We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...

We consider the nonlinear half-Laplacian heat equation $$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$We prove that all blows-up are type I, provided \(n \le 4\) and \( 1<p<p_{*} (n)\) where p_{*} is an explicit exponent which below \(\frac{n+1}{n-1}\), critical Sobolev exponent. Central to our proof a Giga-Kohn monotonicity formu...

We study superharmonic functions related to elliptic equations with structural conditions involving a variable growth exponent. We establish pointwise estimates for such functions in terms of a Wolff type potential. We apply these estimates to prove a variable exponent version of the Hedberg–Wolff theorem on the dual of Sobolev spaces with zero boundary values.