scholarly journals Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t| p–2 t at infinity

2021 ◽  
Vol 10 (1) ◽  
pp. 1039-1060
Author(s):  
Qihan He ◽  
Juntao Lv ◽  
Zongyan Lv
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Pohozaev Identity ◽  
Type Condition ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Harmonic Equation ◽  
Constraint Method ◽  
Harmonic Problem

Abstract We consider the following p-harmonic problem Δ ( | Δ u | p − 2 Δ u ) + m | u | p − 2 u = f ( x , u ) , x ∈ R N , u ∈ W 2 , p ( R N ) , $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2p ≥ 4 and lim t → ∞ f ( x , t ) | t | p − 2 t = l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

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.

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
Image Position

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity

2021 ◽  
pp. 2150002
Author(s):  
Manassés de Souza ◽  
Uberlandio B. Severo ◽  
Thiago Luiz do Rêgo
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
State Level ◽  
Open Interval ◽  
Ground State Solution ◽  
Nodal Solutions ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Fractional Kirchhoff Type Problems ◽  
Kirchhoff Type Problems

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] is a bounded open interval, [Formula: see text] is a continuous potential, the nonlinear term [Formula: see text] has exponential growth of Trudinger–Moser type, [Formula: see text] and [Formula: see text] denotes the standard Gagliardo seminorm of the fractional Sobolev space [Formula: see text]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [Formula: see text] as a positive parameter, we study the behavior of the nodal solutions as [Formula: see text].

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.

New Super-quadratic Conditions on Ground State Solutions for Superlinear Schrödinger Equation

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
X. H. Tang
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
Nontrivial Solutions ◽  
Subcritical Nonlinearity ◽  
A New Technique ◽  
Cerami Sequences ◽  
Semilinear Schrödinger Equation ◽  
Funct Anal

AbstractConsider the semilinear Schrödinger equationwhere f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ + V. Based on the work of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θfor the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009] and Liu [Calc. Var. 45: 1-9, 2012].

Existence and Concentration of Solutions for Choquard Equations with Steep Potential Well and Doubly Critical Exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yong-Yong Li ◽  
Gui-Dong Li ◽  
Chun-Lei Tang
Keyword(s):  
Critical Exponents ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Riesz Potential ◽  
Nehari Manifold ◽  
Linking Theorem ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Steep Potential Well

AbstractIn this paper, we investigate the non-autonomous Choquard equation-\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N},where N\geq 4, \lambda>0, V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, I_{\alpha} is the Riesz potential of order \alpha\in(0,N) and F(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}}, in which 2_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2_{*}^{\alpha}=\frac{N+\alpha}{N} are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in \mathbb{R}^{N}; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in \mathbb{R}^{N}.

scholarly journals Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

2021 ◽  
Author(s):  
Xilin Dou ◽  
xiaoming he
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Concentration Compactness Principle ◽  
Whole Space ◽  
Compactness Principle ◽  
Vanishing Potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

scholarly journals Elliptic problem driven by different types of nonlinearities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Debajyoti Choudhuri ◽  
Dušan D. Repovš
Keyword(s):  
Nontrivial Solution ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Different Types ◽  
Exponential Critical Growth

AbstractIn this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: $$\begin{aligned} \begin{aligned} (-\Delta )^{\frac{1}{2}}u+u+\bigl(\ln \vert \cdot \vert * \vert u \vert ^{2}\bigr)&=f(u)+\mu \vert u \vert ^{- \gamma -1}u,\quad \text{in }\mathbb{R}, \end{aligned} \end{aligned}$$ ( − Δ ) 1 2 u + u + ( ln | ⋅ | ∗ | u | 2 ) = f ( u ) + μ | u | − γ − 1 u , in  R , where $\mu >0$ μ > 0 , $(*)$ ( ∗ ) is the convolution operation between two functions, $0<\gamma <1$ 0 < γ < 1 , f is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity f is of exponential critical growth.

scholarly journals Existence of multiple nontrivial solutions of the nonlinear Schrödinger-Korteweg-de Vries type system

2021 ◽  
Vol 11 (1) ◽  
pp. 636-654
Author(s):  
Qiuping Geng ◽  
Jun Wang ◽  
Jing Yang
Keyword(s):  
Positive Solution ◽  
Bifurcation Theory ◽  
Type System ◽  
Ground State Solution ◽  
Energy Estimates ◽  
Nontrivial Solutions ◽  
Interesting Phenomenon ◽  
Nonlinear Schrödinger ◽  
De Vries ◽  
Constraint Method

Abstract In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

scholarly journals Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

2021 ◽  
Vol 19 (1) ◽  
pp. 297-305
Author(s):  
Yuting Zhu ◽  
Chunfang Chen ◽  
Jianhua Chen ◽  
Chenggui Yuan
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Multiple Solutions ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Petviashvili Equation

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

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