Nonlinear equations and weighted norm inequalities

Abstract

Abstract. We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem −∆u = v u q + w, u ≥ 0 on Ω,on a regular domain Ω in R n in the "superlinear case" q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space.The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary "integration by parts" argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

Existence results for semilinear elliptic equations with small measure data

“…These equations have been widely studied. Especially in [1,2,11], the authors give a sufficient and necessary condition for the existence of a solution of equations closed to (1.2) in the case p = 2, but their method doesn't extend to p = 2. See also [15] for the case of an eigenvalue problem.…”

Quasilinear elliptic equations with exponential nonlinearity and measure data

“…The problem of existence and pointwise estimates of solutions to Equation involving nonlinearity and measure data is a heavily studied topic. Equation with P ( u )= u q has been investigated by numerous authors . In particular, Phuc and Verbitsky, extending early works, established necessary and sufficient conditions involving Bessel capacities or Wolff potentials for the existence of solutions to Equation in the superlinear case q > p −1, sharp pointwise and integral estimates for solutions to corresponding nonlinear inequalities also be given.…”

On the Ptolemy constant of Lorentz sequence spaces and their duals

“…Moreover, the Ptolemy constant C p (X ) turns out to be useful in the study of the equivalence of Green's functions of second-order linear elliptic operators 13 . It is also a useful tool in the study of the existence of positive solutions of certain nonlinear equations 14 . As mentioned above, it is thus meaningful to calculate the exact value of some constants in some concrete spaces [15][16][17] .…”