ABSTRACT

In this paper, we consider a fractional p-Laplacian system with both concave–convex nonlinearities and sign-changing weight functions in bounded domains: $\begin{array}{rl}(-\mathrm{\Delta }{)}_p^{s}u& =\lambda f\left(x\right)|u{|}^{q-2}u+\frac{2\alpha }{\alpha +\text{β}}h\left(x\right)|u{|}^{\alpha -2}u|v{|}^{\text{β}}\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}_p^{s}v& =\mu g\left(x\right)|v{|}^{q-2}v+\frac{2\text{β}}{\alpha +\text{β}}h\left(x\right)|u{|}^{\alpha }|v{|}^{\text{β}-2}v\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =v=0\mathrm{i}\mathrm{n}{\mathbb{R}}^n\setminus \mathrm{\Omega }.\end{array}$ By using the Nehari manifold, together with Ekeland's variational principle, we prove that the above system has at least two nontrivial solutions when the pair of the parameters $(\lambda ,\mu )$ belongs to a certain subset of ${\mathbb{R}}^2$.

中文翻译：

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涉及凹凸非线性和符号变化权重函数的分数 p-Laplacian 系统的 Nehari 流形

摘要

在本文中，我们考虑在有界域中具有凹凸非线性和符号变化权重函数的分数p- Laplacian 系统：$\begin{array}{rl}(-\mathrm{\Delta }{)}_{磷}^{秒}你& =\lambda F\left(X\right)|你{|}^{q-2}你+\frac{2\alpha }{\alpha +\text{β}}H\left(X\right)|你{|}^{\alpha -2}你|v{|}^{\text{β}}\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}_{磷}^{秒}v& =\mu G\left(X\right)|v{|}^{q-2}v+\frac{2\text{β}}{\alpha +\text{β}}H\left(X\right)|你{|}^{\alpha }|v{|}^{\text{β}-2}v\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你& =v=0\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^n\setminus \mathrm{\Omega }.\end{array}$ 通过使用 Nehari 流形，结合 Ekeland 变分原理，我们证明上述系统至少有两个非平凡解，当参数对 $(\lambda ,\mu )$ 属于某个子集 ${\mathbb{电阻}}^2$.