scholarly journals Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1792
Author(s):  
Yun-Ho Kim
Keyword(s):  
Potential Function ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Laplacian Operator ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Continuous Potential ◽  
The Given ◽  
Laplacian Equations

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

scholarly journals Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝn

2018 ◽  
Vol 7 (3) ◽  
pp. 247-257 ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Continuous Function ◽  
Potential Function ◽  
Multiple Solutions ◽  
Laplacian Operator ◽  
Multiplicity Result ◽  
Kirchhoff Type ◽  
Perturbation Term ◽  
New Criterion ◽  
Laplacian Equations

AbstractIn this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff typeM\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x)in {\mathbb{R}^{n}}, where (-Δ)_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡\infty, ps¡n, f : \mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R} is a continuous function, V : \mathbb{R}^{n}\to\mathbb{R}^{+} is a potential function and g : \mathbb{R}^{n}\to\mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L^{p^{\prime}}(\mathbb{R}^{n}), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions.

Solutions of nonlinear problems involving p(x)-Laplacian operator

2015 ◽  
Vol 4 (4) ◽  
pp. 285-293 ◽  
Author(s):  
Zehra Yücedağ
Keyword(s):  
Variational Principle ◽  
Sobolev Space ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Nonlocal Problem ◽  
Nonlinear Problems ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Fountain Theorem ◽  
Existence And Multiplicity

AbstractIn the present paper, by using variational principle, we obtain the existence and multiplicity of solutions of a nonlocal problem involving p(x)-Laplacian. The problem is settled in the variable exponent Sobolev space W01,p(x)(Ω), and the main tools are the Mountain-Pass theorem and Fountain theorem.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

scholarly journals Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Defang Ma ◽  
Zhan Zhou
Keyword(s):  
Difference Equations ◽  
Mountain Pass Theorem ◽  
Sufficient Conditions ◽  
Homoclinic Solutions ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Special Cases ◽  
Discrete Nonlinear Schrödinger Equations ◽  
The Difference

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

scholarly journals Existence and multiplicity results for quasilinear elliptic equations

2005 ◽  
Vol 71 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Wei Dong
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Laplacian Operator ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Bounded Open Subset ◽  
Multiplicity Of Positive Solutions ◽  
Quasilinear Elliptic ◽  
Laplacian Operators

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f(x, s), we show the follwing problem: , where Ω is a bounded open subset of RN, N ≥ 2, with smooth boundary, λ is a positive parameter and ∆p is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large λ.

scholarly journals Existence of Small-Energy Solutions to Nonlocal Schrödinger-Type Equations for Integrodifferential Operators in ℝN

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 5
Author(s):  
Jun Ik Lee ◽  
Yun-Ho Kim ◽  
Jongrak Lee
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Functional Method ◽  
Fountain Theorem ◽  
Small Energy ◽  
Localization Method ◽  
Dual Version ◽  
Symmetric Mountain Pass Theorem ◽  
Integrodifferential Operators ◽  
Modified Functional

We are concerned with the following elliptic equations: ( − Δ ) p , K s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p , K s is the nonlocal integrodifferential equation with 0 < s < 1 < p < + ∞ , s p < N the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies a Carathéodory condition. The present paper is devoted to the study of the L ∞ -bound of solutions to the above problem by employing De Giorgi’s iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose L ∞ -norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem.

scholarly journals Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Lv ◽  
Chun-Lei Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Second Order Hamiltonian Systems ◽  
Superquadratic Potential

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

scholarly journals Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2465-2481 ◽  
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Bitao Cheng ◽  
Huxiao Luo
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Compact Embedding ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Unbounded Potential ◽  
Superquadratic Growth

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations (-?u + V(x)u = K(x)f(u), x ? RN, u ? D1,2(RN), where N ? 3, V(x) and K(x) are both unbounded potential functions and f is a function with a superquadratic growth. Firstly, we prove the existence of infinitely many solutions with compact embedding and by means of symmetric mountain pass theorem. Moreover, we prove the existence of nontrivial solutions without compact embedding in weighted Sobolev spaces and by means of mountain pass theorem. Our results extend and generalize some existing results.

scholarly journals Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Chun Li ◽  
Zeng-Qi Ou ◽  
Chun-Lei Tang
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Linking Theorem ◽  
Nontrivial Solutions ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Local Linking ◽  
Existence And Multiplicity ◽  
Fourth Order Elliptic Equations

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

scholarly journals Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Lizhen Chen ◽  
Anran Li ◽  
Chongqing Wei
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Negative Energy ◽  
Mountain Pass ◽  
Poisson System ◽  
Fountain Theorem ◽  
Symmetric Mountain Pass Theorem ◽  
Dual Fountain Theorem

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.

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