Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation

2016/06/02 by Gilles Évéquoz, Gilles Evéquoz, Evéquoz, Gilles +2
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.AP

paper · pdf · doi:10.48550/arxiv.1606.00606

Revised version; to appear in 'Journal of Fixed Point Theory and Applications', Special issue in honour of Paul Rabinowitz

openalex publication_date 2016/06/02 · arxiv created 2016/10/04 · arxiv updated 2016/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the nonlinear stationary Schrödinger equation -Δu -λu= Q(x)|u|^p-2u, in ℝ^N in the case where N ≥ 3, p is a superlinear, subcritical exponent, Q is a bounded, nonnegative and nontrivial weight function with compact support in ℝ^N and λ∈ ℝ is a parameter. Under further restrictions either on the exponent p or on the shape of Q, we establish the existence of a continuous branch C of nontrivial solutions to this equation which intersects \λ\ × L^s(ℝ^N) for every λ∈ (-∞, λ_Q) and s> (2N)/(N-1). Here λ_Q>0 is an explicit positive constant which only depends on N and diam(supp Q). In particular, the set of values λ along the branch enters the essential spectrum of the operator -Δ.

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