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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.

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