A semilinear problem for the Heisenberg Laplacian
The authors study the Dirichlet problem in a smooth domain \(\Omega\) of \(\mathbb{R}^{2n+1} (x_1,\dots,x_n,y_1,\dots,y_n,t)\) for the equation \(-\Delta_H+ (q-\lambda)u=au^p\), where \[ \Delta_H=\sum^n_{i=1} {\partial^2\over{\partial x^2_i}}+ {\partial^2\over{\partial y^2_i}}+4y_i {\partial^2\over{\partial x_i\partial t}}-4x_i {\partial^2\over{\partial y_i\partial t}}+4(x^2_i+y^2_i) {\partial^2\over{\partial t^2}} \] is the sub-elliptic operator usually called Heisenberg Laplacian, and \(a\) changes sign in \(\Omega\). Some necessary and sufficient conditions are given on the function \(a\) and on \(\lambda\) for the existence of positive solutions \(u\), \(u|_{\partial\Omega}=0\), in terms of the principal eigenvalue and eigensolution of \(-\Delta_H+q\) for \(1<p<1+2/n\). The proof is based on variational methods, including a Hopf type lemma and an embedding theorem.
scientific article; zbMATH DE number 889836
scientific article