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A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary

Content Provider	Springer Nature Link
Author	Yang, Yunyan
Copyright Year	2010
Abstract	Let (M, g) be an N-dimensional compact Riemannian manifold without boundary. When m is a positive integer strictly smaller than N, we prove that $$\sup_{\\|u\\|_{m,{N/m}}\leq 1}\int\limits_Me^{\alpha_{N,m}\|u\|^{N/(N-m)}}{\rm d}v_g< \infty,$$ where \|\|u\|\|m,N/m is the usual Sobolev norm of $${u\in W^{m,N/m}(M)}$$ , and α N,m is the best constant in Adams’ original inequality (Ann Math 128:385–398, 1988). This is a modified version of Adams’ inequality on compact Riemannian manifold which has been proved by Fontana (Comment Math Helv 68:415–454, 1993). Using the above inequality in the case when m = 1, we establish sufficient conditions under which the quasi-linear equation $$-\Delta_Nu+\tau\|u\|^{N-2}u=f(x,u)$$ has a nontrivial positive weak solution in W 1,N (M), where $${-\Delta_N u{\,=\,}-{\rm div}(\|\nabla u\|^{N-2}\nabla u), \tau >0 }$$ , and f (x,u) behaves like $${e^{\gamma \|u\|^{N/(N-1)}}}$$ as $${\|u\|\rightarrow \infty}$$ for some γ > 0.
Ending Page	334
Page Count	18
Starting Page	317
File Format	PDF
ISSN	0232704X
e-ISSN	15729060
Journal	Annals of Global Analysis and Geometry
Issue Number	3
Volume Number	38
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-07-18
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	N-Laplacian Theoretical, Mathematical and Computational Physics Trudinger–Moser inequality Nonlinear elliptic equations Variational principles Geometry Statistics for Business/Economics/Mathematical Finance/Insurance Analysis Group Theory and Generalizations Critical growth Elliptic equations on manifolds, general theory Critical exponents Adams inequality
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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