Taiwanese Journal of Mathematics

A Critical Nonlinear Elliptic Equation with Nonlocal Regional Diffusion

César E. Torres Ledesma

Full-text: Open access

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 $\varepsilon \to 0$.

Article information

Source
Taiwanese J. Math., Volume 22, Number 4 (2018), 909-930.

Dates
Received: 7 June 2017
Accepted: 19 September 2017
First available in Project Euclid: 14 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1507946428

Digital Object Identifier
doi:10.11650/tjm/170905

Mathematical Reviews number (MathSciNet)
MR3830827

Zentralblatt MATH identifier
06965403

Subjects
Primary: 45G05: Singular nonlinear integral equations 35A15: Variational methods 35J60: Nonlinear elliptic equations 35B25: Singular perturbations

Keywords
non local regional Laplacian critical exponent fractional Sobolev spaces ground state solutions

Citation

Torres Ledesma, César E. A Critical Nonlinear Elliptic Equation with Nonlocal Regional Diffusion. Taiwanese J. Math. 22 (2018), no. 4, 909--930. doi:10.11650/tjm/170905. https://projecteuclid.org/euclid.twjm/1507946428


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