Non-Variational Elliptic Systems in Dimension Two: A Priori Bounds and Existence of Positive Solutions

<abstract>This article discusses the existence and uniqueness of radial solution for the elliptic equation system <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left \{ \begin{array}{ll} -\triangle u = f(|x|, \ u, \ v, \ |\nabla u|), \; \; x\in \Omega, \\[10pt] -\triangle v = g(|x|, \ u, \ v, \ |\nabla v|), \; \; x\in \Omega, \\[10pt] u|_{\partial \Omega} = 0, \; v|_{\partial \Omega} = 0, \end{array} \right. $\end{document} </tex-math></disp-formula> where $ \Omega = \{x\in \mathbb{R}^{N}:\; r_1 < |x| < r_2\}, \; N\ge 3, \; f, \; g:[r_1, \; r_2]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^+\to \mathbb{R} $ are continuous. Due to the appearance of the gradient term in the nonlinearity, the equation system has no variational structure and the variational method cannot be applied to it directly. We will give the correlation conditions of $ f $ and $ g $, that is, $ f $ and $ g $ are superlinear or sublinear, and prove the existence and uniqueness of radial solutions by using Leray-Schauder fixed point theorem.</abstract>

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Non-Variational Elliptic Systems in Dimension Two: A Priori Bounds and Existence of Positive Solutions

Cited by 2 publications

References 17 publications

Nonvariational Semilinear Elliptic Systems

Nonvariational Semilinear Elliptic Systems

Existence and uniqueness of radial solution for the elliptic equation system in an annulus

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