scholarly journals Existence and concentration of positive solutions for a p-fractional Choquard equation

2021 ◽  
Vol 6 (11) ◽  
pp. 12929-12951
Author(s):  
Xudong Shang ◽  
Keyword(s):  
Small Parameter ◽  
Positive Solutions ◽  
Ground State ◽  
Global Minimum ◽  
Ground State Solution ◽  
Global Maximum ◽  
Choquard Equation ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
Fractional Choquard Equation

<abstract><p>In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 &lt; p &lt; \infty $, $ N &gt; ps $, $ 0 &lt; \mu &lt; ps $, $ p &lt; q &lt; \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon &gt; 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.</p></abstract>

Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents

2020 ◽  
Vol 120 (3-4) ◽  
pp. 199-248
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Dongdong Qin ◽  
Bitao Cheng
Keyword(s):  
Asymptotic Behavior ◽  
Ground State ◽  
Multiple Solutions ◽  
Global Minimum ◽  
Global Maximum ◽  
Quasilinear Schrödinger Equation ◽  
Critical Sobolev Exponents ◽  
Wave Solutions ◽  
Change Of Variable ◽  
Concentration Behavior

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3, ε > 0, 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ), V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

2019 ◽  
Vol 19 (4) ◽  
pp. 779-795
Author(s):  
Guangze Gu ◽  
Xianhua Tang
Keyword(s):  
Continuous Function ◽  
Small Parameter ◽  
Ground State ◽  
Sobolev Inequality ◽  
Ground States ◽  
Ground State Solution ◽  
Positive Continuous Function ◽  
Primitive Function ◽  
Concentration Behavior ◽  
Kirchhoff Type Problems

AbstractIn this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity\left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right.where {\varepsilon>0} is a small parameter, {a,b>0}, {\mu\in(0,3)}, {V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for {\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as {\varepsilon\to 0}.

Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent

2019 ◽  
Vol 21 (06) ◽  
pp. 1850027 ◽  
Author(s):  
Zhipeng Yang ◽  
Yuanyang Yu ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Critical Exponent ◽  
Ground State ◽  
Local Minimum ◽  
Fractional Laplacian ◽  
Ground State Solution ◽  
Critical Nonlinearity ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Behavior

We are concerned with the existence and concentration behavior of ground state solutions of the fractional Schrödinger–Poisson system with critical nonlinearity [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text] denotes the fractional Laplacian of order [Formula: see text] and satisfies [Formula: see text]. The potential [Formula: see text] is continuous and positive, and has a local minimum. We obtain a positive ground state solution for [Formula: see text] small, and we show that these ground state solutions concentrate around a local minimum of [Formula: see text] as [Formula: see text].

scholarly journals Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 151
Author(s):  
Huxiao Luo ◽  
Shengjun Li ◽  
Chunji Li
Keyword(s):  
Perturbation Method ◽  
Potential Function ◽  
Ground State ◽  
Fractional Laplacian ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Choquard Equation ◽  
Nonlinear Choquard Equation ◽  
Mass Case ◽  
Fractional Choquard Equation

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Existence and Concentration of Positive Ground State Solutions for Schrödinger-Poisson Systems

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Jun Wang ◽  
Lixin Tian ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Small Parameter ◽  
Ground State ◽  
Ground State Solution ◽  
Real Parameter ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Subcritical Nonlinearity

AbstractIn this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution

scholarly journals Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang ◽  
Qiongfen Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Sobolev Inequality ◽  
Existence Of Solutions ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Pohozaev Identity ◽  
Choquard Equation ◽  
Minimax Methods ◽  
Variable Potential

AbstractIn this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.

Positive solutions for a class of quasilinear problems with critical growth in ℝN

2015 ◽  
Vol 145 (2) ◽  
pp. 411-444
Author(s):  
Jun Wang ◽  
Tianqing An ◽  
Fubao Zhang
Keyword(s):  
Positive Solutions ◽  
Ground State ◽  
Sufficient Conditions ◽  
Ground State Solution ◽  
Laplacian Operator ◽  
Minimax Theorems ◽  
Global Minima ◽  
Concentration Phenomena ◽  
Subcritical Nonlinearity ◽  
Quasilinear Problems

In this paper, we study the existence, multiplicity and concentration of positive solutions for a class of quasilinear problemswhere —Δp is the p-Laplacian operator for is a small parameter, f(u) is a superlinear and subcritical nonlinearity that is continuous in u. Using a variational method, we first prove that for sufficiently small ε > 0 the system has a positive ground state solution uε with some concentration phenomena as ε → 0. Then, by the minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials. Finally, we obtain some sufficient conditions for the non-existence of ground state solutions.

Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent

2018 ◽  
Vol 61 (2) ◽  
pp. 353-369 ◽  
Author(s):  
Dongdong Qin ◽  
Yubo He ◽  
Xianhua Tang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Ground State ◽  
Multiple Solutions ◽  
Type Equation ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Kirchhoff Type ◽  
The Nehari Manifold

AbstractIn this paper, we consider the following critical Kirchhoff type equation:By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when 3 < q < 5. The relation between the number of maxima of Q and the number of positive solutions for the problem is also investigated.

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