On Solutions for Strongly Coupled Critical Elliptic Systems on Compact Riemannian Manifolds

Abstract

In this paper, by using variational methods we investigate the existence of solutions for the following system of elliptic equations

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _{g}u + a(x)u + b(x)v&{}=&{} \dfrac{\alpha }{2^{*}}f(x)u|u|^{\alpha - 2} |v|^{\beta } \ \ \text{ in } \ \ M,\\ -\Delta _{g}v+ b(x)u + c(x)v &{}=&{} \dfrac{\beta }{2^{*}}f(x)v|v|^{\beta - 2} |u|^{\alpha } \ \ \text{ in } \ \ M, \end{array}\right. \end{aligned}$$

where (M, g) is a smooth closed Riemannian manifold of dimension $n\ge 3, \Delta _{g}$ is the Laplace–Beltrami operator, a, b and c are functions Hölder continuous in M, f is a smooth function and $\alpha>1, \beta >1$ are two real numbers such that $\alpha + \beta = 2^*,$ where $2^{*}= 2n/(n-2)$ denotes the critical Sobolev exponent. We get these results by assuming sufficient conditions on the function $h= \frac{\alpha }{2^{*}} a + \frac{2\sqrt{\alpha \beta }}{2^{*}}b + \frac{\beta }{2^{*}} c$ related to the linear geometric potential $\frac{n-2}{4(n-1)}R_{g}$, where $R_{g}$ is the scalar curvature associated to the metric g.