Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability

By Xinfu Li. Advances in Nonlinear Analysis, 2021

57 citations recorded by the source corpus.

Abstract 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3550d2301b0ca23d2bd27d4793861c45e7bf1e41