scholarly journals Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

2021 ◽  
Author(s):  
Gurpreet Singh
Keyword(s):  
Nehari Manifold ◽  
Nonlocal Term ◽  
Choquard Equation ◽  
Nodal Set ◽  
Alpha Beta ◽  
Sign Changing Solutions ◽  
Least Energy

AbstractWe investigate the following problem $$\begin{aligned} -\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta }*\frac{|u|^{b}}{|x|^{\alpha }}\right) \frac{|u|^{b-2}}{|x|^{\alpha }}u+\lambda \left( |x|^{-\gamma }*\frac{|u|^{c}}{|x|^{\beta }}\right) \frac{|u|^{c-2}}{|x|^{\beta }}u \quad \text { in }{\mathbb {R}}^{N}, \end{aligned}$$ - div ( v ( x ) | ∇ u | m - 2 ∇ u ) + V ( x ) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where $$b, c, \alpha , \beta >0$$ b , c , α , β > 0 , $$\theta ,\gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$N\ge 3$$ N ≥ 3 , $$2\le m< \infty$$ 2 ≤ m < ∞ and $$\lambda \in {\mathbb {R}}$$ λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

Nonlocal perturbations of the fractional Choquard equation

2017 ◽  
Vol 8 (1) ◽  
pp. 694-706 ◽  
Author(s):  
Gurpreet Singh
Keyword(s):  
Nehari Manifold ◽  
Choquard Equation ◽  
Minimization Method ◽  
Nodal Set ◽  
Alpha Beta ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Fractional Choquard Equation

Abstract We study the equation (-\Delta)^{s}u+V(x)u=(I_{\alpha}*\lvert u\rvert^{p})\lvert u\rvert^{p-2}u+% \lambda(I_{\beta}*\lvert u\rvert^{q})\lvert u\rvert^{q-2}u\quad\text{in }{% \mathbb{R}}^{N}, where {I_{\gamma}(x)=\lvert x\rvert^{-\gamma}} for any {\gamma\in(0,N)} , {p,q>0} , {\alpha,\beta\in(0,N)} , {N\geq 3} , and {\lambda\in{\mathbb{R}}} . First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

scholarly journals On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

2019 ◽  
Vol 149 (5) ◽  
pp. 1163-1173
Author(s):  
Vladimir Bobkov ◽  
Sergey Kolonitskii
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Nodal Set ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

scholarly journals The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jiangyan Yao ◽  
Wei Han
Keyword(s):  
Linear Growth ◽  
Nehari Manifold ◽  
Doubling Property ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Type Problems

In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a (2p-1)-linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that Mb≠∅. The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter b↘0.

Sign-Changing Solutions of Fractional 𝑝-Laplacian Problems

2019 ◽  
Vol 19 (1) ◽  
pp. 29-53 ◽  
Author(s):  
Xiaojun Chang ◽  
Zhaohu Nie ◽  
Zhi-Qiang Wang
Keyword(s):  
Degree Theory ◽  
Nehari Manifold ◽  
Invariant Sets ◽  
Brouwer Degree ◽  
Minimax Theory ◽  
Existence And Multiplicity ◽  
The Nehari Manifold ◽  
Sign Changing Solutions ◽  
Least Energy Solutions ◽  
Least Energy

Abstract In this paper, we obtain the existence and multiplicity of sign-changing solutions of the fractional p-Laplacian problems by applying the method of invariant sets of descending flow and minimax theory. In addition, we prove that the problem admits at least one least energy sign-changing solution by combining the Nehari manifold method with the constrained variational method and Brouwer degree theory. Furthermore, the least energy of sign-changing solutions is shown to exceed twice that of the least energy solutions.

scholarly journals Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wen Guan ◽  
Hua-Bo Zhang
Keyword(s):  
Continuous Function ◽  
Fourth Order ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Order Equation ◽  
Biharmonic Operator ◽  
Nontrivial Solutions ◽  
Fourth Order Equation ◽  
Sign Changing Solutions ◽  
Least Energy

AbstractThe purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: $$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$ Δ 2 u − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = K ( x ) f ( u ) in  R N , where $5\leq N\leq 7$ 5 ≤ N ≤ 7 , $\Delta ^{2}$ Δ 2 denotes the biharmonic operator, $K(x), V(x)$ K ( x ) , V ( x ) are positive continuous functions which vanish at infinity, and $f(u)$ f ( u ) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.

scholarly journals Nodal solutions of a nonlocal Choquard equation in a bounded domain

2019 ◽  
pp. 1950067
Author(s):  
Changfeng Gui ◽  
Hui Guo
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Nonlocal Term ◽  
Local Perturbation ◽  
Nodal Solution ◽  
Choquard Equation ◽  
Nodal Solutions ◽  
Ground State Solutions ◽  
Variational Functional ◽  
Least Energy

In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane–Emden equation [Formula: see text] in [Formula: see text]. The problem has a variational functional with a nonlocal term [Formula: see text]. The appearance of the nonlocal term makes the variational functional very different from the local case [Formula: see text] for which the problem has ground state solutions and least energy nodal solutions if [Formula: see text]. The problem may also be viewed as a nonlocal Choquard equation with a local perturbation term when [Formula: see text]. For [Formula: see text], we show that although ground state solutions always exist, the existence of least energy nodal solution depends on [Formula: see text]: for [Formula: see text], there does not exist a least energy nodal solution while for [Formula: see text], such a solution exists. Note that [Formula: see text] is a critical value. In the case of a linear local perturbation, i.e. [Formula: see text], if [Formula: see text], the problem has a positive ground state and a least energy nodal solution. However, if [Formula: see text], the problem has a ground state which changes sign. Hence, it is also a least energy nodal solution.

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.

scholarly journals Multiple and least energy sign-changing solutions for Schr¨odinger-Poisson equations in R3 with restraint

2017 ◽  
Vol 13 (3) ◽  
pp. 4763-4778
Author(s):  
Zhaohong Sun
Keyword(s):  
Existence Results ◽  
Invariant Sets ◽  
Poisson Equations ◽  
Sign Changing Solutions ◽  
Least Energy

In this paper, we study the existence of multiple sign-changing solutions with a prescribed Lp+1−norm and theexistence of least energy sign-changing restrained solutions for the following nonlinear Schr¨odinger-Poisson system:−△u + u + ϕ(x)u = λ|u|p−1u, in R3,−△ϕ(x) = |u|2, in R3.By choosing a proper functional restricted on some appropriate subset to using a method of invariant sets of descending flow,we prove that this system has infinitely many sign-changing solutions with the prescribed Lp+1−norm and has a least energy forsuch sign-changing restrained solution for p ∈ (3, 5). Few existence results of multiple sign-changing restrained solutions areavailable in the literature. Our work generalize some results in literature.

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