JournalsaihpcVol. 14, No. 3pp. 295–308

Liouville theorems for semilinear equations on the Heisenberg group

  • I. Birindelli

    Università “La Sapienza”, Dip. Matematica, P.zza A.Moro, 2-00185 Roma, Italy

  • I. Capuzzo Dolcetta

    Università “La Sapienza”, Dip. Matematica, P.zza A.Moro, 2-00185 Roma, Italy

  • A. Cutrì

    Università “Tor Vergata”, Dip. Matematica, V.le Ricerca Scientifica, 00133 Roma, Italy
Liouville theorems for semilinear equations on the Heisenberg group cover
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Abstract

In this paper we consider problems of the type

{ΔHu+h(x)up≤0,inD⊂R2n+1,u≥0inD,\left\{\begin{matrix} \Delta _{H}u + h\left(x\right)u^{p} \leq 0, & \text{in}D \subset ℝ^{2n + 1}, \\ u \geq 0 & \text{in}D, \\ \end{matrix}\right.

where ΔH is the Heisenberg Laplacian, D is an unbounded domain and h is a non negative function.

We prove that, under suitable conditions on h, p and D, the only solution of (1) is u ≡ 0.

Résumé

Dans ce travail nous considérons des problèmes du type

{ΔHu+h(x)up≤0,dansD⊂R2n+1,u≥0dansD,\left\{\begin{matrix} \Delta _{H}u + h\left(x\right)u^{p} \leq 0, & \text{dans}D \subset ℝ^{2n + 1}, \\ u \geq 0 & \text{dans}D, \\ \end{matrix}\right.

oùΔH est le Laplacien de Heisenberg, D est un domaine non borné et h est une fonction positive.

Nous démontrons que sous certaines hypothèses sur h, p et D, la seule solution de (1) est u ≡ 0.

Cite this article

I. Birindelli, I. Capuzzo Dolcetta, A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 3, pp. 295–308

DOI 10.1016/S0294-1449(97)80138-2