Abstract
Let B be the unit ball in \({\mathbb{R}^N}\), N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to
where λ ≥ 0. For positive p we assume 5 ≤ N ≤ 12 and \({p > \frac{N+4}{N-4}}\), or N ≥ 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 ≤ N ≤ 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 λ S > 0 such that if λ = λ S there exists a radial weakly singular solution. For λ = λ S there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as λ → λ S .
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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)
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)
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)
Belickiĭ G.R.: Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class. Funct. Anal. Appl. 7, 268–277 (1973)
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)
Berchio E., Gazzola F., Weth T.: Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems. J. Reine Angew. Math. 620, 165–183 (2008)
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)
Chicone, C.: Ordinary differential equations with applications, 2nd edn. Texts in Applied Mathematics, vol. 34. Springer, New York (2006)
Choi Y.S., Xu X.: Nonlinear biharmonic equations with negative exponents. J. Differ. Equ. 246(1), 216–234 (2009)
Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, Inc., New York (1955)
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)
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)
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
Ferrero A., Grunau H.-C.: The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity. J. Differ. Equ. 234(2), 582–606 (2007)
Ferrero A., Grunau H.-C., Karageorgis P.: Supercritical biharmonic equations with power-type nonlinearity. Ann. Mat. Pura Appl. 188, 171–185 (2009)
Ferrero A., Warnault G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70(8), 2889–2902 (2009)
Flores I.: A resonance phenomenon for ground states of an elliptic equation of Emden-Fowler type. J. Differ. Equ. 198(1), 1–15 (2004)
Flores I.: Singular solutions of the Brezis-Nirenberg problem in a ball. Commun. Pure Appl. Anal. 8(2), 673–682 (2009)
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)
Gazzola F., Grunau H.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334(4), 905–936 (2006)
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)
Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001)
Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)
Grunau H., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307(4), 589–626 (1997)
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)
Guo Z., Wei J.: On a fourth order nonlinear elliptic equation with negative exponent. SIAM J. Math. Anal. 40(5), 2034–2054 (2009)
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)
Guo, Z., Wei, J.: Entire solutions and global bifurcations for a biharmonic equation with singular nonlinearity II (preprint)
Hartman, P.: Ordinary differential equations. Classics in Applied Mathematics, vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)
Joseph D.D., Lundgren T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1972)
Karageorgis, P.: Stability and intersection properties of solutions to the nonlinear biharmonic equation (preprint)
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)
Lions P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24(4), 441–467 (1982)
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)
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)
Palis J.: On Morse-Smale dynamical systems. Topology 8, 385–404 (1968)
Pelesko J.A., Bernstein A.A.: Modeling MEMS and NEMS. Chapman Hall and CRC Press, Boca Raton (2002)
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)
Ruelle D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, Inc., Boston (1989)
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Dávila, J., Flores, I. & Guerra, I. Multiplicity of solutions for a fourth order equation with power-type nonlinearity. Math. Ann. 348, 143–193 (2010). https://doi.org/10.1007/s00208-009-0476-8
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DOI: https://doi.org/10.1007/s00208-009-0476-8