Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian

2021 ◽  
Vol 33 (1) ◽  
pp. 141-153
Author(s):  
N. Ustinov
Keyword(s):  
Ground State ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Ground State Solution ◽  
Content Type ◽  
Local Case ◽  
Neumann Laplacian ◽  
Laplacian Operators ◽  
Semilinear Problem

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset \mathbb {R}^n , n ≥ 3 n \geq 3 , s ∈ ( 0 , 1 ) s \in (0, 1) , 2 s ∗ = 2 n / ( n − 2 s ) 2^*_s = 2n/(n-2s) . For the local case where s = 1 s = 1 , corresponding results were obtained earlier for the Neumann Laplacian and Neumann p p -Laplacian operators.

Existence of semiclassical ground-state solutions for semilinear elliptic systems

2012 ◽  
Vol 142 (4) ◽  
pp. 867-895 ◽  
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Ground State ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Limit Problem ◽  
State Solution ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Small Positive Parameter ◽  
Image Position

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

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.

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].

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.

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 Choquard equations with critical nonlinearities

2019 ◽  
Vol 22 (04) ◽  
pp. 1950023 ◽  
Author(s):  
Xinfu Li ◽  
Shiwang Ma
Keyword(s):  
Critical Exponents ◽  
Ground State ◽  
Sobolev Inequality ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Type Problem ◽  
Pohozaev Identity ◽  
Choquard Equation ◽  
Critical Nonlinearities ◽  
Constraint Method

In this paper, we study the Brezis–Nirenberg type problem for Choquard equations in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] are the critical exponents in the sense of Hardy–Littlewood–Sobolev inequality and [Formula: see text] is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasing ground-state solution in [Formula: see text] for the problem. To the end, the regularity and the Pohožaev identity of solutions to a general Choquard equation are obtained.

Ground State for a Coupled Elliptic System with Critical Growth

2018 ◽  
Vol 18 (1) ◽  
pp. 1-15
Author(s):  
Huiling Wu ◽  
Yongqing Li
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Content Type ◽  
Critical Nonlinearities ◽  
Differentiable Functions ◽  
Graphic Content

Abstract We study the following coupled elliptic system with critical nonlinearities: \left\{\begin{aligned} &\displaystyle-\triangle{u}+u=f(u)+\beta h(u)K(v),&&% \displaystyle x\in{\mathbb{R}}^{N},\\ &\displaystyle-\triangle{v}+v=g(v)+\beta H(u)k(v),&&\displaystyle x\in{\mathbb% {R}}^{N},\\ &\displaystyle u,v\in H^{1}({\mathbb{R}}^{N}),\end{aligned}\right. where {\beta>0} ; f, g are differentiable functions with critical growth; and {H,K} are primitive functions of h and k, respectively. Under some assumptions on f, g, h and k, we obtain the existence of a positive ground state solution of this system for {N\geq 2} .

Kirchhoff–Schrödinger equations in ℝ2 with critical exponential growth and indefinite potential

2020 ◽  
pp. 2050030
Author(s):  
Marcelo F. Furtado ◽  
Henrique R. Zanata
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Exponential Growth ◽  
Nonlocal Problem ◽  
Type Inequality ◽  
Ground State Solution ◽  
Type Function ◽  
Local Case ◽  
Critical Exponential Growth ◽  
Indefinite Potential

We prove the existence of ground state solution for the nonlocal problem [Formula: see text] where [Formula: see text] is a Kirchhoff type function, [Formula: see text] may be negative and noncoercive, [Formula: see text] is locally bounded and the function [Formula: see text] has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case [Formula: see text]. In the proofs, we apply Variational Methods besides a new Trudinger–Moser type inequality.

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