One-dimensional symmetry of positive bounded solutions to the nonlinear Schrödinger equation in the half-space in two and three dimensions

Christos Sourdis ()
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Christos Sourdis: National and Kapodistrian University of Athens

Partial Differential Equations and Applications, 2021, vol. 2, issue 6, 1-6

Abstract: Abstract We are concerned with the half-space Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta v+v=|v|^{p-1}v &{} \text {in}\ \mathbb {R}^N_+, \\ v=c\ \text {on}\ \partial \mathbb {R}^N_+, &{}\lim _{x_N\rightarrow \infty }v(x',x_N)=0\ \text {uniformly in}\ x'\in \mathbb {R}^{N-1}, \end{array}\right. \end{aligned}$$ - Δ v + v = | v | p - 1 v in R + N , v = c on ∂ R + N , lim x N → ∞ v ( x ′ , x N ) = 0 uniformly in x ′ ∈ R N - 1 , where $$\mathbb {R}^N_+=\{x\in \mathbb {R}^N \ : \ x_N>0\}$$ R + N = { x ∈ R N : x N > 0 } for some $$N\ge 2$$ N ≥ 2 , and $$p>1$$ p > 1 , $$c>0$$ c > 0 are constants. It was shown recently by Fernandez and Weth (Math Ann, 2021) that there exists an explicit number $$c_p\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, such that for $$0 c_p$$ c > c p there are no bounded positive solutions. If $$N=2, \ 3$$ N = 2 , 3 , we show that in the case where $$c = c_p$$ c = c p there is no other bounded positive solution besides the one-dimensional one.

Keywords: Semilinear elliptic problems; One-dimensional symmetry; Positive solutions; Entire solutions; Nonlinear Schrödinger equation; 35B08; 35B50; 35J15; 35J61 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00125-4

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