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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 & 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.   Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 313.   Google Scholar

[3]

E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems,, J. Differential Equations, 229 (2006), 1.  doi: doi:10.1016/j.jde.2006.04.003.  Google Scholar

[4]

H. Brezis and W. A. Strauss, Semi-linear second order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: doi:10.2969/jmsj/02540565.  Google Scholar

[5]

J. Chabrowski and João Marcos do Ó, On some fourth-order semilinear elliptic problems in $\R$,, Nonlinear Anal., 49 (2002), 861.  doi: doi:10.1016/S0362-546X(01)00144-4.  Google Scholar

[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.   Google Scholar

[7]

Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equations,, Adv. Differential Equations, 2 (1997), 361.   Google Scholar

[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.  doi: doi:10.1006/jdeq.1996.0138.  Google Scholar

[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.  doi: doi:10.1142/S0219199702000695.  Google Scholar

[10]

J. Duoandikoetxea, "Fourier Analysis,", Graduate Studies in Math., 29 (2004).   Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differentialerential Equations of Second Order,", Springer-Verlag, (2001).   Google Scholar

[12]

H. C. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations,, Reaction Differentialusion systems (Trieste, 194 (1998), 163.   Google Scholar

[13]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $R$,, Indiana Univ. Math. J., 54 (2005), 443.  doi: doi:10.1512/iumj.2005.54.2502.  Google Scholar

[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.  doi: doi:10.1007/BF01765635.  Google Scholar

[15]

Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms,, J. Differential Equations, 235 (2007), 435.  doi: doi:10.1016/j.jde.2007.01.006.  Google Scholar

[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.  doi: doi:10.1016/j.jde.2006.09.018.  Google Scholar

[17]

T. Sato, Positive solutions with weak isolated singularities to some semilinear elliptic equations,, Tohoku Math. J., 47 (1995), 55.  doi: doi:10.2748/tmj/1178225635.  Google Scholar

[18]

T. Sato, Positive solutions to some semilinear elliptic equations with nonnegative forcing terms,, preprint., ().   Google Scholar

[19]

G. Sweers, No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems,, Math. Nachr., 246/247 (2002), 202.  doi: doi:10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G.  Google Scholar

[20]

T. Watanabe, Two positive solutions for an inhomogeneous scalar field equation,, to appear in J. Nonlinear and Convex Analysis., ().   Google Scholar

[21]

H. F. Weinberger, "Variational Methods for Eigenvalue Approximation,", Regional Conference Series in Applied Mathematics, 15 (1994).   Google Scholar

[22]

M. Willem, Minimax theorems,, in, 24 (1996).   Google Scholar

[23]

X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation,, J. Differential Equations, 92 (1991), 163.  doi: doi:10.1016/0022-0396(91)90045-B.  Google Scholar

show all references

References:
[1]

N. Aronszajn and K. T. Smith, Theory of Bessel potentials I,, Ann. Inst. Fourier, 11 (1961), 385.   Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 313.   Google Scholar

[3]

E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems,, J. Differential Equations, 229 (2006), 1.  doi: doi:10.1016/j.jde.2006.04.003.  Google Scholar

[4]

H. Brezis and W. A. Strauss, Semi-linear second order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: doi:10.2969/jmsj/02540565.  Google Scholar

[5]

J. Chabrowski and João Marcos do Ó, On some fourth-order semilinear elliptic problems in $\R$,, Nonlinear Anal., 49 (2002), 861.  doi: doi:10.1016/S0362-546X(01)00144-4.  Google Scholar

[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.   Google Scholar

[7]

Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equations,, Adv. Differential Equations, 2 (1997), 361.   Google Scholar

[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.  doi: doi:10.1006/jdeq.1996.0138.  Google Scholar

[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.  doi: doi:10.1142/S0219199702000695.  Google Scholar

[10]

J. Duoandikoetxea, "Fourier Analysis,", Graduate Studies in Math., 29 (2004).   Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differentialerential Equations of Second Order,", Springer-Verlag, (2001).   Google Scholar

[12]

H. C. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations,, Reaction Differentialusion systems (Trieste, 194 (1998), 163.   Google Scholar

[13]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $R$,, Indiana Univ. Math. J., 54 (2005), 443.  doi: doi:10.1512/iumj.2005.54.2502.  Google Scholar

[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.  doi: doi:10.1007/BF01765635.  Google Scholar

[15]

Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms,, J. Differential Equations, 235 (2007), 435.  doi: doi:10.1016/j.jde.2007.01.006.  Google Scholar

[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.  doi: doi:10.1016/j.jde.2006.09.018.  Google Scholar

[17]

T. Sato, Positive solutions with weak isolated singularities to some semilinear elliptic equations,, Tohoku Math. J., 47 (1995), 55.  doi: doi:10.2748/tmj/1178225635.  Google Scholar

[18]

T. Sato, Positive solutions to some semilinear elliptic equations with nonnegative forcing terms,, preprint., ().   Google Scholar

[19]

G. Sweers, No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems,, Math. Nachr., 246/247 (2002), 202.  doi: doi:10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G.  Google Scholar

[20]

T. Watanabe, Two positive solutions for an inhomogeneous scalar field equation,, to appear in J. Nonlinear and Convex Analysis., ().   Google Scholar

[21]

H. F. Weinberger, "Variational Methods for Eigenvalue Approximation,", Regional Conference Series in Applied Mathematics, 15 (1994).   Google Scholar

[22]

M. Willem, Minimax theorems,, in, 24 (1996).   Google Scholar

[23]

X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation,, J. Differential Equations, 92 (1991), 163.  doi: doi:10.1016/0022-0396(91)90045-B.  Google Scholar

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