Given $(M,g)$ a smooth compact Riemannian manifold of dimension $n \ge 3$, there exist $A, B > 0$ such that for any $u \in H_1^2(M)$, $$\Vert u\Vert_{2^\star}^2 \le A\Vert\nabla u\Vert_2^2 + B\Vert u\Vert_1^2 , $$ where $H_1^2(M)$ is the standard Sobolev space consisting of functions in $L^2(M)$ whose gradient is also in $L^2(M)$. The best possible $A$ in this inequality is $K_n^2$, where $K_n$ is the sharp constant $K$ in the Euclidean Sobolev inequality $\Vert u\Vert_{2^\star} \le K\Vert\nabla u\Vert_2$. Thanks to previous work by Druet-Hebey-Vaugon and Hebey, it turns out that the above inequality with $A = K_n^2$ is always true when $n = 3$, in other words without any assumption on the manifold, and true when $n = 4$ if the scalar curvature is everywhere negative, or the scalar curvature is nonpositive and the manifold is conformally flat, or the sectional curvature is nonpositive and a local isoperimetric inequality as in the Cartan-Hadamard conjecture holds. On the contrary, thanks to previous works by Druet-Hebey-Vaugon, the inequality with $A = K_n^2$ is false when $n \ge 4$ and the scalar curvature is positive somewhere. Independent considerations give that for any $\varepsilon > 0$ there exists $B_\varepsilon$ such that for any $u\in H_1^2(M)$, $$\Vert u\Vert_{2^\star}^2 \le (K_n^2+\varepsilon)\Vert\nabla u\Vert_2^2 + B_\varepsilon\Vert u\Vert_1^2 . $$ Defining $B_\varepsilon(g)$ as the smallest $B_\varepsilon$ in this inequality, the difficult question we are concerned with in this article is to describe the asymptotic behavior of $B_\varepsilon(g)$ as $\varepsilon\to 0$. A complete answer to this question is given.

For a wide class of nonlinearities $f(u)$ satisfying \[\mbox{ $f(0)=f(a)=0$, $f(u)>0$ in $(0,a)$ and $f(u) < 0$ in $(a,\infty)$,}\] we study the quasilinear equation $-\Delta_p u=\lambda c(x)f(u)$ over the entire $\mathbb{R}^N$ or over a bounded smooth domain $\Omega$. Such equation covers various models from chemical reaction theory and population biology. We show that any nonnegative solution on the entire $\mathbb{R}^N$ must be a constant, and for large $\lambda$, any positive solution on $\Omega$ must approach $a$ in compact subsets of $\Omega$, no matter whether or not the solution has a prescribed behavior near the boundary of $\Omega$. We also determine the flat cores of the positive solutions and show that the flat cores enlarge to the whole $\Omega$ as $\lambda$ goes to infinity. Our proof of these results demonstrates the usefulness of boundary blow-up solutions in various classical problems.

We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate for the semigroup $(e^{tA})$ and it is strictly weaker than the assumption that the $e^{tA}$ are integral operators whose kernels satisfy Gaussian estimates. As an application we present new results for the maximal regularity of Schr\"odinger operators with singular potentials, elliptic higher order operators with bounded measurable coefficients, and elliptic second order operators with singular lower order terms. Moreover, we prove a similar result for maximal regularity of the discrete time evolution equation $ u_{n+1} - Tu_n = f_n ,$ $n\in\mathbb N_0 ,$ $u_0=0 $.