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

• Autorzy: Jleli Mohamed

Identyfikatory

DOI

10.1515/dema-2022-0193

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

exterior problem; Heisenberg group; nonnegative radial solution; critical value

PL

grupa Heisenberga; wielkość krytyczna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220193
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Jleli Mohamed

• Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Bibliografia

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