On weak solutions for $p$-Laplacian equations with weights

Abstract

Partial differential equations.-On the regularity of weak solutions to H-systems, by ROBERTA MUSINA. ABSTRACT.-We prove that every weak solution to the H-surface equation is locally bounded, provided the prescribed mean curvature H satisfies a suitable condition at infinity. No smoothness assumption is required on H. We also consider the Dirichlet problem for the H-surface equation on a bounded regular domain with L ∞ boundary data and the H-bubble problem. Under the same assumptions on H , we prove that every weak solution is globally bounded.

Combined effects in nonlinear problems arising in the study of anisotropic continuous media

“…It remains to argue that the non-negative weak solution u is, in fact, positive. Indeed, using similar arguments as in Pucci and Servadei [36], which are based on the Moser iteration, we obtain that u ∈ L ∞ (Ω). Next, by bootstrap regularity, u is a classical solution of problem (P λ ) − .…”

Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights

“…The main results of this paper and of [41] were presented in a unified way, but without proofs, in the survey [40].…”

Eigenvalue problems with unbalanced growth: Nonlinear patterns and standing wave solutions

“…Next, as in the proof of Theorem 3(b) and using Theorem 1(ii) of Pucci and Servadei [24] in combination with the Moser iteration, we deduce that U ∈ L ∞ loc (Ω). This regularity property implies that U ∈ C 1,µ (Ω ∩ B R (0)), where µ = µ(R) ∈ (0, 1) Applying the generalized Pucci-Serrin maximum principle, as in the proof of Theorem 3(c), we conclude that U > 0 in Ω.…”