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On a variational property of integral functionals and related conjectures

100%
Ricceri B.
Banach Center Publications
|
1996
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tom 35
|
nr 1
237-242
EN
The aim of this paper is to give the proofs of those results that in [4] were only announced, and, at the same time, to propose some possible developments, indicating some of the most significant open problems.
2
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Another fixed point theorem for nonexpansive potential operators

100%
Ricceri B.
Studia Mathematica
|
2012
|
tom 211
|
nr 2
147-151
EN
We prove the following result: Let X be a real Hilbert space and let J: X → ℝ be a C¹ functional with a nonexpansive derivative. Then, for each r > 0, the following alternative holds: either J' has a fixed point with norm less than r, or $sup_{||x||=r}J(x) = sup_{||u||_{L²([0,1],X)}=r} ∫_{0}^{1} J(u(t))dt$.
3
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A limit involving functions in $W^{1,p}_0(Ω)$

100%
Ricceri B.
Colloquium Mathematicum
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1999
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tom 82
|
nr 2
219-222
EN
We point out the following fact: if Ω ⊂ $ℝ^n$ is a bounded open set, δ>0, and p>1, then $lim_{𝜀 → 0^+} inf_{𝑢 ∈ V_𝜀} ∫_Ω |∇𝑢(x)|^p dx=∞$, where $V_𝜀={𝑢 ∈ W^{1,p}_0(Ω): meas ({x ∈ Ω:|𝑢(x)|>δ})>𝜀}.$
4
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A bifurcation theory for some nonlinear elliptic equations

100%
Ricceri B.
Colloquium Mathematicum
|
2003
|
tom 95
|
nr 1
139-151
EN
We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{λ}$) ⎩ $u_{∣∂Ω} = 0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{λ}$) admits a non-zero, non-negative strong solution $u_{λ} ∈ ⋂_{p≥2}W^{2,p}(Ω)$ such that $lim_{λ→0⁺} ||u_{λ}||_{W^{2,p}(Ω)} = 0$ for all p ≥ 2. Moreover, the function $λ ↦ I_{λ}(u_{λ})$ is negative and decreasing in ]0,λ*[, where $I_{λ}$ is the energy functional related to ($P_{λ}$).
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