On a p-Kirchhoff problem involving a critical nonlinearity

2014 ◽  
Vol 352 (4) ◽  
pp. 295-298 ◽  
Author(s):  
Anass Ourraoui
Keyword(s):  
Critical Nonlinearity ◽  
Kirchhoff Problem

scholarly journals A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

2017 ◽  
Vol 8 (1) ◽  
pp. 645-660 ◽  
Author(s):  
Alessio Fiscella
Keyword(s):  
Singular Term ◽  
Nonlocal Problem ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Subset ◽  
Sobolev Exponent ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Kirchhoff Problem

Abstract In this paper, we consider the following critical nonlocal problem: \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is an open bounded subset of {\mathbb{R}^{N}} with continuous boundary, dimension {N>2s} with parameter {s\in(0,1)} , {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, {\lambda>0} is a real parameter, {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while {(-\Delta)^{s}} is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

scholarly journals On an Elliptic Problem with Critical Nonlinearity in Expanding Annuli

1999 ◽  
Vol 163 (1) ◽  
pp. 29-62 ◽  
Author(s):  
Mohameden O. Ahmedou ◽  
Khalil O. El Mehdi
Keyword(s):  
Elliptic Problem ◽  
Critical Nonlinearity

scholarly journals On a p x -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hafid Lebrimchi ◽  
Mohamed Talbi ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Problem

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

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