EXISTENCE OF MANY NON-RADIAL SOLUTIONS OF AN ELLIPTIC SYSTEM

Abstract

In this paper, we consider the following elliptic system

\(\begin{cases} − ∆u = µ_{1}u^{p-1} + βu^{\frac{p}{2}-1} v^{\frac{p}{2}}, \; x ∈ Ω \\ − ∆v = µ_{2}v^{p-1} + βu^{\frac{p}{2}} v^{\frac{p}{2}-1}, \; x ∈ Ω \\ u, v > 0, x ∈ Ω, u= v = 0, x ∈ ∂Ω, \end{cases}\)

where Ω ⊂ \(\mathbb{R}^{N}\) (N > 4) is an annulus, µ₁, µ₂ > 0, β > 0 and 2 < p < \(\frac{2N-2}{N-3}\).  By variational and rescaling method, we prove that the system has many non-radial solutions.