们使用nehari歧管方法研究了以下非局部椭圆系统的多样性，该解决方案涉及可变指数和凹形convex非线性：\ begin {equation*} \; \; \; \; \; \; \; \ begin {array} {rl}（-Δ）_ {p（\ cdot）}^{s} u＆= λ〜a（x）| u |^{q（x）-2} u++\ frac {α（x）} {α（x）+β（x）} c（x）| u |^{α（x）-2} u | V | ^{β（x）}，\ hspace {2mm} x \ inω; \\（-Δ）_ {p（\ cdot）}^{s} v＆= μ〜b（x）| v |^{q（x）-2} v+\ frac {α（x）} {α（x）+β（x）} c（x）| v |^{α（x）-2} v | u | ^{β（x）}，\ hspace {2.5mm} x \ inω; \\ u = v＆= 0，\ hspace {1cm} x \ inω^c：= \ Mathbb r^n \setMinusΩ，\ end end {array} \ end {quare} \ end {qore {qore*}，其中$ω\ subset \ subset \ subset \ subbb r^n，〜n，〜n，〜n，〜n，〜n \ geq2 $ a a $ a a $ and a $ ainm a in a $ andain， $ s \ in（0,1），$ $ p \ in C（\ mathbb r^n \ times \ times \ mathbb r^n，（1，\ infty）））$和$ q，α，α，β\ in c（\overlineΩ，（1，\ infty））$是可变的导向主管，$ a，b，c \ infty c \ infty c \ in c c \ in c in c c \ in c c \ in c c \ in c in c in c c \ in c（非负重功能。我们表明存在$λ> 0 $，因此对于所有$λ+μ<λ$，在某些假设上，在$ q，α，β$的某些假设下存在两个非平凡和非负解。
In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} \;\;\; \begin{array}{rl} (-Δ)_{p(\cdot)}^{s} u&=λ~ a(x)| u|^{q(x)-2}u+\frac{α(x)}{α(x)+β(x)}c(x)| u|^{α(x)-2}u| v| ^{β(x)},\hspace{2mm} x\in Ω; \\ (-Δ)_{p(\cdot)}^{s} v&=μ~ b(x)| v|^{q(x)-2}v+\frac{α(x)}{α(x)+β(x)}c(x)| v|^{α(x)-2}v| u| ^{β(x)},\hspace{2.5mm} x\in Ω; \\ u=v&=0 ,\hspace{1cm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $Ω\subset\mathbb R^N,~N\geq2$ is a smooth bounded domain, $λ,μ>0$ are the parameters, $s\in(0,1),$ $p\in C(\mathbb R^N\times \mathbb R^N,(1,\infty))$ and $q,α,β\in C(\overlineΩ,(1,\infty))$ are the variable exponents and $a,b,c\in C(\overlineΩ,[0,\infty))$ are the non-negative weight functions. We show that there exists $Λ>0$ such that for all $λ+μ<Λ$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,α,β$.
