Nehari manifold for a Schrödinger equation with magnetic potential involving sign-changing weight function

• Published in: Applicable analysis Vol. 103; no. 6; pp. 1036 - 1063
• Main Authors: de Paiva, Francisco Odair, de Souza Lima, Sandra Machado, Miyagaki, Olímpio Hiroshi
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 12.04.2024; Taylor & Francis Ltd
• Subjects: fibering map; magnetic potential; Manifolds; Nehari manifold; Schrodinger equation; Sign-changing weight function; Weighting functions
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2023.2230257

We consider the following class of elliptic problems \[ - \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,\quad x\in {\mathbb{R}}^N, \] − Δ A u + u = a λ ( x ) | u | q − 2 u + b μ ( x ) | u | p − 2 u , x ∈ R N , where $ 1<q<2<p<2^*= \frac {2N}{N-2} $ 1 < q < 2 < p < 2 ∗ = 2 N N − 2 $ N\geq ~3 $ N ≥ 3 , $ a_{\lambda }(x) $ a λ ( x ) is a sign-changing weight function, $ b_{\mu }(x) $ b μ ( x ) is continuous, $ \lambda > 0 $ λ > 0 and $ \mu > 0 $ μ > 0 are real parameters, $ u \in H^1_A({\mathbb {R}}^N) $ u ∈ H A 1 ( R N ) and $ A:{\mathbb {R}}^N \rightarrow {\mathbb {R}}^N $ A : R N → R N is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions.