### Abstract

This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem Δu + f(u) = h(x) + cφ(x) on the unit ball Ω ⊃ R^{N} with boundary condition u = 0 on ∂ω. Here 0(x) is a positive function and /(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u) < n, where p, is the smallest positive eigenvalue for Δ1 + μ = 0 in ω with -0 on ∂ω. It is shown that, given any integer k > 0, the value c may be chosen so large that there are 2k + 1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

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

Pages (from-to) | 1919-1945 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 351 |

Issue number | 5 |

State | Published - 1999 |

### Fingerprint

### Keywords

- Critical exponent
- Dirichlet problem
- Nodal curves
- Radially symmetric
- Superlinear jumping nonlinearity

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*351*(5), 1919-1945.

**On the number of radially symmetric solutions to dirichlet problems with jumping nonlinearities of superlinear order.** / Castro, Alfonso; Kuiper, Hendrik J.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 351, no. 5, pp. 1919-1945.

}

TY - JOUR

T1 - On the number of radially symmetric solutions to dirichlet problems with jumping nonlinearities of superlinear order

AU - Castro, Alfonso

AU - Kuiper, Hendrik J.

PY - 1999

Y1 - 1999

N2 - This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem Δu + f(u) = h(x) + cφ(x) on the unit ball Ω ⊃ RN with boundary condition u = 0 on ∂ω. Here 0(x) is a positive function and /(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u) < n, where p, is the smallest positive eigenvalue for Δ1 + μ = 0 in ω with -0 on ∂ω. It is shown that, given any integer k > 0, the value c may be chosen so large that there are 2k + 1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

AB - This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem Δu + f(u) = h(x) + cφ(x) on the unit ball Ω ⊃ RN with boundary condition u = 0 on ∂ω. Here 0(x) is a positive function and /(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u) < n, where p, is the smallest positive eigenvalue for Δ1 + μ = 0 in ω with -0 on ∂ω. It is shown that, given any integer k > 0, the value c may be chosen so large that there are 2k + 1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

KW - Critical exponent

KW - Dirichlet problem

KW - Nodal curves

KW - Radially symmetric

KW - Superlinear jumping nonlinearity

UR - http://www.scopus.com/inward/record.url?scp=33646942692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646942692&partnerID=8YFLogxK

M3 - Article

VL - 351

SP - 1919

EP - 1945

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -