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Compactness results for divergence type nonlinear elliptic equations

Authors:

Institution:

(1) Departamento de Matemática, Universidade Federal do Paraná, Caixa Postal 019081, Curitiba, PR, 81531-990, Brazil;(2) Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, MG, 30123-970, Brazil

Abstract:

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation

$-\frac{1}{p} {\rm div}_{g_\alpha}\left( \nabla^\xi_{g_\alpha}F_\alpha(\nabla_{g_\alpha} u) \right) + a_\alpha(x) |u|^{p-2} u = |u|^{p^* - 2}u\quad {\rm in} \ \ M.$

This question is directly connected to the study of the first best constant ${A^\alpha_{\rm opt}}$ associated with the Riemannian F _α-Sobolev inequality

$\left(\ \int\limits_M |u|^{p^*} dv_{g_\alpha} \right)^{\frac{p}{p^*}}\leq A \int\limits_M F_\alpha(\nabla_{g_\alpha} u) dv_{g_\alpha} + B \int\limits_M |u|^p dv_{g_\alpha}.$

Precisely, we need to know the dependence of ${A^\alpha_{\rm opt}}$ under F _α and g _α. For that, we obtain its value as the supremum on M of best constants associated with certain homogeneous Sobolev inequalities on each tangent space and show that ${A^\alpha_{\rm opt}}$ is attained on M. We then establish the continuous dependence of ${A^\alpha_{\rm opt}}$ in relation to F _α and g _α. The tools used here are based on convex analysis, blow-up, and variational approach.

Keywords:

Critical Sobolev exponents Divergence type equations Compactness of solutions

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