A resonance phenomenon for ground states of an elliptic equation of Emden–Fowler type.

*(English)*Zbl 1055.34031The author studies the existence of radial solutions of the elliptic equation
\[
\Delta u+u^p+u^q=0\tag{1}
\]
in \(\mathbb R^N\), \(N\geq 3\), with \(0<u(x)\to+\infty\) as \(| x| \to+\infty\). By C. S. Lin and W. M. Ni [Proc. Am. Math. Soc. 102, 271–277 (1988; Zbl 0652.35085)] it was proved that if \(1<p<\frac{N+2}{N-2}<q\) and \(q=2p-1\), then there exists an explicit radial solution of the form \(u(r)=(\frac{A}{B+r^2})^{\frac{1}{p-1}}\).

In the present paper, this result is generalized. It is proved that if the range of \(p\) is further restricted to \(p>\frac{N+2\sqrt{N-1}}{N+2\sqrt{N-1}-4}\), then for \(q=2p-1\), equation (1) has infinitely many radial solutions with fast decay \(O(r^{2-N})\) besides the explicit solution. If \(q\) is close to \(2p-1\), then equation (1) also has a large number of such solutions.

It is known that the existence of positive radial solutions of (1) is equivalent to the existence of solutions of the ordinary differential equation \[ u''+\frac{N-1}{r}u'+u_+^p+u_+^q=0 \] with \(u'(0)=0\), \(0<u(r)\to+\infty\) where \(u_+=\max\{u, 0\}\). This equation can be further reduced to the second-order differential equation \[ x''-\alpha x'+x_+^p+e^{-\gamma t}x_+^q-\beta x=0 \] by using the classical transformation \(x(t)=r^{\frac{2}{p-1}}u(r)| _{r=e^t}\). On the basis of this preparation, the invariant manifold theory is used to prove the existence of solutions \(x(t)\) satisfying \(x(t)\to 0\) as \(t\to\pm\infty\).

In the present paper, this result is generalized. It is proved that if the range of \(p\) is further restricted to \(p>\frac{N+2\sqrt{N-1}}{N+2\sqrt{N-1}-4}\), then for \(q=2p-1\), equation (1) has infinitely many radial solutions with fast decay \(O(r^{2-N})\) besides the explicit solution. If \(q\) is close to \(2p-1\), then equation (1) also has a large number of such solutions.

It is known that the existence of positive radial solutions of (1) is equivalent to the existence of solutions of the ordinary differential equation \[ u''+\frac{N-1}{r}u'+u_+^p+u_+^q=0 \] with \(u'(0)=0\), \(0<u(r)\to+\infty\) where \(u_+=\max\{u, 0\}\). This equation can be further reduced to the second-order differential equation \[ x''-\alpha x'+x_+^p+e^{-\gamma t}x_+^q-\beta x=0 \] by using the classical transformation \(x(t)=r^{\frac{2}{p-1}}u(r)| _{r=e^t}\). On the basis of this preparation, the invariant manifold theory is used to prove the existence of solutions \(x(t)\) satisfying \(x(t)\to 0\) as \(t\to\pm\infty\).

Reviewer: Zaihong Wang (Beijing)

##### MSC:

34B15 | Nonlinear boundary value problems for ordinary differential equations |

35J60 | Nonlinear elliptic equations |

35B34 | Resonance in context of PDEs |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34C45 | Invariant manifolds for ordinary differential equations |

PDF
BibTeX
XML
Cite

\textit{I. Flores}, J. Differ. Equations 198, No. 1, 1--15 (2004; Zbl 1055.34031)

Full Text:
DOI

##### References:

[1] | Bamon, R.; Flores, I.; del Pino, M., Positive solutions of elliptic equations in \(R\^{}\{N\}\) with a super-subcritical nonlinearity, C.R. acad. sci. Paris, 330, 187-191, (2000) · Zbl 0943.35025 |

[2] | R. Bamon, I. Flores, M. del Pino, Ground states of semilinear elliptic equations: a geometric approach, Ann. Inst. H. Poincare Anal. Non Lineaire, in preparation. · Zbl 0988.35054 |

[3] | Belitskij, G.R., Equivalence and normal forms of germs of smooth mappings, Russ. math. surv., 33, 101-177, (1978) · Zbl 0398.58009 |

[4] | G.R. Belitskij, Normal Forms, Invariants and Local Mappings, Naukova Dumka, Kiev, 1979, p. 176. |

[5] | Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure appl. math., 42, 3, 271-297, (1989) · Zbl 0702.35085 |

[6] | Fowler, R., Further studies of Emden’s and similar differential equations, Quart. J. math., 2, 259-288, (1931) · JFM 57.0523.02 |

[7] | Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003 |

[8] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001 |

[9] | M. Hirsch, C. Pugh, M. Schub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer, New York, 1977. |

[10] | Johnson, R.; Pan, X.; Yi, Y., Positive solutions of super-critical elliptic equations and asymptotics, Comm. partial differential equations, 18, 977-1019, (1993) · Zbl 0793.35029 |

[11] | Lin, C.-S.; Ni, W.-M., A counterexample to the nodal line conjecture and a related semilinear equation, Proc. amer. math. soc., 102, 2, 271-277, (1988) · Zbl 0652.35085 |

[12] | Palis, J.; de Melo, W., Geometric theory of dynamical systemsan introduction, (1982), Springer New York, Heidelberg, Berlin |

[13] | Zou, H., Symmetry of ground states of semilinear elliptic equations with mixed Sobolev growth, Indiana univ. math. J., 45, 221-240, (1996) · Zbl 0864.35009 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.