Abstract
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.
Highlights
1 Introduction This paper is concerned with the existence and multiplicity of non-negative solutions to the semilinear equation on the Heisenberg group HN of the form
Proof For any u ∈ Nμ, the assumption (A ) and the Sobolev inequality imply that u –μ
Let w be a non-negative solution of
Summary
1 Introduction This paper is concerned with the existence and multiplicity of non-negative solutions to the semilinear equation on the Heisenberg group HN of the form There are several papers studying the existence and nonexistence of solutions of semilinear equations with Kohn Laplacian in the past two decades. We do not see any multiplicity results as regards the semilinear equation with critical exponent on the Heisenberg group with general bounded domain. The purpose of the present paper is to prove that under suitable assumptions on K(ξ ) and μ, the problem under consideration has at least two non-negative solutions.
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