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August, 2018 A Critical Nonlinear Elliptic Equation with Nonlocal Regional Diffusion
César E. Torres Ledesma
Taiwanese J. Math. 22(4): 909-930 (August, 2018). DOI: 10.11650/tjm/170905

Abstract

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent \[ \epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha} u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \quad \textrm{in $\mathbb{R}^{n}$}, \quad u \in H^{\alpha}(\mathbb{R}^{n}), \] where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q \in (1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*} = 2n/(n-2\alpha)$ is the critical Sobolev exponent, $\lambda \gt 0$ is a parameter and $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional Laplacian, whose range of scope is a ball with radius $\rho(x) \gt 0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon \to 0$.

Citation

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César E. Torres Ledesma. "A Critical Nonlinear Elliptic Equation with Nonlocal Regional Diffusion." Taiwanese J. Math. 22 (4) 909 - 930, August, 2018. https://doi.org/10.11650/tjm/170905

Information

Received: 7 June 2017; Accepted: 19 September 2017; Published: August, 2018
First available in Project Euclid: 14 October 2017

zbMATH: 06965403
MathSciNet: MR3830827
Digital Object Identifier: 10.11650/tjm/170905

Subjects:
Primary: 35A15 , 35B25 , 35J60 , 45G05

Keywords: Critical exponent , fractional Sobolev spaces , ground state solutions , non local regional Laplacian

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 4 • August, 2018
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