# Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

Colloquium Mathematicae (2003)

• Volume: 97, Issue: 2, page 221-231
• ISSN: 0010-1354

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## Abstract

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We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-{\Delta }_{p}u+\lambda \left(x\right){|u|}^{p-2}u=\mu f\left(x,u\right)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $\Omega \subset {ℝ}^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $\lambda \in {L}^{\infty }\left(\Omega \right)$ with $essin{f}_{x\in \Omega }\lambda \left(x\right)>0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

## How to cite

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Giovanni Anello, and Giuseppe Cordaro. "Infinitely many positive solutions for the Neumann problem involving the p-Laplacian." Colloquium Mathematicae 97.2 (2003): 221-231. <http://eudml.org/doc/284810>.

@article{GiovanniAnello2003,
abstract = {We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-Δ_\{p\}u + λ(x)|u|^\{p-2\}u = μf(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω ⊂ ℝ^\{N\}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ ∈ L^\{∞\}(Ω)$ with $essinf_\{x∈Ω\} λ(x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.},
author = {Giovanni Anello, Giuseppe Cordaro},
journal = {Colloquium Mathematicae},
keywords = {-Laplacian; small solutions; positive solutions},
language = {eng},
number = {2},
pages = {221-231},
title = {Infinitely many positive solutions for the Neumann problem involving the p-Laplacian},
url = {http://eudml.org/doc/284810},
volume = {97},
year = {2003},
}

TY - JOUR
AU - Giovanni Anello
AU - Giuseppe Cordaro
TI - Infinitely many positive solutions for the Neumann problem involving the p-Laplacian
JO - Colloquium Mathematicae
PY - 2003
VL - 97
IS - 2
SP - 221
EP - 231
AB - We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-Δ_{p}u + λ(x)|u|^{p-2}u = μf(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω ⊂ ℝ^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ ∈ L^{∞}(Ω)$ with $essinf_{x∈Ω} λ(x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.
LA - eng
KW - -Laplacian; small solutions; positive solutions
UR - http://eudml.org/doc/284810
ER -

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