Lane-Emden equations and the geometry of Sobolev-Poincaré inequalities

This thesis is addressed to the investigation of the optimal constants in the Sobolev-Poincaré inequalities and in the Hardy inequalities, as well as, to the study of the Lane-Emden equation for the $p-$Laplacian, with subhomogeneous power $q$ in the right-hand side. First, we prove a comparison principle for positive supersolutions and subsolutions of the Lane-Emden equations, then, as applications, we give some results concerning such solutions when defined on convex sets. By exploiting the previous results, we discuss the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function, thanks to a preliminary study for the sharp constants in Morrey-type and Hardy-type inequalities for general open sets. In turn, this analysis permits to study the asymptotic behaviour of both the optimal constants in the Sobolev-Poincaré inequalities and the positive solutions of the Lane-Emden equation, as the exponent $p$ diverges to $\infty$, under optimal assumptions on the open set. We also give some geometric lower bounds for sharp Sobolev-Poincaré constants $\lambda_{p,q}$, when $q < p$, on the class of convex bounded open sets: indeed, we prove that $\lambda_{p,q}$ can be bounded from below both in terms of the norm of the distance function in a suitable Lebesgue space, and in terms of a certain power of the inradius of the set. The results so obtained generalize the Makai inequality and the Hersch-Protter inequality, respectively. Finally, in the last part of the thesis, we study the sharp constant in the Hardy inequality in the setting of fractional Sobolev spaces $W^{s,p}$ defined on general open sets, for every $0<s<1$. We first list some properties of such a constant and we give a variational characterization, which extends an analogous well-known result for the local case. Then, focusing on the class of convex open sets, we compute the sharp fractional Hardy constant in the regime $s\,p \ge 1$, by constructing suitable supersolutions for the associated equation by means of the distance function. We note that we can avoid the claimed restriction on $s\,p$ when the convex set is an half-space.