Publication Date:
2016-12-20
Description:
We study the nonexistence of nontrivial solutions for the nonlinear elliptic system $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta _x)^{\alpha /2}u+|x|^{2\delta } (-\Delta _y)^{\beta /2}u+|x|^{2\eta }|y|^{2\theta } (-\Delta _z)^{\gamma /2}u&{}=&{} v^p,\\ \\ (-\Delta _x)^{\mu /2}v+|x|^{2\delta } (-\Delta _y)^{\nu /2}v+|x|^{2\eta }|y|^{2\theta } (-\Delta _z)^{\sigma /2}v&{}=&{} u^q, \\ \end{array} \right. \end{aligned}$$ where \((x,y,z)\in \mathbb {R}^{N_1}\times \mathbb {R}^{N_2}\times \mathbb {R}^{N_3}\) , \(0〈\alpha ,\beta ,\gamma ,\mu , \nu , \sigma \le 2\) , \(\delta , \eta ,\theta \ge 0\) , and \(p,q〉1\) . Here, \((-\Delta _x)^{\alpha /2}\) , \(0〈\alpha 〈2\) , is the fractional Laplacian operator of order \(\alpha /2\) with respect to the variable \(x\in \mathbb {R}^{N_1}\) , \((-\Delta _y)^{\beta /2}\) , \(0〈\beta 〈2\) , is the fractional Laplacian operator of order \(\beta /2\) with respect to the variable \(y\in \mathbb {R}^{N_2}\) , and \((-\Delta _z)^{\gamma /2}\) , \(0〈\gamma 〈2\) , is the fractional Laplacian operator of order \(\gamma /2\) with respect to the variable \(z\in \mathbb {R}^{N_3}\) . Using a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.
Print ISSN:
2193-5343
Electronic ISSN:
2193-5351
Topics:
Mathematics
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