Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

Abstract

We show that every weak supersolution of a variable exponent p-Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically Hölder continuous. As a technical tool we derive Harnacktype estimates for possibly unbounded supersolutions.

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

“…The variable Harnack inequality in the above form was proved by Alkhutov [9] (see also Alkhutov and Krasheninnikova [10]) and subsequently improved to embrace the case of unbounded solutions by Harjulehto et al [36,Theorem 3.9]. There, c H depends only on n, p and the L q s (B(w, 4r ))-norm of u for 1 < q < n n−1 and…”

“…Indeed, by Theorem 3.5 in Adamowicz et al [3], we have that u * is a (quasicontinuous) supersolution. Furthermore, Corollary 4.7 in Harjulehto et al [36] implies that the p(·)-capacity of the polar set of u * is zero, i.e., C p(·) ({u * = ∞}) = 0, see Definition 2.1, also [3] and [36] for further discussion. Thus, m < ∞.…”

Harnack's inequality for general solutions with nonstandard growth

“…Our result is optimal in the sense that even all known proofs for Harnack's inequality of the p(x)-Laplace equation end up with the additional term R in (1.1), see e.g. [8], [9] and Chapter 13 of [3]. Notice also that the additional R-term appears for the non-homogenous equations even in the case of constant exponent p, see [15] and [6], Chapter 7.5.…”

“…The following Caccioppoli estimate is the key result of this paper; it corresponds to Lemma 3.2 of [8].…”

“…In his paper the constant corresponding to C in (1.1) depends also on the L ∞ (B(x, 4R))-norm of the function. Later, Harjulehto, Kinnunen and Lukkari [8] were able to improve the argumentation so that their constant C depends on the L q (B(x, 4R))-norm of u, 0 < q < ∞, instead of the L ∞ (B(x, 4R))-norm. Here the exponent q can be made arbitrarily small by choosing R small enough in (1.1).…”