### Abstract

We study the multiplicity of nonnegative solutions to the problem,. (P_{λ})-Δu=λa(x)u^{p}+f(u)inΩ,u=0on ∂Ω, where Ω is a smooth bounded domain in R^{N}, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving u^{p} may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (P_{λ}) is strongly influenced by u^{p} and depends on λ whenever one of the following two cases holds:. •p≤1 and f oscillates near the origin;•p≥1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a∈L^{∞}(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various H_{0}^{1}- and L^{∞}-norm estimates of solutions are also given.

Original language | English |
---|---|

Pages (from-to) | 555-570 |

Number of pages | 16 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 94 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2010 |

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### Keywords

- Critical and supercritical growth
- Dirichlet problem
- Indefinite potential
- Oscillatory nonlinearity
- Variational method

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

*Journal des Mathematiques Pures et Appliquees*,

*94*(6), 555-570. https://doi.org/10.1016/j.matpur.2010.03.005

**New competition phenomena in Dirichlet problems.** / Kristály, A.; MoroŞanu, Gheorghe.

Research output: Contribution to journal › Article

*Journal des Mathematiques Pures et Appliquees*, vol. 94, no. 6, pp. 555-570. https://doi.org/10.1016/j.matpur.2010.03.005

}

TY - JOUR

T1 - New competition phenomena in Dirichlet problems

AU - Kristály, A.

AU - MoroŞanu, Gheorghe

PY - 2010/12

Y1 - 2010/12

N2 - We study the multiplicity of nonnegative solutions to the problem,. (Pλ)-Δu=λa(x)up+f(u)inΩ,u=0on ∂Ω, where Ω is a smooth bounded domain in RN, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving up may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (Pλ) is strongly influenced by up and depends on λ whenever one of the following two cases holds:. •p≤1 and f oscillates near the origin;•p≥1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a∈L∞(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various H01- and L∞-norm estimates of solutions are also given.

AB - We study the multiplicity of nonnegative solutions to the problem,. (Pλ)-Δu=λa(x)up+f(u)inΩ,u=0on ∂Ω, where Ω is a smooth bounded domain in RN, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving up may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (Pλ) is strongly influenced by up and depends on λ whenever one of the following two cases holds:. •p≤1 and f oscillates near the origin;•p≥1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a∈L∞(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various H01- and L∞-norm estimates of solutions are also given.

KW - Critical and supercritical growth

KW - Dirichlet problem

KW - Indefinite potential

KW - Oscillatory nonlinearity

KW - Variational method

UR - http://www.scopus.com/inward/record.url?scp=78349306348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78349306348&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2010.03.005

DO - 10.1016/j.matpur.2010.03.005

M3 - Article

AN - SCOPUS:78349306348

VL - 94

SP - 555

EP - 570

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 6

ER -