Demonstratio Mathematica (Feb 2023)
On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
Abstract
We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form ΔHmu(q)+λψ(q)K(r(q))f(r2−Q(q),u(q))=0{\Delta }_{{{\mathbb{H}}}^{m}}u\left(q)+\lambda \psi \left(q)K\left(r\left(q))f\left({r}^{2-Q}\left(q),u\left(q))=0 in B1c{B}_{1}^{c}, under the Dirichlet boundary conditions u=0u=0 on ∂B1\partial {B}_{1} and limr(q)→∞u(q)=0{\mathrm{lim}}_{r\left(q)\to \infty }u\left(q)=0. Here, λ≥0\lambda \ge 0 is a parameter, ΔHm{\Delta }_{{{\mathbb{H}}}^{m}} is the Kohn Laplacian on the Heisenberg group Hm=R2m+1{{\mathbb{H}}}^{m}={{\mathbb{R}}}^{2m+1}, m>1m\gt 1, Q=2m+2Q=2m+2, B1{B}_{1} is the unit ball in Hm{{\mathbb{H}}}^{m}, B1c{B}_{1}^{c} is the complement of B1{B}_{1}, and ψ(q)=∣z∣2r2(q)\psi \left(q)=\frac{| z{| }^{2}}{{r}^{2}\left(q)}. Namely, under certain conditions on KK and ff, we show that there exists a critical parameter λ∗∈(0,∞]{\lambda }^{\ast }\in \left(0,\infty ] in the following sense. If 0≤λ<λ∗0\le \lambda \lt {\lambda }^{\ast }, the above problem admits a unique nonnegative radial solution uλ{u}_{\lambda }; if λ∗<∞{\lambda }^{\ast }\lt \infty and λ≥λ∗\lambda \ge {\lambda }^{\ast }, the problem admits no nonnegative radial solution. When 0≤λ<λ∗0\le \lambda \lt {\lambda }^{\ast }, a numerical algorithm that converges to uλ{u}_{\lambda } is provided and the continuity of uλ{u}_{\lambda } with respect to λ\lambda , as well as the behavior of uλ{u}_{\lambda } as λ→λ∗−\lambda \to {{\lambda }^{\ast }}^{-}, are studied. Moreover, sufficient conditions on the the behavior of f(t,s)f\left(t,s) as s→∞s\to \infty are obtained, for which λ∗=∞{\lambda }^{\ast }=\infty or λ∗<∞{\lambda }^{\ast }\lt \infty . Our approach is based on partial ordering methods and fixed point theory in cones.
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