Multiple non-negative solutions to a semilinear equation on Heisenberg group with indefinite nonlinearity

This paper is concerned with the existence and multiplicity of non-negative solutions to the semilinear equation $-\Delta_{H} u = K(\xi)\vert u\vert ^{2^{\sharp}-2}u + \mu \vert \xi \vert _{H}^{\alpha}u$ in a bounded domain $\Omega\subset\mathbb{H}^{N}$ with Dirichlet boundary conditions. Here $\mathbb{H}^{N}$ is the Heisenberg group and $2^{\sharp}= 2q/(q-2)$ is the critical exponent of the Sobolev embedding on the Heisenberg group. The function $K(\xi)$ may be sign changing on Ω. Using the variational method, we prove that this problem has at least two non-negative solutions provided μ, α, and $K(\xi)$ satisfy some conditions.