Quasilinear elliptic equations with sub-natural growth terms in bounded domains

Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

“…In previous work [24], the author proved a decomposition theorem of measures and provided an existence theorem for equations of the type (1.1). The sufficient condition was given by the L p (w)-L q (σ) trace inequality of Sobolev type…”

“…Claims that are equivalent to these were first given by Seesanea and Verbitsky [42] for linear uniformly elliptic operators. In [24], their arguments were applied to weighted (p, w)-Laplace operators, and Theorem 1.1 was proved with a different formulation. We give a direct proof of the existence theorem for (1.1) and further discuss what occurs when condition (1.2) is relaxed.…”

“…Their work shows a connection between the existence of positive solutions to (1.4), energy conditions of the type of (4.1) (or equivalent (1.2)) and certain weighted norm inequalities. An extension of their result to (p, w)-Laplace equations on bounded domains was given by the author [24] (see also [25]). The counterpart of the Cascante-Ortega-Verbitsky theorem was also presented.…”

“…Next, we introduce classes of smooth measures. For detail, see [24] and the references therein. Let S 0 (Ω) be the set of all Radon measures µ satisfying…”

“…For detail, see [24]. From the argument in [24, Theorem 3.1], the following comparison principle holds.…”