Multiple solutions to a perturbed Neumann problem

Giuseppe Cordaro. "Multiple solutions to a perturbed Neumann problem." Studia Mathematica 178.2 (2007): 167-175. <http://eudml.org/doc/284813>.

@article{GiuseppeCordaro2007,
abstract = {We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in $ℝ^\{N\}$ with boundary of class C², $α ∈ L^\{∞\}(Ω)$ with $ess inf_\{Ω\}α > 0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $sup_\{|s|≤t\} |g(⋅,s)| ∈ L^\{p\}(Ω)$ and $g(⋅,t) ∈ L^\{∞\}(Ω)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2 ξ² - ∫_\{0\}^\{ξ\} f(t)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above Neumann problem has at least M+integer part of M/2 distinct strong solutions in $W^\{2,p\}(Ω)$.},
author = {Giuseppe Cordaro},
journal = {Studia Mathematica},
keywords = {Neumann problem; connected component; multiplicity of solutions; weak solution; strong solution},
language = {eng},
number = {2},
pages = {167-175},
title = {Multiple solutions to a perturbed Neumann problem},
url = {http://eudml.org/doc/284813},
volume = {178},
year = {2007},
}

TY - JOUR
AU - Giuseppe Cordaro
TI - Multiple solutions to a perturbed Neumann problem
JO - Studia Mathematica
PY - 2007
VL - 178
IS - 2
SP - 167
EP - 175
AB - We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in $ℝ^{N}$ with boundary of class C², $α ∈ L^{∞}(Ω)$ with $ess inf_{Ω}α > 0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $sup_{|s|≤t} |g(⋅,s)| ∈ L^{p}(Ω)$ and $g(⋅,t) ∈ L^{∞}(Ω)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2 ξ² - ∫_{0}^{ξ} f(t)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above Neumann problem has at least M+integer part of M/2 distinct strong solutions in $W^{2,p}(Ω)$.
LA - eng
KW - Neumann problem; connected component; multiplicity of solutions; weak solution; strong solution
UR - http://eudml.org/doc/284813
ER -