On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group

Abstract

Abstract We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form Δ H m u ( q ) + λ ψ ( q ) K ( r ( q ) ) f ( r 2 − 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 B 1 c {B}_{1}^{c} , under the Dirichlet boundary conditions u = 0 u=0 on ∂ B 1 \partial {B}_{1} and lim r ( q ) → ∞ u ( q ) = 0 {\mathrm{lim}}_{r\left(q)\to \infty }u\left(q)=0 . Here, λ ≥ 0 \lambda \ge 0 is a parameter, Δ H m {\Delta }_{{{\mathbb{H}}}^{m}} is the Kohn Laplacian on the Heisenberg group H m = R 2 m + 1 {{\mathbb{H}}}^{m}={{\mathbb{R}}}^{2m+1} , m > 1 m\gt 1 , Q = 2 m + 2 Q=2m+2 , B 1 {B}_{1} is the unit ball in H m {{\mathbb{H}}}^{m} , B 1 c {B}_{1}^{c} is the complement of B 1 {B}_{1} , and ψ ( q ) = ∣ z ∣ 2 r 2 ( q ) \psi \left(q)=\frac{| z{| }^{2}}{{r}^{2}\left(q)} . Namely, under certain conditions on K K and f f , 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 u λ {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 u λ {u}_{\lambda } is provided and the continuity of u λ {u}_{\lambda } with respect to λ \lambda , as well as the behavior of u λ {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.