×

Multiplicity of solutions for a fourth order equation with power-type nonlinearity. (English) Zbl 1220.35047

Authors’ abstract: “Let \(B\) be the unit ball in \(\mathbb R^N\), \(N\geq 3\), and \(n\) be the exterior unit normal vector on the boundary. We consider radial solutions to
\[ \Delta^2 u = \lambda(1+ {\text{sign}}(p)u)^{p} \quad \text{in }B, \quad u=0, \quad \frac{\partial u}{\partial n} = 0 \quad \text{on }\partial B, \]
where \(\lambda \geq 0\). For positive \(p\) we assume \(5 \leq N \leq 12\) and \(p > \frac{N+4}{N-4}\), or \(N \geq 13\) and \(\frac{N+4}{N-4} < p < p_c\), where \(p _c\) is a constant depending on \(N\). For negative \(p\) we assume \(4 \leq N \leq 12\) and \(p < p _c\), or \(N = 3\) and \(p_c^+< p < p_c\), where \(p_c^+\) is a constant. We show that there is a unique \(\lambda_S> 0\) such that if \(\lambda = \lambda_S\) there exists a radial weakly singular solution. For \(\lambda = \lambda_S\) there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as \(\lambda \rightarrow \lambda_S\).”

MSC:

35J60 Nonlinear elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35B32 Bifurcations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arioli G., Gazzola F., Grunau H.-C.: Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity. J. Differ. Equ. 230(2), 743–770 (2006) · Zbl 1152.35360 · doi:10.1016/j.jde.2006.05.015
[2] Arioli G., Gazzola F., Grunau H.-C., Mitidieri E.: A semilinear fourth order elliptic problem with exponential nonlinearity. SIAM J. Math. Anal. 36(4), 1226–1258 (2005) · Zbl 1162.35339 · doi:10.1137/S0036141002418534
[3] Bamón R., Flores I., del Pino M.: Ground states of semilinear elliptic equations: a geometric approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(5), 551–581 (2000) · Zbl 0988.35054 · doi:10.1016/S0294-1449(00)00126-8
[4] Belickiĭ G.R.: Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class. Funct. Anal. Appl. 7, 268–277 (1973)
[5] Berchio, E., Gazzola, F.: Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities. Electron. J. Differ. Equ. vol. 34, p. 20 (2005) · Zbl 1129.35349
[6] Berchio E., Gazzola F., Weth T.: Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems. J. Reine Angew. Math. 620, 165–183 (2008) · Zbl 1182.35109 · doi:10.1515/CRELLE.2008.052
[7] Cassani D., do Ó J.M., Ghoussoub N.: On a fourth order elliptic problem with a singular nonlinearity. Adv. Nonlinear Stud. 9(1), 177–197 (2009) · Zbl 1181.35114
[8] Chicone, C.: Ordinary differential equations with applications, 2nd edn. Texts in Applied Mathematics, vol. 34. Springer, New York (2006) · Zbl 1120.34001
[9] Choi Y.S., Xu X.: Nonlinear biharmonic equations with negative exponents. J. Differ. Equ. 246(1), 216–234 (2009) · Zbl 1165.35014 · doi:10.1016/j.jde.2008.06.027
[10] Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, Inc., New York (1955) · Zbl 0064.33002
[11] Cowan, C., Esposito, P., Ghoussoub, N., Moradifam, A.: The critical dimension for a fourth order elliptic problem with singular nonlinearity. Arch. Ration. Mech. Anal. (2009, to appear) · Zbl 1225.35093
[12] Dávila J., Dupaigne L., Guerra I., Montenegro M.: Stable solutions for the bilaplacian with exponential nonlinearity. SIAM J. Math. Anal. 39(2), 565–592 (2007) · Zbl 1138.35022 · doi:10.1137/060665579
[13] Dávila, J., Flores, I.,Guerra, I.: Multiplicity of solutions for a fourth order problem with exponential nonlinearity. J. Differ. Equ. (2009). doi: 10.1016/j.jde.2009.07.023 · Zbl 1190.34017
[14] Ferrero A., Grunau H.-C.: The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity. J. Differ. Equ. 234(2), 582–606 (2007) · Zbl 1189.35099 · doi:10.1016/j.jde.2006.11.007
[15] Ferrero A., Grunau H.-C., Karageorgis P.: Supercritical biharmonic equations with power-type nonlinearity. Ann. Mat. Pura Appl. 188, 171–185 (2009) · Zbl 1179.35125 · doi:10.1007/s10231-008-0070-9
[16] Ferrero A., Warnault G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70(8), 2889–2902 (2009) · Zbl 1171.35374 · doi:10.1016/j.na.2008.12.041
[17] Flores I.: A resonance phenomenon for ground states of an elliptic equation of Emden-Fowler type. J. Differ. Equ. 198(1), 1–15 (2004) · Zbl 1055.34031 · doi:10.1016/S0022-0396(02)00015-3
[18] Flores I.: Singular solutions of the Brezis-Nirenberg problem in a ball. Commun. Pure Appl. Anal. 8(2), 673–682 (2009) · Zbl 1170.35401 · doi:10.3934/cpaa.2009.8.673
[19] Dolbeault J., Flores I.: Geometry of phase space and solutions of semilinear elliptic equations in a ball. Trans. Am. Math. Soc. 359(9), 4073–4087 (2007) · Zbl 1170.35039 · doi:10.1090/S0002-9947-07-04397-8
[20] Gazzola F., Grunau H.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334(4), 905–936 (2006) · Zbl 1152.35034 · doi:10.1007/s00208-005-0748-x
[21] Gelfand I.M.: Some problems in the theory of quasilinear equations. Section 15, due to G.I. Barenblatt. Am. Math. Soc. Trans. II Ser. 29, 295–381 (1963)
[22] Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001) · Zbl 1042.35002
[23] Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125
[24] Grunau H., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307(4), 589–626 (1997) · Zbl 0892.35031 · doi:10.1007/s002080050052
[25] Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Applied Mathematical Sciences, vol. 42. Springer, New York (1990)
[26] Guo Z., Wei J.: On a fourth order nonlinear elliptic equation with negative exponent. SIAM J. Math. Anal. 40(5), 2034–2054 (2009) · Zbl 1175.35144 · doi:10.1137/070703375
[27] Guo Z., Wei J.: Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in \({\mathbb{R}^{3}}\) . Adv. Differ. Equ. 13(7–8), 753–780 (2008) · Zbl 1203.35018
[28] Guo, Z., Wei, J.: Entire solutions and global bifurcations for a biharmonic equation with singular nonlinearity II (preprint)
[29] Hartman, P.: Ordinary differential equations. Classics in Applied Mathematics, vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002) · Zbl 1009.34001
[30] Joseph D.D., Lundgren T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1972) · Zbl 0266.34021
[31] Karageorgis, P.: Stability and intersection properties of solutions to the nonlinear biharmonic equation (preprint) · Zbl 1172.31005
[32] Lin F., Yang Y.: Nonlinear non-local elliptic equation modelling electrostatic actuation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2081), 1323–1337 (2007) · Zbl 1143.78001 · doi:10.1098/rspa.2007.1816
[33] Lions P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24(4), 441–467 (1982) · Zbl 0511.35033 · doi:10.1137/1024101
[34] McKenna, P.J., Reichel, W.: Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry. Electron. J. Differ. Equ., no. 37, 13 pp. (2003) · Zbl 1109.35321
[35] Mitidieri, E., Pokhozhaev, S.I.: A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 3–383 (2001). English translation: Proc. Steklov Inst. Math. 234, 1–362 (2001) · Zbl 0987.35002
[36] Palis J.: On Morse-Smale dynamical systems. Topology 8, 385–404 (1968) · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X
[37] Pelesko J.A., Bernstein A.A.: Modeling MEMS and NEMS. Chapman Hall and CRC Press, Boca Raton (2002) · Zbl 1031.74003
[38] Rellich, F.: Halbbeschränkte Differentialoperatoren höherer Ordnung. In: Gerretsen, J.C.H., et al. (eds.) Proceedings of the international congress of mathematicians Amsterdam 1954, vol. III, pp. 243–250. Nordhoff, Groningen (1956) · Zbl 0074.06703
[39] Ruelle D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, Inc., Boston (1989) · Zbl 0684.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.