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 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].

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.

scholarly journals Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

2019 ◽  
Vol 19 (1) ◽  
pp. 113-132 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Giovany M. Figueiredo ◽  
Teresa Isernia ◽  
Giovanni Molica Bisci
Keyword(s):  
Careful Analysis ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Deformation Lemma ◽  
Fractional Schrödinger Equations ◽  
Fractional Spaces ◽  
The Nehari Manifold ◽  
Sign Changing Solutions

Abstract We consider the following class of fractional Schrödinger equations: (-\Delta)^{\alpha}u+V(x)u=K(x)f(u)\quad\text{in }\mathbb{R}^{N}, where {\alpha\in(0,1)} , {N>2\alpha} , {(-\Delta)^{\alpha}} is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

Degree, quaternions and periodic solutions

2021 ◽  
Vol 379 (2191) ◽  
pp. 20190378
Author(s):  
Jean Mawhin
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Topological Degree ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Brouwer Degree ◽  
Homogeneous Polynomials ◽  
Theme Issue ◽  
Existence And Multiplicity

The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

scholarly journals Regularity and multiplicity results for fractional (p,q)-Laplacian equations

2019 ◽  
Vol 22 (08) ◽  
pp. 1950065 ◽  
Author(s):  
Divya Goel ◽  
Deepak Kumar ◽  
K. Sreenadh
Keyword(s):  
Weak Solutions ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Subcritical Case ◽  
Existence Of A Solution ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold ◽  
Weak Harnack Inequality ◽  
Laplacian Equations

This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.

scholarly journals Existence and Multiplicity of Positive Solutions for Schrödinger-Kirchhoff-Poisson System with Singularity

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Mengjun Mu ◽  
Huiqin Lu
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Nehari Manifold ◽  
Poisson System ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold

We study a singular Schrödinger-Kirchhoff-Poisson system by the variational methods and the Nehari manifold and we prove the existence, uniqueness, and multiplicity of positive solutions of the problem under different conditions.

scholarly journals Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

2021 ◽  
pp. 1-21
Author(s):  
Shubin Yu ◽  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Type System ◽  
Invariant Sets ◽  
Smooth Domain ◽  
Positive Parameter ◽  
Poisson System ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Poisson Type

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in   Ω , − Δ ϕ = u 2 in   Ω , u = ϕ = 0 on   ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .

scholarly journals Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Sihua Liang ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
State Energy ◽  
Convergence Property ◽  
Degree Theory ◽  
Critical Sobolev Exponent ◽  
Topological Degree Theory ◽  
Nodal Solutions ◽  
Sobolev Exponent ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where $$N >sp$$ N > s p with $$s \in (0, 1)$$ s ∈ ( 0 , 1 ) , $$p>1$$ p > 1 , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$ [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , $$p_s^*=Np/(N-ps)$$ p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N $$(N\ge 3)$$ ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and $$\lambda $$ λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$ u b . Moreover, for any $$\lambda > 0$$ λ > 0 , we show that the energy of $$u_b$$ u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$ b → 0 .

scholarly journals Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

Sign in / Sign up

Export Citation Format

Share Document