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January  2011, 10(1): 245-268. doi: 10.3934/cpaa.2011.10.245

## Singular positive solutions for a fourth order elliptic problem in $R$

 1 Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan 2 Osaka City University Advanced Mathematical Institute, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

Received  January 2010 Revised  May 2010 Published  November 2010

In this paper, we consider the following fourth order elliptic problem in $R^N$:

$\Delta^2 u-c_1\Delta u+c_2 u=u^p+\kappa \sum_{i=1}^m \alpha_i \delta_{a_i}$ in $\mathcal D'(R^N),$

$u(x)>0, u(x) \rightarrow 0$ as $|x| \rightarrow \infty.$

We will prove if $0 < \kappa < \kappa^*$ for some $\kappa^*\in (0,\infty)$, then this problem has at least two singular positive solutions.

Citation: Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure and Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245
##### References:
 [1] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. [2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. [3] E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23. doi: doi:10.1016/j.jde.2006.04.003. [4] H. Brezis and W. A. Strauss, Semi-linear second order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: doi:10.2969/jmsj/02540565. [5] J. Chabrowski and João Marcos do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal., 49 (2002), 861-884. doi: doi:10.1016/S0362-546X(01)00144-4. [6] C. C. Chen and C. S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246. [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equations, Adv. Differential Equations, 2 (1997), 361-382. [8] Y. Deng and Y. Li, Existence and bifurcation of the positive solutions for a semilinear equation with critical exponent, J. Differential Equations, 130 (1996), 179-200. doi: doi:10.1006/jdeq.1996.0138. [9] Z. Djadli, A. Malchiodi and M. Ould Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere I: A perturbation result, Commun. Contemp. Math., 4 (2002), 375-408. doi: doi:10.1142/S0219199702000695. [10] J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Math., 29, 2004, AMS. Providence. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differentialerential Equations of Second Order," Springer-Verlag, Berlin, 2001. [12] H. C. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations, Reaction Differentialusion systems (Trieste, 1995), 163-182 Lect. Notes in Pure and Appl. Math., 194 (1998), Dekker, New York. [13] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J., 54 (2005), 443-464. doi: doi:10.1512/iumj.2005.54.2502. [14] R. A. Johnson, X. Pan and Y. Yi, Singular solutions of the elliptic equation $\Delta u-u+u^p=0$, Ann. Mat. Pura Appl., 166 (1994), 203-225. doi: doi:10.1007/BF01765635. [15] Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms, J. Differential Equations, 235 (2007), 435-483. doi: doi:10.1016/j.jde.2007.01.006. [16] S. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations, 233 (2007), 151-180. doi: doi:10.1016/j.jde.2006.09.018. [17] T. Sato, Positive solutions with weak isolated singularities to some semilinear elliptic equations, Tohoku Math. J., 47 (1995), 55-80. doi: doi:10.2748/tmj/1178225635. [18] T. Sato, Positive solutions to some semilinear elliptic equations with nonnegative forcing terms, preprint. [19] G. Sweers, No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems, Math. Nachr., 246/247 (2002), 202-206. doi: doi:10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G. [20] T. Watanabe, Two positive solutions for an inhomogeneous scalar field equation, to appear in J. Nonlinear and Convex Analysis. [21] H. F. Weinberger, "Variational Methods for Eigenvalue Approximation," Regional Conference Series in Applied Mathematics, 15, 1994, SIAM, Pliladelphia. [22] M. Willem, Minimax theorems, in "Prog. in Nonlinear Differential Equations and Their Applications," 24 (1996) Birkhäuser, Boston. [23] X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: doi:10.1016/0022-0396(91)90045-B.

show all references

##### References:
 [1] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. [2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. [3] E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23. doi: doi:10.1016/j.jde.2006.04.003. [4] H. Brezis and W. A. Strauss, Semi-linear second order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: doi:10.2969/jmsj/02540565. [5] J. Chabrowski and João Marcos do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal., 49 (2002), 861-884. doi: doi:10.1016/S0362-546X(01)00144-4. [6] C. C. Chen and C. S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246. [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equations, Adv. Differential Equations, 2 (1997), 361-382. [8] Y. Deng and Y. Li, Existence and bifurcation of the positive solutions for a semilinear equation with critical exponent, J. Differential Equations, 130 (1996), 179-200. doi: doi:10.1006/jdeq.1996.0138. [9] Z. Djadli, A. Malchiodi and M. Ould Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere I: A perturbation result, Commun. Contemp. Math., 4 (2002), 375-408. doi: doi:10.1142/S0219199702000695. [10] J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Math., 29, 2004, AMS. Providence. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differentialerential Equations of Second Order," Springer-Verlag, Berlin, 2001. [12] H. C. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations, Reaction Differentialusion systems (Trieste, 1995), 163-182 Lect. Notes in Pure and Appl. Math., 194 (1998), Dekker, New York. [13] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J., 54 (2005), 443-464. doi: doi:10.1512/iumj.2005.54.2502. [14] R. A. Johnson, X. Pan and Y. Yi, Singular solutions of the elliptic equation $\Delta u-u+u^p=0$, Ann. Mat. Pura Appl., 166 (1994), 203-225. doi: doi:10.1007/BF01765635. [15] Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms, J. Differential Equations, 235 (2007), 435-483. doi: doi:10.1016/j.jde.2007.01.006. [16] S. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations, 233 (2007), 151-180. doi: doi:10.1016/j.jde.2006.09.018. [17] T. Sato, Positive solutions with weak isolated singularities to some semilinear elliptic equations, Tohoku Math. J., 47 (1995), 55-80. doi: doi:10.2748/tmj/1178225635. [18] T. Sato, Positive solutions to some semilinear elliptic equations with nonnegative forcing terms, preprint. [19] G. Sweers, No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems, Math. Nachr., 246/247 (2002), 202-206. doi: doi:10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G. [20] T. Watanabe, Two positive solutions for an inhomogeneous scalar field equation, to appear in J. Nonlinear and Convex Analysis. [21] H. F. Weinberger, "Variational Methods for Eigenvalue Approximation," Regional Conference Series in Applied Mathematics, 15, 1994, SIAM, Pliladelphia. [22] M. Willem, Minimax theorems, in "Prog. in Nonlinear Differential Equations and Their Applications," 24 (1996) Birkhäuser, Boston. [23] X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: doi:10.1016/0022-0396(91)90045-B.
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