scholarly journals Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

2021 ◽  
Vol 10 (1) ◽  
pp. 1328-1355
Author(s):  
Yuxi Meng ◽  
Xinrui Zhang ◽  
Xiaoming He
Keyword(s):  
Fractional Laplacian ◽  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Poisson System ◽  
Real Numbers ◽  
Least Energy Solution ◽  
Sobolev Exponent ◽  
Fractional Laplacian Operator ◽  
Vanishing Potentials ◽  
Least Energy

Abstract In this paper, we study the fractional Schrödinger-Poisson system ( − Δ ) s u + V ( x ) u + K ( x ) ϕ | u | q − 2 u = h ( x ) f ( u ) + | u | 2 s ∗ − 2 u , in   R 3 , ( − Δ ) t ϕ = K ( x ) | u | q , in   R 3 , $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$ where s, t ∈ (0, 1), 3 < 4s < 3 + 2t, q ∈ (1, 2 s ∗ $\begin{array}{} \displaystyle 2^*_s \end{array}$ /2) are real numbers, (−Δ) s stands for the fractional Laplacian operator, 2 s ∗ := 6 3 − 2 s $\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$ is the fractional critical Sobolev exponent, K, V and h are non-negative potentials and V, h may be vanish at infinity. f is a C 1-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth

2020 ◽  
Vol 20 (3) ◽  
pp. 579-597
Author(s):  
Guozhen Lu ◽  
Yansheng Shen
Keyword(s):  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Laplacian Equation ◽  
Concentration Compactness Principle ◽  
Sobolev Exponent ◽  
Laplacian Operators ◽  
Special Case

AbstractIn this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:\left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right.where {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, {p>1}, {s\in(0,1)}, {ps<N}, {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with {p<q\leq p^{\ast}_{s}} and {Q_{u}} and {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., {s=1}) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian {(-\Delta)^{s}} (i.e., {p=2} and {0<s<1}) has not been studied in the literature before.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

scholarly journals Remarks on the Generalized Fractional Laplacian Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 320 ◽  
Author(s):  
Chenkuan Li ◽  
Changpin Li ◽  
Thomas Humphries ◽  
Hunter Plowman
Keyword(s):  
Differential Equations ◽  
Fractional Laplacian ◽  
Laplacian Operator ◽  
Generalized Convolution ◽  
Riesz Fractional Derivative ◽  
Derivative Operator ◽  
Delta Sequence ◽  
Remote Sites ◽  
Definition Of ◽  
Fractional Laplacian Operator

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [Formula: see text].

scholarly journals On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth

2002 ◽  
Vol 7 (10) ◽  
pp. 547-561 ◽  
Author(s):  
Marco A. S. Souto
Keyword(s):  
Bounded Domain ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Dirichlet Problems ◽  
Growth Problem ◽  
Sobolev Exponent ◽  
Least Energy Solutions ◽  
Least Energy

We study the location of the peaks of solution for the critical growth problem−ε 2Δu+u=f(u)+u 2*−1,u>0inΩ,u=0on∂Ω, whereΩis a bounded domain;2*=2N/(N−2),N≥3, is the critical Sobolev exponent andfhas a behavior likeup,1<p<2*−1.

scholarly journals Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Sihua Liang ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
State Energy ◽  
Convergence Property ◽  
Degree Theory ◽  
Critical Sobolev Exponent ◽  
Topological Degree Theory ◽  
Nodal Solutions ◽  
Sobolev Exponent ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where $$N >sp$$ N > s p with $$s \in (0, 1)$$ s ∈ ( 0 , 1 ) , $$p>1$$ p > 1 , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$ [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , $$p_s^*=Np/(N-ps)$$ p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N $$(N\ge 3)$$ ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and $$\lambda $$ λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$ u b . Moreover, for any $$\lambda > 0$$ λ > 0 , we show that the energy of $$u_b$$ u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$ b → 0 .

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