2011/09/21 by Gilles Évéquoz, Évéquoz, Gilles, Tobias Weth +1
Computer Science · Mathematics · #35J10 (Primary) 35A15 #35J60 #47H11 (Secondary) #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #math.AP #msc:35A15 #msc:35J10 #msc:35J60 #msc:47H11
paper · pdf · doi:10.48550/arxiv.1109.4550
31 pages
arxiv created 2011/09/21 · openalex publication_date 2011/09/21 · arxiv updated 2011/09/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider the stationary semilinear Schrödinger equation -Δu + a(x) u = f(x,u), u∈ H1(\RN), where a and f are continuous functions converging to some limits a_∞>0 and f_∞=f_∞(u) as |x|→∞. In the indefinite setting where the Schrödinger operator -Δ+a has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on a and f. We stress that for some of our results we also allow zero to lie in the spectrum of -Δ+ a.