Studies of normalized solutions to Schrödinger equations with Sobolev critical exponent and combined nonlinearities

2021/04/27 by Li, Xinfu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.12997

Abstract

We consider the Sobolev critical Schrödinger equation with combined nonlinearities \begincases -Δu=λu+|u|^{2^*-2}u+μ|u|^q-2u, x∈ℝ^N,
u∈ H¹(ℝ^N), ∫_ℝ^N|u|²dx=a, \endcases where N≥ 3, μ>0, λ∈ ℝ, a>0 and q∈ (2,2^*). We prove in this paper (1) Multiplicity and stability of solutions for q∈ (2,2+(4)/(N)) and μa^{(q(1-γ_q))/(2)}≤ (2K)^{(qγ_q-2^*)/(2^*-2)} with γ_q:=(N)/(2)-(N)/(q) and K being some positive constant. This result extends the results obtained in Jeanjean et al. \citeJEANJEAN-JENDREJ and Jeanjean and Le \citeJeanjean-Le for the case μa^{(q(1-γ_q))/(2)}

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