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March  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 and 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, SIAM J. Math. Anal., 36 (2005), 1226-1258. doi: 10.1137/S0036141002418534. [2] E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic Journal of Differential Equations, 34 (2005), pp. 120. [3] T. Boggio, Sulle funzioni di Green drdine m, Rend. Circ. Mat. Palermo, (1905), 97-135. [4] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Diff. Eq., 1 (1996), 73-90. [5] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [6] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math., 63 (2010), 1362-1380. doi: 10.1002/cpa.20327. [7] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733. 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, Adv. Nonlinear Stud., 9 (2009), 177-197. [9] L. B. Chaabane, On the extremal solutions of semilinear elliptic problems, Abstr. Appl. Anal., 1 (2005), 1-9. doi: 10.1155/AAA.2005.1. [10] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Advanced Nonlinear Studies, 11 (2011), p. 695. [11] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033. [12] C. Cowan and M. Fazly, Uniqueness of solutions for a nonlocal elliptic eigenvalue problem, Math. Res. Lett., 19 (2012), 613-626. 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, Arch. Rat. Mech. Anal., 58 (1975), 207-218. [14] J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. doi: 10.1137/060665579. [15] J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Henri Poincare, 10 (2010), 1311-1333, doi: 10.1007/s00023-009-0016-9. [16] P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-353. 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, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189. [18] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803. [19] T. Hashimoto, Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equaions, May 24-27, 2002, Wilmington, NC, USA, pp 393-402. [20] X. Luo, Uniqueness of weak extremal solution to biharmonic equation with logarithmically convex nonlinearities, Journal of PDEs, 23 (2010), 315-329. [21] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [22] J. McGough, On solution continua of supercritical quasilinear elliptic problems, Differential Integral Equations, 7 (1994), 1453-1471. [23] J. McGough and J. Mortensen, Pohozaev obstructions on non-starlike domains, Calc. Var. Partial Differential Equations, 18 (2003), 189-205. doi: 10.1007/s00526-002-0188-3. [24] J. McGough, J. Mortensen, C. Rickett and G. Stubbendieck, Domain geometry and the Pohozaev identity, Electron. J. Differential Equations, 32 (2005), 16 pp. [25] F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. doi: 10.1080/03605308008820155. [26] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248. [27] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S. I Math., 330 (2000), 997-1002. doi: 10.1016/S0764-4442(00)00289-5. [28] R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations, 5 (2000), 1201-1220. [29] K. Schmitt, Positive solutions of semilinear elliptic boundary value problems, Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472, Kluwer Academic Publishers, Dordrecht, 1995, pp. 447-500.

show all references

##### References:
 [1] G. Arioli, F. Gazzola, H.-C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal., 36 (2005), 1226-1258. doi: 10.1137/S0036141002418534. [2] E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic Journal of Differential Equations, 34 (2005), pp. 120. [3] T. Boggio, Sulle funzioni di Green drdine m, Rend. Circ. Mat. Palermo, (1905), 97-135. [4] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Diff. Eq., 1 (1996), 73-90. [5] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [6] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math., 63 (2010), 1362-1380. doi: 10.1002/cpa.20327. [7] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733. 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, Adv. Nonlinear Stud., 9 (2009), 177-197. [9] L. B. Chaabane, On the extremal solutions of semilinear elliptic problems, Abstr. Appl. Anal., 1 (2005), 1-9. doi: 10.1155/AAA.2005.1. [10] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Advanced Nonlinear Studies, 11 (2011), p. 695. [11] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033. [12] C. Cowan and M. Fazly, Uniqueness of solutions for a nonlocal elliptic eigenvalue problem, Math. Res. Lett., 19 (2012), 613-626. 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, Arch. Rat. Mech. Anal., 58 (1975), 207-218. [14] J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. doi: 10.1137/060665579. [15] J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Henri Poincare, 10 (2010), 1311-1333, doi: 10.1007/s00023-009-0016-9. [16] P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-353. 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, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189. [18] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803. [19] T. Hashimoto, Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equaions, May 24-27, 2002, Wilmington, NC, USA, pp 393-402. [20] X. Luo, Uniqueness of weak extremal solution to biharmonic equation with logarithmically convex nonlinearities, Journal of PDEs, 23 (2010), 315-329. [21] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [22] J. McGough, On solution continua of supercritical quasilinear elliptic problems, Differential Integral Equations, 7 (1994), 1453-1471. [23] J. McGough and J. Mortensen, Pohozaev obstructions on non-starlike domains, Calc. Var. Partial Differential Equations, 18 (2003), 189-205. doi: 10.1007/s00526-002-0188-3. [24] J. McGough, J. Mortensen, C. Rickett and G. Stubbendieck, Domain geometry and the Pohozaev identity, Electron. J. Differential Equations, 32 (2005), 16 pp. [25] F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. doi: 10.1080/03605308008820155. [26] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248. [27] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S. I Math., 330 (2000), 997-1002. doi: 10.1016/S0764-4442(00)00289-5. [28] R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations, 5 (2000), 1201-1220. [29] K. Schmitt, Positive solutions of semilinear elliptic boundary value problems, Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472, Kluwer Academic Publishers, Dordrecht, 1995, pp. 447-500.
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