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

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  • 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}
    Mathematics Subject Classification: 35J55, 35B05, 35K50, 35K55.


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  • [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.


    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.


    T. Boggio, Sulle funzioni di Green drdine m, Rend. Circ. Mat. Palermo, (1905), 97-135.


    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.


    H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.


    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.


    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.


    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.


    L. B. Chaabane, On the extremal solutions of semilinear elliptic problems, Abstr. Appl. Anal., 1 (2005), 1-9.doi: 10.1155/AAA.2005.1.


    C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Advanced Nonlinear Studies, 11 (2011), p. 695.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    X. Luo, Uniqueness of weak extremal solution to biharmonic equation with logarithmically convex nonlinearities, Journal of PDEs, 23 (2010), 315-329.


    Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168.


    J. McGough, On solution continua of supercritical quasilinear elliptic problems, Differential Integral Equations, 7 (1994), 1453-1471.


    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.


    J. McGough, J. Mortensen, C. Rickett and G. Stubbendieck, Domain geometry and the Pohozaev identity, Electron. J. Differential Equations, 32 (2005), 16 pp.


    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.


    M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.doi: 10.1112/S0024609305004248.


    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.


    R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations, 5 (2000), 1201-1220.


    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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