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Indefinite semi-linear equations on the Heisenberg group: A priori bounds and existence. (English) Zbl 0965.35019

The authors consider a semilinear equation of the form \(\Delta_H u+ f(\xi,u)= 0\), where \(\Delta_H= \sum^n_{i=1} X^2_i+ Y^2_i\) with \(X_i= \partial/\partial x_i+2y_i\partial/\partial t\), \(Y_i= \partial/\partial y_i- 2x_i\partial/\partial t\), \(i= 1,2,\dots, n\), \(\xi= (x_1,\dots, x_n; y_1,\dots, y_n;t)\in \mathbb{R}^{2n+1}\) and \(f\) has superlinear polynomial growth when \(u\) tends to infinity and is allowed a change of sign. They give an a priori bound independent of the constant \(\tau> 0\) for the supremum norm of any solution \(u\in\Gamma^2(\Omega)\cap C(\overline\Omega)\) of the Dirichlet problem \(u> 0\), \(\Delta_Hu+ f(\xi,u)+ \tau=0\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), in a bounded domain \(\Omega\), and prove the existence of solutions of the Dirichlet problem \(u\in T^2(\Omega)\cap C(\overline\Omega)\), \(u>0\), \(\Delta_Hu+ f(\xi, u)=0\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) by using the above a priori estimate and topological degree arguments combined with regularity theory and the maximum principle.

MSC:

35B45 A priori estimates in context of PDEs
58J05 Elliptic equations on manifolds, general theory
35B50 Maximum principles in context of PDEs
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