Acta Applicandae Mathematicae ( IF 1.2 ) Pub Date : 2021-04-02 , DOI: 10.1007/s10440-021-00402-9 Sarika Goyal , Tuhina Mukherjee
In this article, we deal with the existence of non-negative solutions of the class of following non local problem
$$ \left \{ \textstyle\begin{array}{l} \quad - M\left (\displaystyle \int _{\mathbb{R}^{n}}\int _{\mathbb{R}^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right ) (-\Delta )^{s}_{n/s} u=\left (\displaystyle \int _{\Omega }\frac{G(y,u)}{|x-y|^{\mu }}~dy \right )g(x,u) \; \text{in}\; \Omega , \\ \quad \quad u =0\quad \text{in} \quad \mathbb{R}^{n} \setminus \Omega , \end{array}\displaystyle \right . $$where \((-\Delta )^{s}_{n/s}\) is the \(n/s\)-fractional Laplace operator, \(n\geq 1\), \(s\in (0,1)\) such that \(n/s\geq 2\), \(\Omega \subset \mathbb{R}^{n}\) is a bounded domain with Lipschitz boundary, \(M:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) and \(g:\Omega \times \mathbb{R}\rightarrow \mathbb{R}\) are continuous functions, where \(g\) behaves like \(\exp ({|u|^{\frac{n}{n-s}}})\) as \(|u|\rightarrow \infty \). The key feature of this article is the presence of Kirchhoff model along with convolution type nonlinearity having exponential growth which appears in several physical and biological models.
中文翻译：
分数阶Laplacian的具有Choquad指数型非线性的Kirchhoff方程
在本文中，我们处理以下非局部问题一类的非负解的存在
$$ \ left \ {\ textstyle \ begin {array} {l} \ quad-M \ left（\ displaystyle \ int _ {\ mathbb {R} ^ {n}} \ int _ {\ mathbb {R} ^ { n}} \ frac {| u（x）-u（y）| ^ {\ frac {n} {s}}} {| xy | ^ {2n}}〜dxdy \ right）（-\ Delta）^ { s} _ {n / s} u = \ left（\ displaystyle \ int _ {\ Omega} \ frac {G（y，u）} {| xy | ^ {\ mu}}〜dy \ right）g（x ，u）\; \ text {in} \; \ Omega，\\ \ quad \ quad u = 0 \ quad \ text {in} \ quad \ mathbb {R} ^ {n} \ setminus \ Omega，\ end {array} \ displaystyle \ right。$$其中\（（-\ Delta）^ {s} _ {n / s} \）是\（n / s \）-分式Laplace运算符\（n \ geq 1 \），\（s \ in（0 ，1）\），这样\（n / s \ geq 2 \），\（\ Omega \ subset \ mathbb {R} ^ {n} \）是一个具有Lipschitz边界\（M：\ mathbb { R} ^ {+} \ rightarrow \ mathbb {R} ^ {+} \）和\（g：\ Omega \ times \ mathbb {R} \ rightarrow \ mathbb {R} \）是连续函数，其中\（g \）的行为就像\（\ exp（{| u | ^ {\ frac {n} {ns}}}）\）就像\（| u | \ rightarrow \ infty \）。本文的主要特点是存在基尔霍夫模型以及具有指数增长的卷积型非线性，这种非线性出现在几种物理和生物模型中。
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