-
Previous Article
The stability of the equilibrium for a perturbed asymmetric oscillator
- CPAA Home
- This Issue
-
Next Article
Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system
Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian
1. | Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Av. Diagonal 647. 08028 Barcelona |
2. | Centro de Matemática, Universidade de Coimbra, Apartado 3008, 3001–454 Coimbra, Portugal |
Under some assumptions on $f$ that make its growth comparable to $u^m$, we prove that every semi-stable solution is bounded if $m < m_{c s}$. Here, $m_{c s}=m_{c s}(N,p)$ is an explicit exponent which is optimal for the boundedness of semi-stable solutions. In particular, it is bigger than the critical Sobolev exponent $p^\star-1$.
We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and $1 < p < 2$.
[1] |
Avner Friedman, Kang-Ling Liao. The role of the cytokines IL-27 and IL-35 in cancer. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1203-1217. doi: 10.3934/mbe.2015.12.1203 |
[2] |
Francisco Pedroche, Regino Criado, Esther García, Miguel Romance, Victoria E. Sánchez. Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks and Heterogeneous Media, 2015, 10 (1) : 101-125. doi: 10.3934/nhm.2015.10.101 |
[3] |
Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1031-1041. doi: 10.3934/dcdss.2020061 |
[4] |
Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48 |
[5] |
Thomas Hagen, Andreas Johann, Hans-Peter Kruse, Florian Rupp, Sebastian Walcher. Dynamical systems and geometric mechanics: A special issue in Honor of Jürgen Scheurle. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : i-iii. doi: 10.3934/dcdss.20204i |
[6] |
Piotr Jaworski, Marcin Pitera. The 20-60-20 rule. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1149-1166. doi: 10.3934/dcdsb.2016.21.1149 |
[7] |
Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050 |
[8] |
Pierluigi Colli, Gianni Gilardi, Dietmar Hömberg, Pavel Krejčí, Elisabetta Rocca. Preface: Special issue dedicated to Jürgen Sprekels on the occasion of his 65th birthday. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : i-ii. doi: 10.3934/dcds.2015.35.6i |
[9] |
Vladimir Ejov, Anatoli Torokhti. How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293 |
[10] |
Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial and Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779 |
[11] |
Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun. Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021056 |
[12] |
Peter E. Kloeden, Yuan Lou. Preface for the special issue "20 years of DCDS-B". Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : i-ii. doi: 10.3934/dcdsb.2020372 |
[13] |
Guoliang Xue, Weiyi Zhang, Tie Wang, Krishnaiyan Thulasiraman. On the partial path protection scheme for WDM optical networks and polynomial time computability of primary and secondary paths. Journal of Industrial and Management Optimization, 2007, 3 (4) : 625-643. doi: 10.3934/jimo.2007.3.625 |
[14] |
Robert Stephen Cantrell, Suzanne Lenhart, Yuan Lou, Shigui Ruan. Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : i-ii. doi: 10.3934/dcdsb.2014.19.1i |
[15] |
Guohong Zhang, Xiaoli Wang. Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3755-3786. doi: 10.3934/dcdsb.2018076 |
[16] |
Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305 |
[17] |
Carlo Cercignani. The Boltzmann equation in the 20th century. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 83-94. doi: 10.3934/dcds.2009.24.83 |
[18] |
Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074 |
[19] |
Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic and Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373 |
[20] |
Vladimir V. Marchenko, Klavdii V. Maslov, Dmitry Shepelsky, V. V. Zhikov. E.Ya.Khruslov. On the occasion of his 70th birthday. Networks and Heterogeneous Media, 2008, 3 (3) : 647-650. doi: 10.3934/nhm.2008.3.647 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]