# Bifurcation theorems for nonlinear problems with lack of compactness

• Volume: 82, Issue: 1, page 77-85
• ISSN: 0066-2216

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## Abstract

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We deal with a bifurcation result for the Dirichlet problem ⎧$-{\Delta }_{p}u=\mu /{|x|}^{p}{|u|}^{p-2}u+\lambda f\left(x,u\right)$ a.e. in Ω, ⎨ ⎩${u}_{|\partial \Omega }=0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $\lambda {*}_{\mu }$ such that for every $\lambda \in \right]0,\lambda {*}_{\mu }\left[$ the above problem admits a nonzero weak solution ${u}_{\lambda }$ in $W{₀}^{1,p}\left(\Omega \right)$ satisfying $li{m}_{\lambda \to 0⁺}||{u}_{\lambda }||=0$.

## How to cite

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Francesca Faraci, and Roberto Livrea. "Bifurcation theorems for nonlinear problems with lack of compactness." Annales Polonici Mathematici 82.1 (2003): 77-85. <http://eudml.org/doc/280183>.

@article{FrancescaFaraci2003,
abstract = {We deal with a bifurcation result for the Dirichlet problem ⎧$-Δ_\{p\}u = μ/|x|^\{p\} |u|^\{p-2\}u + λf(x,u)$ a.e. in Ω, ⎨ ⎩$u_\{|∂Ω\} = 0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $λ*_\{μ\}$ such that for every $λ ∈ ]0,λ*_\{μ\}[$ the above problem admits a nonzero weak solution $u_\{λ\}$ in $W₀^\{1,p\}(Ω)$ satisfying $lim_\{λ→0⁺\} ||u_\{λ\}|| = 0$.},
author = {Francesca Faraci, Roberto Livrea},
journal = {Annales Polonici Mathematici},
keywords = {bifurcation point; -Laplacian; critical points},
language = {eng},
number = {1},
pages = {77-85},
title = {Bifurcation theorems for nonlinear problems with lack of compactness},
url = {http://eudml.org/doc/280183},
volume = {82},
year = {2003},
}

TY - JOUR
AU - Francesca Faraci
AU - Roberto Livrea
TI - Bifurcation theorems for nonlinear problems with lack of compactness
JO - Annales Polonici Mathematici
PY - 2003
VL - 82
IS - 1
SP - 77
EP - 85
AB - We deal with a bifurcation result for the Dirichlet problem ⎧$-Δ_{p}u = μ/|x|^{p} |u|^{p-2}u + λf(x,u)$ a.e. in Ω, ⎨ ⎩$u_{|∂Ω} = 0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $λ*_{μ}$ such that for every $λ ∈ ]0,λ*_{μ}[$ the above problem admits a nonzero weak solution $u_{λ}$ in $W₀^{1,p}(Ω)$ satisfying $lim_{λ→0⁺} ||u_{λ}|| = 0$.
LA - eng
KW - bifurcation point; -Laplacian; critical points
UR - http://eudml.org/doc/280183
ER -

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