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

Tokushi Sato; Tatsuya Watanabe

doi:10.3934/cpaa.2011.10.245

Article Contents

2011, Volume 10, Issue 1: 245-268. Doi: 10.3934/cpaa.2011.10.245

This issue Previous Article Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$ Next Article Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials

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

Tokushi Sato^1, and
Tatsuya Watanabe^2,

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

Received: December 31, 2009

Revised: April 30, 2010

Published: December 2010

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J35; Secondary: 35J60.

Citation:

Related Papers

Cited by

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 $\R$, 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 $R$, 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.