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- Publisher
- Springer Journals
- Copyright
- Copyright © Tusi Mathematical Research Group (TMRG) 2020
- ISSN
- 2639-7390
- eISSN
- 2008-8752
- DOI
- 10.1007/s43034-020-00101-w
- Publisher site
- See Article on Publisher Site
Abstract
We are concerned with the following p-Laplacian equation -Δpu+|u|p-2u=μu+|u|s-2u,inRN,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} -\varDelta _{p} u+|u|^{p-2}u=\mu u+|u|^{s-2}u,~\text {in}~{\mathbb {R}}^{N}, \end{aligned}$$\end{document}where -Δpu=div(|∇u|p-2∇u)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\varDelta _{p}u=div(|\nabla u|^{p-2}\nabla u)$$\end{document}, 1<p<N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1<p<N$$\end{document}, μ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu \in {\mathbb {R}}$$\end{document}, s∈(N+2Np,p∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s\in (\frac{N+2}{N}p,p^{*})$$\end{document}, p∗=NpN-p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{*}=\frac{Np}{N-p}$$\end{document} is the critical Sobolev exponent. Using constrained variational methods, we prove that the above problem has a normalized solution. Our contribution is that we can deal with the L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{2}$$\end{document}-supercritical case (N+2Np,p∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\frac{N+2}{N}p ,p^{*})$$\end{document} by a mountain-pass argument on the prescribed L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{2}$$\end{document}-norm constraint.
Journal
Annals of Functional Analysis – Springer Journals
Published: Nov 19, 2020