scholarly journals The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator

2016 ◽  
pp. 201-217 ◽  
Author(s):  
Abdeljab ar Ghanmi ◽  
Kamel Saoudi
Keyword(s):  
Elliptic Equation ◽  
Laplace Operator ◽  
Nehari Manifold ◽  
Singular Elliptic Equation ◽  
The Nehari Manifold ◽  
Fractional Laplace Operator

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Li Wang ◽  
Tao Han ◽  
Kun Cheng ◽  
Jixiu Wang
Keyword(s):  
Growth Condition ◽  
Ground State ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Nonlocal Term ◽  
Ground State Solutions ◽  
Cerami Sequences ◽  
The Nehari Manifold ◽  
Fractional Laplace Operator

AbstractIn this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities:$$\left\{\begin{array}{ll}\left(a+b{\left[u\right]}_{s}^{2}\right)\,{\left(-{\Delta}\right)}^{s}u+u+\phi \left(x\right)u=\left({\vert x\vert }^{-\mu }\ast F\left(u\right)\right)f\left(u\right)\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \\ {\left(-{\Delta}\right)}^{t}\phi \left(x\right)={u}^{2}\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \end{array}\right.$$where$${\left[u\right]}_{s}^{2}={\int }_{{\mathrm{&#x211d;}}^{3}}{\vert {\left(-{\Delta}\right)}^{\frac{s}{2}}u\vert }^{2}\,\mathrm{d}x={\iint }_{{\mathrm{&#x211d;}}^{3}{\times}{\mathrm{&#x211d;}}^{3}}\frac{{\vert u\left(x\right)-u\left(y\right)\vert }^{2}}{{\vert x-y\vert }^{3+2s}}\,\mathrm{d}x\mathrm{d}y\,\text{,}$$$s,t\in \left(0,1\right)$ with $2t+4s{ >}3,0{< }\mu {< }3-2t,$$f:{\mathrm{&#x211d;}}^{3}{\times}\mathrm{&#x211d;}\to \mathrm{&#x211d;}$ satisfies a Carathéodory condition and (−Δ)s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

On the fractional Lazer-McKenna conjecture with critical growth

2021 ◽  
pp. 1-33
Author(s):  
Qi Li ◽  
Shuangjie Peng
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Reduction Method ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Positive Eigenfunction ◽  
Image Position ◽  
Fractional Laplace Operator

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$ . Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

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