Nonlocal ▫pp▫-Kirchhoff equations with singular and critical nonlinearity terms

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ▫begin{cases} ([u]_{s,p}^p)^{sigma-1}(-Delta)^s_p u = frac{lambda}{u^{gamma}}+u^{ p_s^{*}-1} & quad text{in }Omega,\ u>0, & quad text{in }Omega,\ u=0, & quad text{in }mathbb{R}^{N}setminus Omega, end{cases}▫ where ▫$Omega$▫ is a bounded domain in ▫$mathbb{R}^N$▫ with the smooth boundary ▫$partial Omega$▫, ▫$0 sp$, $1<sigma<p^*_s/p,$▫ with ▫$p_s^{*}=frac{Np}{N-ps},$▫ ▫$(- Delta )_p^s$▫ is the nonlocal ▫$p$▫-Laplace operator and ▫$[u]_{s,p}$▫ is the Gagliardo $p$-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem

Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities

In this article, we study certain critical Schrödinger-Kirchhoff-type systems involving the fractional $p$-Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such systems

Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities

In this article, we study certain critical Schrödinger-Kirchhoff-type systems involving the fractional pp-Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such systems

The Nehari manifold for a boundary value problem involving Riemann–Liouville fractional derivative

Abstract We aim to investigate the following nonlinear boundary value problems of fractional differential equations: (Pλ){−tD1α(|0Dtα(u(t))|p−20Dtαu(t))=f(t,u(t))+λg(t)|u(t)|q−2u(t)(t∈(0,1)),u(0)=u(1)=0, $$\begin{aligned} (\mathrm{P}_{\lambda}) \left \{ \textstyle\begin{array}{l} -_{t}D_{1}^{\alpha} ( \vert {}_{0}D_{t}^{\alpha}(u(t)) \vert ^{p-2} {}_{0}D_{t}^{\alpha}u(t) ) \\ \quad=f(t,u(t))+\lambda g(t) \vert u(t) \vert ^{q-2}u(t)\quad (t\in(0,1)),\\ u(0)=u(1)=0, \end{array}\displaystyle \right . \end{aligned}$$ where λ is a positive parameter, 2<r<p<q $2< r< p< q$, 12<α<1 $\frac{1}{2}<\alpha < 1$, g∈C([0,1]) $g\in C([0,1])$, and f∈C([0,1]×R,R) $f\in C([0,1]\times\mathbb{R},\mathbb{R})$. Under appropriate assumptions on the function f, we employ the method of Nehari manifold combined with the fibering maps in order to show the existence of solutions to the boundary value problem for the nonlinear fractional differential equations with Riemann–Liouville fractional derivative. We also present an example as an application

Large-scale participation in policy design: citizen proposals for rural development in Tunisia

International audienceMore and more literature and practice recommend involving the public at the early stages of the policy cycle, i.e. issue identification, definition of the policy objectives and policy design. Policy design involves, among others, identifying solutions, ideas or alternatives which may address the policy objectives. Three main arguments are often put forward to advocate for the involvement of stakeholders, or the public, in policy design: a "user-centered " argument (i.e. for the policy to better meet people's priorities), an innovation argument (i.e. to conceive new solutions) and a collective argument (i.e. to identify collective actions and better tackle environmental problems). However, in both research and practice these arguments have been challenged. Research has insufficiently generated evidence of the influence of large-scale participation in policy design on resulting proposed actions. The objective of this paper is to analyze whether a large-scale participatory process leads to action proposals that fit people's priorities and that are innovative and collective. It draws from a land management and rural development policy design experiment conducted in six vulnerable areas of Tunisia. 4,300 direct participants were involved and 11,583 action proposals were collected. Our results highlight the influence of the local circumstances on innovation and the interest towards collective actions. Our results also show that whether policy design is made individually or in group influences the outcomes. The results also suggest that appropriate facilitation can help fostering more collective and innovative actions. We conclude the paper by opening up the idea of hybridizing policy design methods with methods from political and agricultural sciences in order to better understand the drivers and rationalities behind participants' action proposals