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
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
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.
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do Ó, J.M., Yang, Y. A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary. Ann Glob Anal Geom 38, 317–334 (2010). https://doi.org/10.1007/s10455-010-9218-0
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DOI: https://doi.org/10.1007/s10455-010-9218-0