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2016, 15(2): 519-533. doi: 10.3934/cpaa.2016.15.519

Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter

1. 

Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

Received  April 2015 Revised  September 2015 Published  January 2016

We examine the equation $$ \Delta^2 u = \lambda f(u) \qquad \Omega, $$ with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for the sytem \begin{eqnarray} &-\Delta u = \lambda f(v) \qquad \Omega, \\ &-\Delta v = \gamma g(u) \qquad \Omega, \\ &u= v = 0 \qquad \partial \Omega. \end{eqnarray}
Citation: Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure & Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519
References:
[1]

G. Arioli, F. Gazzola, H.-C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity,, \emph{SIAM J. Math. Anal.}, 36 (2005), 1226. doi: 10.1137/S0036141002418534.

[2]

E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities,, \emph{Electronic Journal of Differential Equations}, 34 (2005).

[3]

T. Boggio, Sulle funzioni di Green drdine m,, \emph{Rend. Circ. Mat. Palermo}, (1905), 97.

[4]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited,, \emph{Adv. Diff. Eq.}, 1 (1996), 73.

[5]

H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, \emph{Rev. Mat. Univ. Complut. Madrid}, 10 (1997), 443.

[6]

X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1362. doi: 10.1002/cpa.20327.

[7]

X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations,, \emph{J. Funct. Anal.}, 238 (2006), 709. doi: 10.1016/j.jfa.2005.12.018.

[8]

D. Cassani, J. M. do O and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity,, \emph{Adv. Nonlinear Stud.}, 9 (2009), 177.

[9]

L. B. Chaabane, On the extremal solutions of semilinear elliptic problems,, \emph{Abstr. Appl. Anal.}, 1 (2005), 1. doi: 10.1155/AAA.2005.1.

[10]

C. Cowan, Regularity of the extremal solutions in a Gelfand system problem,, \emph{Advanced Nonlinear Studies}, 11 (2011).

[11]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033.

[12]

C. Cowan and M. Fazly, Uniqueness of solutions for a nonlocal elliptic eigenvalue problem,, \emph{Math. Res. Lett.}, 19 (2012), 613. doi: 10.4310/MRL.2012.v19.n3.a9.

[13]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems,, \emph{Arch. Rat. Mech. Anal.}, 58 (1975), 207.

[14]

J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity,, \emph{SIAM J. Math. Anal.}, 39 (2007), 565. doi: 10.1137/060665579.

[15]

J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations,, \emph{Ann. Henri Poincare}, 10 (2010), 1311. doi: 10.1007/s00023-009-0016-9.

[16]

P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS,, \emph{Methods Appl. Anal.}, 15 (2008), 341. doi: 10.4310/MAA.2008.v15.n3.a6.

[17]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1731. doi: 10.1002/cpa.20189.

[18]

N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case,, \emph{SIAM J. Math. Anal.}, 38 (2007), 1423. doi: 10.1137/050647803.

[19]

T. Hashimoto, Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations,, \emph{Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equaions}, (2002), 24.

[20]

X. Luo, Uniqueness of weak extremal solution to biharmonic equation with logarithmically convex nonlinearities,, \emph{Journal of PDEs}, 23 (2010), 315.

[21]

Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems,, \emph{Houston J. Math.}, 23 (1997), 161.

[22]

J. McGough, On solution continua of supercritical quasilinear elliptic problems,, \emph{Differential Integral Equations}, 7 (1994), 1453.

[23]

J. McGough and J. Mortensen, Pohozaev obstructions on non-starlike domains,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 189. doi: 10.1007/s00526-002-0188-3.

[24]

J. McGough, J. Mortensen, C. Rickett and G. Stubbendieck, Domain geometry and the Pohozaev identity,, \emph{Electron. J. Differential Equations}, 32 (2005).

[25]

F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe,, \emph{Comm. Partial Differential Equations}, 5 (1980), 791. doi: 10.1080/03605308008820155.

[26]

M. Montenegro, Minimal solutions for a class of elliptic systems,, \emph{Bull. London Math. Soc.}, 37 (2005), 405. doi: 10.1112/S0024609305004248.

[27]

G. Nedev, Regularity of the extremal solution of semilinear elliptic equations,, \emph{C. R. Acad. Sci. Paris S. I Math.}, 330 (2000), 997. doi: 10.1016/S0764-4442(00)00289-5.

[28]

R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry,, \emph{Adv. Differential Equations}, 5 (2000), 1201.

[29]

K. Schmitt, Positive solutions of semilinear elliptic boundary value problems,, \emph{Topological Methods in Differential Equations and Inclusions} (Montreal, (1994), 447.

show all references

References:
[1]

G. Arioli, F. Gazzola, H.-C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity,, \emph{SIAM J. Math. Anal.}, 36 (2005), 1226. doi: 10.1137/S0036141002418534.

[2]

E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities,, \emph{Electronic Journal of Differential Equations}, 34 (2005).

[3]

T. Boggio, Sulle funzioni di Green drdine m,, \emph{Rend. Circ. Mat. Palermo}, (1905), 97.

[4]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited,, \emph{Adv. Diff. Eq.}, 1 (1996), 73.

[5]

H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, \emph{Rev. Mat. Univ. Complut. Madrid}, 10 (1997), 443.

[6]

X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1362. doi: 10.1002/cpa.20327.

[7]

X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations,, \emph{J. Funct. Anal.}, 238 (2006), 709. doi: 10.1016/j.jfa.2005.12.018.

[8]

D. Cassani, J. M. do O and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity,, \emph{Adv. Nonlinear Stud.}, 9 (2009), 177.

[9]

L. B. Chaabane, On the extremal solutions of semilinear elliptic problems,, \emph{Abstr. Appl. Anal.}, 1 (2005), 1. doi: 10.1155/AAA.2005.1.

[10]

C. Cowan, Regularity of the extremal solutions in a Gelfand system problem,, \emph{Advanced Nonlinear Studies}, 11 (2011).

[11]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1033. doi: 10.3934/dcds.2010.28.1033.

[12]

C. Cowan and M. Fazly, Uniqueness of solutions for a nonlocal elliptic eigenvalue problem,, \emph{Math. Res. Lett.}, 19 (2012), 613. doi: 10.4310/MRL.2012.v19.n3.a9.

[13]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems,, \emph{Arch. Rat. Mech. Anal.}, 58 (1975), 207.

[14]

J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity,, \emph{SIAM J. Math. Anal.}, 39 (2007), 565. doi: 10.1137/060665579.

[15]

J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations,, \emph{Ann. Henri Poincare}, 10 (2010), 1311. doi: 10.1007/s00023-009-0016-9.

[16]

P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS,, \emph{Methods Appl. Anal.}, 15 (2008), 341. doi: 10.4310/MAA.2008.v15.n3.a6.

[17]

P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1731. doi: 10.1002/cpa.20189.

[18]

N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case,, \emph{SIAM J. Math. Anal.}, 38 (2007), 1423. doi: 10.1137/050647803.

[19]

T. Hashimoto, Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations,, \emph{Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equaions}, (2002), 24.

[20]

X. Luo, Uniqueness of weak extremal solution to biharmonic equation with logarithmically convex nonlinearities,, \emph{Journal of PDEs}, 23 (2010), 315.

[21]

Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems,, \emph{Houston J. Math.}, 23 (1997), 161.

[22]

J. McGough, On solution continua of supercritical quasilinear elliptic problems,, \emph{Differential Integral Equations}, 7 (1994), 1453.

[23]

J. McGough and J. Mortensen, Pohozaev obstructions on non-starlike domains,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 189. doi: 10.1007/s00526-002-0188-3.

[24]

J. McGough, J. Mortensen, C. Rickett and G. Stubbendieck, Domain geometry and the Pohozaev identity,, \emph{Electron. J. Differential Equations}, 32 (2005).

[25]

F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe,, \emph{Comm. Partial Differential Equations}, 5 (1980), 791. doi: 10.1080/03605308008820155.

[26]

M. Montenegro, Minimal solutions for a class of elliptic systems,, \emph{Bull. London Math. Soc.}, 37 (2005), 405. doi: 10.1112/S0024609305004248.

[27]

G. Nedev, Regularity of the extremal solution of semilinear elliptic equations,, \emph{C. R. Acad. Sci. Paris S. I Math.}, 330 (2000), 997. doi: 10.1016/S0764-4442(00)00289-5.

[28]

R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry,, \emph{Adv. Differential Equations}, 5 (2000), 1201.

[29]

K. Schmitt, Positive solutions of semilinear elliptic boundary value problems,, \emph{Topological Methods in Differential Equations and Inclusions} (Montreal, (1994), 447.

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