American Institute of Mathematical Sciences

Advanced
November  2009, 8(6): 1779-1793. doi: 10.3934/cpaa.2009.8.1779

Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$

Daniele Castorina 1, , Pierpaolo Esposito 2, and  Berardino Sciunzi 3,

1. 

Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
2. 

Dipartimento di Matematica, Universitá di Roma Tre, Largo San Leonardo Murialdo, 1, I-00146 Roma, Italy
3. 

Dipartimento di Matematica, Universitá della Calabria, V. P. Bucci, I-87036 Arcavacata di Rende (CS), Italy

Received  June 2008 Revised  April 2009 Published  August 2009

Stability properties for solutions of $-\Delta_m(u)=f(u)$ in $\mathbb{R}^N$ are investigated, where $N\geq 2$ and $m \geq 2$. The aim is to identify a critical dimension $N^\#$ so that every non-constant solution is linearly unstable whenever $2\leq N < N^\#$. For positive, increasing and convex nonlinearities $f(u)$, global bounds on $\frac{f \, f''}{(f')^2}$ allows us to find a dimension $N^\#$, which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with $C^1-$nonlinearities and the dimension $N^\#$ we find is still optimal.
Keywords: critical dimension., p−Laplace operator, linear instability.
Mathematics Subject Classification: Primary: 35J60, 35J70; Secondary: 35B0.
Citation: Daniele Castorina, Pierpaolo Esposito, Berardino Sciunzi. Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1779-1793. doi: 10.3934/cpaa.2009.8.1779
[1]

Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965
[2]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[3]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[4]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[5]

Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421
[6]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[7]

A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709
[8]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078
[9]

Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
[10]

Salvatore A. Marano, Sunra J. N. Mosconi. Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 279-291. doi: 10.3934/dcdss.2018015
[11]

Vladimir Bobkov, Mieko Tanaka. Remarks on minimizers for (p, q)-Laplace equations with two parameters. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1219-1253. doi: 10.3934/cpaa.2018059
[12]

Zheng Zhou. Layered solutions in $R^2$ for a class of $p$-Laplace equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 819-837. doi: 10.3934/cpaa.2010.9.819
[13]

Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024
[14]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[15]

Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013
[16]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177
[17]

Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115
[18]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
[19]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020040
[20]

Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences