Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions

Abstract

The study of the regularity of solutions to the elliptic equations with non-standard growth has been initiated by Zhikov, Marcellini, and Lieberman, and in the last thirty years, the qualitative theory of second-order elliptic and parabolic equations has been actively developed. Equations of this type and systems of such equations arise in various problems of mathematical physics and engineering (e.g. in describing electrorheological fluids, or in image recognition and data denoising). There are two cases of the type of growth. The simple so-called ''logarithmic'' case is studied very well and there are a lot of classical results in this regard. But the so-called ''non-logarithmic'' growth differs substantially from the logarithmic case. The non-logarithmic condition introduced by Zhikov turned out to be a precise condition for the smoothness of finite functions in the corresponding Sobolev space, which makes it extremely interesting to study. But to our knowledge, there are only a few results in this direction. Zhikov and Pastukhova proved higher integrability of the gradient of solutions to the $p(x)$-Laplace equation under the non-logarithmic condition. Interior continuity, continuity up to the boundary, and Harnack's inequality to the $p(x)$-Laplace equation were proved by Alkhutov, Krasheninnikova, and Surnachev. These results were generalized by Skrypnik and Voitovich. The qualitative properties of bounded solutions of $p(x)$-Laplace equation under the non-logarithmic condition were established by Skrypnik and Yevgenieva. As for unbounded solutions, there are just a few results. Ok has proved the boundedness of minimizers of elliptic functionals of the double-phase type under some assumptions on the growth parameters. The obtained condition gives a possibility to improve the regularity results for unbounded minimizers. The weak Harnack inequality for unbounded supersolutions of the corresponding elliptic equations with generalized Orlicz growth under the so-called logarithmic conditions was proved by Benyaiche, Harjulehto, H\"{a}st\"{o} and Karppinen. In the current paper, the weak Harnack inequality for unbounded solutions to the $p(x)$-Laplace equation has been proved under the precise non-logarithmic condition on the function $p(x)$.