Nehari manifold for fractional s($\cdot$)-Laplacian system involving concave-convex nonlinearities with magnetic field
首發時間：2023-06-16
Abstract：This paper is concerned with the nonlocal elliptic system driven by the variable-order fractional magnetic Laplace operator involving concave-convex nonlinearities\begin{equation*}\left\{\begin{array}{rl}(-\Delta)_{A}^{s(\cdot)} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)|u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \hspace{2mm}{\rm in}\ \Omega, \\(-\Delta)_{A}^{s(\cdot)} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\beta(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)}| v| ^{\beta(x)-2}v, \hspace{2.5mm}{\rm in}\ \Omega, \\u=v&=0 , \hspace{1cm} {\rm in}\ \mathbb{R}^N\backslash\Omega,\end{array}\right.\end{equation*}where $\Omega\subset\mathbb R^N, ~N\geq2$ is a smooth bounded domain, $\lambda, \mu>0$ are the parameters,$s\in C(\mathbb R^N\times \mathbb R^N, (0, 1))$ and $q, \alpha, \beta\in C(\overline{\Omega}, (1, \infty))$ are the variable exponents and$a, b, c\in C(\overline{\Omega}, [0, \infty))$ are the non-negative weight functions. $(-\Delta)_{A}^{s(\cdot)}$ is the variable-order fractional magnetic Laplace operator, the magnetic field is $A\in C^{0, \alpha}(\mathbb R^N, \mathbb R^N)$ with $\alpha\in(0, 1]$ and $u:\mathbb R^N\to\mathbb C$. Use Nehari manifold to prove that there exists $\Lambda>0$ such that $\forall\lambda+\mu<\Lambda$, this system obtains at least two non-negative solutions of theabove problem under some assumptions on $q, \alpha, \beta$.
keywords： Fundamental Mathematics Variable exponents Nehari manifold Concave-convex nonlinearities Magnetic field.
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帶凹凸非線性項和磁場分數階~s($\cdot$)-Laplacian~系統的~Nehari~流形
摘要：本文主要考慮帶凹凸非線性項和可變階數型磁場分數階~Laplace~算子的非局部橢圓型系統\begin{equation*}\left\{\begin{array}{rl}(-\Delta)_{A}^{s(\cdot)} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)|u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \hspace{2mm}{\rm in}\ \Omega, \\(-\Delta)_{A}^{s(\cdot)} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\beta(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)}| v| ^{\beta(x)-2}v, \hspace{2.5mm}{\rm in}\ \Omega, \\u=v&=0 , \hspace{1cm} {\rm in}\ \mathbb{R}^N\backslash\Omega,\end{array}\right.\end{equation*}其中~$\Omega\subset\mathbb R^N, ~N\geq2$~是光滑有界域,$\lambda, \mu>0$~是參數,$s\in C(\mathbb R^N\times \mathbb R^N, (0, 1))$,$q, \alpha, \beta\in C(\overline{\Omega}, (1, \infty))$~是可變指數,$a, b, c\in C(\overline{\Omega}, [0, \infty))$~是非負權函數。$(-\Delta)_{A}^{s(\cdot)}$~是可變階數分數磁場~Laplace~算子,$A\in C^{0, \alpha}(\mathbb R^N, \mathbb R^N)$~是磁場,$\alpha\in(0, 1]$,$u:\mathbb R^N\to\mathbb C$。應用~Nehari~流形證明,存在~$\Lambda>0$,使得對于~$\forall\lambda+\mu<\Lambda$,在~$q, \alpha, \beta$~適當假設下,上述問題的系統至少有兩個非負解。
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帶凹凸非線性項和磁場分數階~s($\cdot$)-Laplacian~系統的~Nehari~流形
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