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In this paper, we present necessary conditions for the existence of weak solution of the parabolic-type equations and systems on the Heisenberg group. The main technique for proving the results relies on the method of test functions. [ABSTRACT FROM AUTHOR]

Here, the existence and multiplicity of weak solutions to a generalized (p (⋅) , q (⋅)) -Laplace equation involving Leray–Lions type operators with Hardy potential are studied under Dirichlet boundary conditions on the Heisenberg groups. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions for the noncooperative Schrödinger–Kirchhoff system involving the p-Laplacian operator and critical nonlinearities on the Heisenberg group. Under some suitable conditions, together with the limit index theory and the concentration–compactness principle, we obtain the existence and multiplicity of solutions for this system. To our best knowledge, the existence results for the noncooperative system with p-Laplacian and critical nonlinearities are new on the Heisenberg group. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group H n . We first derive a critical inequality in H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in R n . [ABSTRACT FROM AUTHOR]

The aim of this paper is to study properties of solutions to the fractional p-subLaplace equations on the Heisenberg group. Based on the maximum principles and the generalization of the direct method of moving planes, we obtain the symmetry and monotonicity of the solutions on the whole group and the Liouville property of solutions on a half space. [ABSTRACT FROM AUTHOR]

The existence of at least one positive radial solution of the Neumann problem − Δ H n u + R (ξ) u = a (| ξ | H n) | u | p − 2 u − b (| ξ | H n) | u | q − 2 u , is proved on the Heisenberg group H n , via the variational principle, where a (| ξ | H n) , b (| ξ | H n) are nonnegative radial functions and R (ξ) satisfies suitable conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of quasilinear elliptic equations on Heisenberg group by using the nonsmooth critical point theory. Under some weaker assumptions, the multiplicity and boundedness of solutions for these equations are obtained. [ABSTRACT FROM AUTHOR]

We obtain an explicit smooth kernel which works as the Green's function when applied to circular functions for annular and strip shaped domains in the Heisenberg group Hn by using the Kelvin transform and its generalizations. [ABSTRACT FROM AUTHOR]