Existence and multiplicity of solutions for Schrödinger equations with asymptotically linear nonlinearities allowing interaction with essential spectrum

Linjie Song ()
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Linjie Song: AMSS, Academia Sinica

Partial Differential Equations and Applications, 2022, vol. 3, issue 2, 1-19

Abstract: Abstract We study the nonlinear problem $$-\Delta u + V(x)u = f(u), x \in \mathbb {R}^{N}, \lim _{ |x| \rightarrow \infty } u(x) = 0$$ - Δ u + V ( x ) u = f ( u ) , x ∈ R N , lim | x | → ∞ u ( x ) = 0 , where the Schrödinger operator $$-\Delta + V$$ - Δ + V is positive and f is asymptotically linear. Moreover, $$\lim _{|x| \rightarrow \infty } V(x) = \sigma _{0}$$ lim | x | → ∞ V ( x ) = σ 0 . We allow the interference of essential spectrum, i.e. $$\sup _{t \ne 0}f(t)/t \ge \sigma _{0}$$ sup t ≠ 0 f ( t ) / t ≥ σ 0 . If $$\sup _{t \ne 0}2F(t)/t^{2} \sigma _{0}\}) > 0$$ m e s ( { x ∈ R N : V ( x ) > σ 0 } ) > 0 .

Keywords: Morse theory; Schrödinger equations; Asymptotical linearity; Essential spectrum; 35A15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00162-7

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