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.
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Apr 4, 2023
Changmu Chu + 1
Open Access
