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On the Existence of Positive Weak Solution for Nonlinear System with Singular Weights

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Abstract

In this article, we study the existence results of large positive weak solution for nonlinear system with singular weights (1.4), where \(\Omega\) is a bounded domain of \(R^{n}\) with boundary \(\partial\Omega\), \(0\in\Omega\), \(1<p,q<n\), \(0\leq r<\frac{n-p}{p}\), \(0\leq s<\frac{n-q}{q}\), and \(\Lambda_{p}u=|u|^{p-2}u\), \(\varrho_{p},\varrho_{q}\), \(\lambda,\mu,\gamma,\delta\) are positive constants and \(a\), \(b\) are weight functions. We prove the existence of a large positive weak solutions for mappings. \(\lambda\), \(\mu\) large when \(\lim\limits_{x\rightarrow+\infty}\frac{f^{\frac{1}{p-1}}(M(g(x))^{\frac{1}{q-1}})}{x}=0\), for every \(M>0\). Here, there is no any sign-changing conditions on \(a\) or \(b\). The proof of the main results is based on the sub-supersolutions method. Application and concluding remark are provided to demonstrate the effectiveness of our results.

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Funding

The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Award No. 38/58/1439-2018.

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Khafagy, S., Serag, H. On the Existence of Positive Weak Solution for Nonlinear System with Singular Weights. J. Contemp. Mathemat. Anal. 55, 259–267 (2020). https://doi.org/10.3103/S1068362320040068

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