Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in RN View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2017-04-03

AUTHORS

Vincenzo Ambrosio, Hichem Hajaiej

ABSTRACT

This paper is concerned with the following fractional Schrödinger equation (-Δ)su+u=k(x)f(u)+h(x)inRNu∈Hs(RN),u>0inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$\end{document}where s∈(0,1),N>2s,(-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,1),N> 2s, (-\Delta )^{s}$$\end{document} is the fractional Laplacian, k is a bounded positive function, h∈L2(RN),h≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0$$\end{document} is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that |h|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|h|_{2}$$\end{document} is sufficiently small. More... »

PAGES

1-25

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10884-017-9590-6

DOI

http://dx.doi.org/10.1007/s10884-017-9590-6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1084518802


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