ETD

• anisotropic
• functional
• minimum
• variational
• warped
• Yamabe

In this thesis we study an anisotropic Yamabe type equation, with critical exponent, using a variational method. We find a positive minimum for the functional associated to the differential equation, which is a weak solution for the equation, and next we prove that this minimum is smooth also and then is a strong solution for the anisotropic equation. Since we have a critical exponent we have a lack of compactness and in this work we find a condition on the functional which ensure compactness for minimizing sequences. We look for an energetic level for the infimum of the functional under which we recover compactness. So we expand the functional arround a particular family of bubble function, depending on a real parameter. We find a Best Constant type inequality for Sobolev embedding in the presence of an anisotropi coefficient. As an application of the main part of the thesis we apply our results to find a solution of a supercritical Yamabe type equation on a warped product Riemannian manifold.