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- Publisher
- Springer Journals
- Copyright
- Copyright © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
- ISSN
- 0126-6705
- eISSN
- 2180-4206
- DOI
- 10.1007/s40840-020-00995-8
- Publisher site
- See Article on Publisher Site
Abstract
We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator LK\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}_K$$\end{document}-LKu=λu+f(x,u),inΩ,u=0,inRn\Ω,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega $$\end{document} be an open-bounded set of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^n$$\end{document} with continuous boundary, λ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda \in {\mathbb {R}}$$\end{document} is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with λ<λ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda <\lambda _1$$\end{document}, where λ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _1$$\end{document} is the first eigenvalue of the nonlocal operator -LK\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-{\mathcal {L}}_K$$\end{document} with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with λ≥λ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda \ge \lambda _1$$\end{document}. As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian (-Δ)αu=λu+f(x,u),inΩ,u=0,inRn\Ω,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$\end{document}where 0<α<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\alpha <1$$\end{document}. The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.
Journal
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Aug 20, 2020