Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term
In this article we study the quasilinear elliptic equation with absorption lower term $$ -\hbox{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0, \quad u\geq 0. $$ Despite of the lack of comparison principle, we prove a priori estimate of Keller-Osserman type. Particularly, und...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/91/abstr.html |
| _version_ | 1817971242007789568 |
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| author | Kateryna Buryachenko |
| author_facet | Kateryna Buryachenko |
| author_sort | Kateryna Buryachenko |
| collection | DOAJ |
| description | In this article we study the quasilinear elliptic equation with absorption
lower term
$$
-\hbox{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0,
\quad u\geq 0.
$$
Despite of the lack of comparison principle, we prove a priori
estimate of Keller-Osserman type. Particularly, under some natural
assumptions on the functions g,f for nonnegative solutions we
prove an estimate of the form
$$
\int_0^{u(x)} f(s)\,ds\leq c\frac{u(x)}{r}g\big(\frac{u(x)}{r}\big),\quad
x\in\Omega, B_{8r}(x)\subset\Omega,
$$
with constant c, independent on u(x). Using this estimate we
give a simple proof of the Harnack inequality. |
| first_indexed | 2024-04-13T20:44:16Z |
| format | Article |
| id | doaj.art-363fb0b57467416bb62d7e474ca587a8 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-13T20:44:16Z |
| publishDate | 2018-04-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-363fb0b57467416bb62d7e474ca587a82022-12-22T02:30:44ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912018-04-01201891,19Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order termKateryna Buryachenko0 Vasyl Stus Donetsk National Univ., Vinnytsia, Ukraine In this article we study the quasilinear elliptic equation with absorption lower term $$ -\hbox{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0, \quad u\geq 0. $$ Despite of the lack of comparison principle, we prove a priori estimate of Keller-Osserman type. Particularly, under some natural assumptions on the functions g,f for nonnegative solutions we prove an estimate of the form $$ \int_0^{u(x)} f(s)\,ds\leq c\frac{u(x)}{r}g\big(\frac{u(x)}{r}\big),\quad x\in\Omega, B_{8r}(x)\subset\Omega, $$ with constant c, independent on u(x). Using this estimate we give a simple proof of the Harnack inequality.http://ejde.math.txstate.edu/Volumes/2018/91/abstr.htmlHarnack inequalityquasilinear elliptic equationKeller-Osserman type estimateabsorption lower term |
| spellingShingle | Kateryna Buryachenko Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term Electronic Journal of Differential Equations Harnack inequality quasilinear elliptic equation Keller-Osserman type estimate absorption lower term |
| title | Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term |
| title_full | Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term |
| title_fullStr | Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term |
| title_full_unstemmed | Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term |
| title_short | Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term |
| title_sort | harnack inequality for quasilinear elliptic equations with p q growth conditions and absorption lower order term |
| topic | Harnack inequality quasilinear elliptic equation Keller-Osserman type estimate absorption lower term |
| url | http://ejde.math.txstate.edu/Volumes/2018/91/abstr.html |
| work_keys_str_mv | AT katerynaburyachenko harnackinequalityforquasilinearellipticequationswithpqgrowthconditionsandabsorptionlowerorderterm |