On Critical Exponent for the Existence and Multiplicity of Positive Weak Solutions for a Class of (p, Q)-laplacian Nonlinear System


Authors

M. B. Ghaemi - Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran G. A. Afrouzi - Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran S. H. Rasouli - Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran M. Choubin - Department of Mathematics, Faculty of Basic Sciences, Payame Noor University, Tehran, Iran


Abstract

In this paper, we prove the existence of positive weak solution for the nonlinear elliptic system \[ \begin{cases} -\Delta_p u=\lambda_1u^a+\mu_1v^b,\,\,\,\,\, x\in\Omega,\\ -\Delta_q v=\lambda_2u^c+\mu_2v^d,\,\,\,\,\, x\in\Omega,\\ u=0=v,\,\,\,\,\, x\in \partial \Omega. \end{cases} \] where \(\Delta_sz=div(|\nabla z|^{s-2}\nabla z), s>1, \lambda_1, \lambda_2, \mu_1\) and \(\mu_2\) are positive parameters, and \(\Omega\) is a bounded domain in \(R^N, a + c < p - 1\) and \(b + d < q - 1\). We also discuss a multiplicity result when \(0 < \lambda_1, \lambda_2, \mu_1, \mu_2<\lambda^* \) for some \(\lambda^* \). We obtain our results via the method of sub - and super solutions.


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