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Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability
Advances in Nonlinear Analysis ( IF 4.2 ) Pub Date : 2022-03-09 , DOI: 10.1515/anona-2022-0230 Xinfu Li 1
Advances in Nonlinear Analysis ( IF 4.2 ) Pub Date : 2022-03-09 , DOI: 10.1515/anona-2022-0230 Xinfu Li 1
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In this article, we consider the upper critical Choquard equation with a local perturbation − Δ u = λ u + ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + μ ∣ u ∣ q − 2 u , x ∈ R N , u ∈ H 1 ( R N ) , ∫ R N ∣ u ∣ 2 = a , \left\{\begin{array}{l}-\Delta u=\lambda u+\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u+\mu | u\hspace{-0.25em}{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1em}{\displaystyle \int }_{{{\mathbb{R}}}^{N}}| u\hspace{-0.25em}{| }^{2}=a,\end{array}\right. where N ≥ 3 N\ge 3 , μ > 0 \mu \gt 0 , a > 0 a\gt 0 , λ ∈ R \lambda \in {\mathbb{R}} , α ∈ ( 0 , N ) \alpha \in \left(0,N) , p = p ¯ ≔ N + α N − 2 p=\bar{p}:= \frac{N+\alpha }{N-2} , q ∈ 2 , 2 + 4 N q\in \left(2,2+\frac{4}{N}\right) , and I α = C ∣ x ∣ N − α {I}_{\alpha }=\frac{C}{| x{| }^{N-\alpha }} with C > 0 C\gt 0 . When μ a q ( 1 − γ q ) 2 ≤ ( 2 K ) q γ q − 2 p ¯ 2 ( p ¯ − 1 ) \mu {a}^{\tfrac{q\left(1-{\gamma }_{q})}{2}}\le {\left(2K)}^{\tfrac{q{\gamma }_{q}-2\bar{p}}{2\left(\bar{p}-1)}} with γ q = N 2 − N q {\gamma }_{q}=\frac{N}{2}-\frac{N}{q} and K K being some positive constant, we prove (1) Existence and orbital stability of the ground states. (2) Existence, positivity, radial symmetry, exponential decay, and orbital instability of the “second class” solutions. This article generalized and improved parts of the results obtained for the Schrödinger equation.
更新日期：2022-03-09
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