### Abstract

In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two sign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is sufficiently thin, and ii) the existence of arbitrarily many sign-changing non-radial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball.

Original language | English (US) |
---|---|

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Electronic Journal of Differential Equations |

Volume | 1998 |

State | Published - Jan 30 1998 |

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### Keywords

- Dirichlet problem
- Sign-changing solution

### ASJC Scopus subject areas

- Analysis

### Cite this

**A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems.** / Castro, Alfonso; Cossio, Jorge; Neuberger, John M.

Research output: Contribution to journal › Article

*Electronic Journal of Differential Equations*, vol. 1998, pp. 1-18.

}

TY - JOUR

T1 - A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems

AU - Castro, Alfonso

AU - Cossio, Jorge

AU - Neuberger, John M

PY - 1998/1/30

Y1 - 1998/1/30

N2 - In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two sign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is sufficiently thin, and ii) the existence of arbitrarily many sign-changing non-radial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball.

AB - In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two sign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is sufficiently thin, and ii) the existence of arbitrarily many sign-changing non-radial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball.

KW - Dirichlet problem

KW - Sign-changing solution

UR - http://www.scopus.com/inward/record.url?scp=3342925295&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3342925295&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3342925295

VL - 1998

SP - 1

EP - 18

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -