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

Jleli Mohamed

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220193

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 in Bc1 , under the Dirichlet boundary conditions u=0 on ∂B1 and limr(q)→∞u(q)=0 . Here, λ≥0 is a parameter, ΔHm is the Kohn Laplacian on the Heisenberg group Hm=R2m+1 , m>1 , Q=2m+2 , B1 is the unit ball in Hm, Bc1 is the complement of B1 , and ψ(q)=∣∣z∣∣2r2(q) . Namely, under certain conditions on K and f , we show that there exists a critical parameter λ∗∈(0,∞] in the following sense. If 0≤λ<λ∗ , the above problem admits a unique nonnegative radial solution uλ ; if λ∗<∞ and λ≥λ∗ , the problem admits no nonnegative radial solution. When 0≤λ<λ∗ , a numerical algorithm that converges to uλ is provided and the continuity of uλ with respect to λ , as well as the behavior of uλ as λ→λ∗− , are studied. Moreover, sufficient conditions on the the behavior of f(t,s) as s→∞ are obtained, for which λ∗=∞ or λ∗<∞ . Our approach is based on partial ordering methods and fixed point theory in cones.