A class of Adams–Fontana type inequalities and related functionals on manifolds

2010

Ann Glob Anal Geom

Self Cite

Let (M, g) be an N-dimensional compact Riemannian manifold without boundary. When m is a positive integer strictly smaller than N, we prove thatwhere u m,N/m is the usual Sobolev norm of u ∈ W m,N/m (M), and α N,m is the best constant in Adams' original inequality (Ann. Math., 1988). This is a modified version of Adams' inequality on compact Riemannian manifold which has been proved by L. Fontana (Comment. Math Helv., 1993). Using the above inequality in the case when m = 1, we establish sufficient conditions under which the quasilinear equationhas a nontrivial positive weak solution in

Section: Let (Mmentioning

confidence: 68%

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

2010

Ann Glob Anal Geom

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“…on the Sobolev space W 1,2 (Σ, R). The existence of nonnegative solutions to equation (1.2) in case that τ k is a positive real number was studied by Zhao and the author [16] by using variational methods. More explicitly, assuming that λ τ = λ τ (Σ) is the first eigenvalue of the operator ∆ g + τ, where τ > 0 is a constant, we proved that the equation ∆ g u + τu = λue u 2 has a nonnegative solution if λ < λ τ .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary

2014

Calc. Var.

Self Cite

“…This problem was studied by Yamabe [24], Trudinger [22], Aubin [4], and completely solved by Schoen [19]. Though there is no background of geometry or physics, there are still some works concerning the problem (2) on Riemannian manifolds, see for examples [28,11,30,27].…”

Section: Introductionmentioning

confidence: 99%

Yamabe type equations on graphs

Grigor’yan

Lin

Journal of Differential Equations

2016

Self Cite

115

Let G = (V, E) be a locally finite graph, Ω ⊂ V be a bounded domain, ∆ be the usual graph Laplacian, and λ 1 (Ω) be the first eigenvalue of −∆ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if α < λ 1 (Ω), then for any p > 2, there exists a positive solution towhere Ω • and ∂Ω denote the interior and the boundary of Ω respectively. Also we consider similar problems involving the p-Laplacian and poly-Laplacian by the same method. Such problems can be viewed as discrete versions of the Yamabe type equations on Euclidean space or compact Riemannian manifolds.