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Minimax Theory and its Applications 04 (2019), No. 1, 161--188
Copyright Heldermann Verlag 2019

Existence of Entire Solutions for Quasilinear Equations in the Heisenberg Group

Patrizia Pucci
Dip. di Matematica e Informatica, Universit di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy


The paper deals with the existence of entire solutions for a quasilinear equation $(\mathcal{E}_{\lambda})$ in $\mathbb{H}^{n}$, depending on a real parameter $\lambda$, which involves a general elliptic operator $\mathbf{A}$ in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value $\lambda _{\ast}>0$ with the property that $(\mathcal{E}_{\lambda})$ admits nontrivial nonnegative entire solutions if and only if $\lambda \geq \lambda _{\ast }$. Furthermore, under the further assumption that the potential $\mathcal{A}$ of $\mathbf{A}$ is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of $(\mathcal{E}_\lambda)$, when $\lambda > \lambda _{\ast }$.

Keywords: Heisenberg group, entire solutions, critical exponents.

MSC: 35J62, 35J70, 35B08; 35J20, 35B09.

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