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Borderline variational problems for fractional Hardy-Schrödinger operators

University of British Columbia

In this thesis, we study properties of the fractional Hardy-Schrödinger operator L_(γ,α)≔(-∆)^(α/(2 ))- γ/〖|x|〗^α both on R^n and on its bounded domains. The following functional inequality is key to our variational approach. C〖(∫_(R^n)〖 〖|u|〗^(2_α^* (s))/〖|x|〗^s dx〗)〗^(2/(2_α^* (s)))≤ ∫_(R^n)〖 〖|(-∆)^(α/(4 )) u|〗^2 dx〗- γ∫_(R^n)〖 〖|u|〗^2/〖|x|〗^α dx,〗 where 0 ≤ s < α < 2, n > α, 2_α^* (s)=(2(n-s))/(n-α) and γ< γ_H(α), the latter being the best fractional Hardy constant on R^n. We address questions regarding the attainability of the best constant C > 0 attached to this inequality. This allows us to establish the existence of non-trivial weak solutions for the following doubly critical problem on R^n, L_(γ,α) u=〖|u|〗^(2_α^*-2)u + (〖|u|〗^(2_α^* (s)-2) u)/〖|x|〗^s in R^n, where 〖2_α^*≔2〗_α^* (0). We then look for least-energy solutions of the following linearly perturbed non-linear boundary value problem on bounded subdomains of R^n containing the singularity 0: (L_(γ,α)-λΙ)u= 〖|u|〗^(2_α^* (s)-1)/〖|x|〗^s on Ω, We show that if γ is below a certain threshold γ_crit(α), then such solutions exist for all 〖0<λ<λ〗_1 (L_(γ,α) ), the latter being the first eigenvalue of L_(γ,α). On the other hand, for γ_crit (α)<γ<γ_H (α), we prove existence of such solutions only for those λ in (0,λ_1 (L_(γ,α))) for which Ω has a positive fractional Hardy- Schrödinger mass m_(γ,λ)^α (Ω). This latter notion is introduced by way of an invariant of the linear equation (L_(γ,α)-λΙ)u= 0 on Ω. We then study the effect of non-linear perturbation 〖h(x)u〗^(q-1), where h∈C^0 (¯Ω), h≥0 and 2<q<2_α^*. Our analysis shows that the existence of solutions is guaranteed by this perturbation whenever 0≤γ≤γ_crit (α), while for γ_crit (α)<γ<γ_H (α), it depends on both the perturbation and the geometry of the domain.

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2017-07-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/62397

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/