Critical exponent for half-Laplacian in the whole space

2015/10/03 by Giacomoni, Jacques, Mishra, Pawan, Sreenadh, Konijeti
#36J65 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.00804

Abstract

We study the existence of weak solutions for fractional elliptic equations of the type, (-Δ)^(1)/(2) u+ V(x) u= h(u), ugt; 0 \textrmin \mathbb R, %where 12, 10, K(x)>0, f is continuous and sign changing. where h is a real valued function that behaves like e^u2 as u→ ∞ and V(x) is a positive, continuous unbounded function. Here (-Δ)^(1)/(2) is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t=0. We also study the corresponding critical exponent problem for the Kirchhoff equation m(∫_{\mathbb R}|(-Δ)^(1)/(2)u|² dx+ ∫_\mb R u² V(x)dx)((-Δ)^(1)/(2) u+ V(x) u)= f(u) in \mathbb R where f(u) behaves like e^u2 as u→ ∞ and f(u)∼ u³ as u→ 0.

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