scholarly journals Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tianfang Wang ◽  
Wen Zhang
Keyword(s):  
Ground State ◽  
Nehari Manifold ◽  
Existence Results ◽  
Infinitely Many Solutions ◽  
Choquard Equation ◽  
Ground State Solutions ◽  
Existence And Multiplicity ◽  
Variational Tools ◽  
Nonlinear Choquard Equation ◽  
The Nehari Manifold

AbstractIn this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , where $N\geq 3$ N ≥ 3 , $0<\mu <N$ 0 < μ < N , $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ 2 N − μ N ≤ p < 2 N − μ N − 2 , ∗ represents the convolution between two functions. We assume that the potential function $V(x)$ V ( x ) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. C. DE ALBUQUERQUE ◽  
JOÃO MARCOS DO Ó ◽  
EDCARLOS D. SILVA
Keyword(s):  
Large Class ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Nehari Manifold ◽  
Coupled Systems ◽  
Coupling Term ◽  
Ground State Solutions ◽  
Image Position ◽  
The Nehari Manifold

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

scholarly journals On the Existence of Ground State Solutions of the Periodic Discrete Coupled Nonlinear Schrödinger Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Meihua Huang ◽  
Zhan Zhou
Keyword(s):  
Ground State ◽  
Nehari Manifold ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
The Nehari Manifold

We study the existence of ground state solutions of the periodic discrete coupled nonlinear Schrödinger lattice by using the Nehari manifold approach combined with periodic approximations. We show that both of the components of the ground state solutions are not zero.

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.

scholarly journals Multiple solutions for superlinear double phase Neumann problems

2021 ◽  
Vol 116 (1) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Youpei Zhang
Keyword(s):  
Neumann Problem ◽  
Ground State ◽  
Multiple Solutions ◽  
Nehari Manifold ◽  
Ground State Solutions ◽  
Neumann Problems ◽  
Double Phase ◽  
The Nehari Manifold ◽  
Superlinear Reaction

AbstractWe study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

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 Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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