scholarly journals Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in RN

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 436 ◽  
Author(s):  
Jae-Myoung Kim ◽  
Yun-Ho Kim ◽  
Jongrak Lee
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Term ◽  
The Other ◽  
Large Energy ◽  
Small Energy ◽  
Multiplicity Results ◽  
Carathéodory Function

We herein discuss the following elliptic equations: M ∫ R N ∫ R N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y ( - Δ ) p s u + V ( x ) | u | p - 2 u = λ f ( x , u ) i n R N , where ( - Δ ) p s is the fractional p-Laplacian defined by ( - Δ ) p s u ( x ) = 2 lim ε ↘ 0 ∫ R N ∖ B ε ( x ) | u ( x ) - u ( y ) | p - 2 ( u ( x ) - u ( y ) ) | x - y | N + p s d y , x ∈ R N . Here, B ε ( x ) : = { y ∈ R N : | x - y | < ε } , V : R N → ( 0 , ∞ ) is a continuous function and f : R N × R → R is the Carathéodory function. Furthermore, M : R 0 + → R + is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function M and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L ∞ -norm.

scholarly journals Multiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractional p-Laplacian of Order s

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
M. Khiddi
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nehari Manifold ◽  
Lusternik Schnirelmann ◽  
The Nehari Manifold

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s. We show the above result when λ>0 is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Giuseppina D’Aguí
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Asymptotic Condition ◽  
Multiplicity Results ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
Finite Dimensional ◽  
Superlinear Growth ◽  
Discrete Problems ◽  
Minimum Theorem

AbstractIn this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

Existence of solution for quasilinear equations involving local conditions

2019 ◽  
Vol 150 (6) ◽  
pp. 3074-3086
Author(s):  
Patricio Cerda ◽  
Leonelo Iturriaga
Keyword(s):  
Continuous Function ◽  
Nonlinear Term ◽  
Quasilinear Equation ◽  
Existence Of Solution ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Local Conditions ◽  
Existence Of Weak Solutions ◽  
Carathéodory Function ◽  
Image Position

AbstractIn this paper, we study the existence of weak solutions of the quasilinear equation \begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.

scholarly journals Existence Results for A Perturbed Fourth-Order Equation

2017 ◽  
Vol 23 (2) ◽  
pp. 55-65
Author(s):  
Mohammad Reza Heidari Tavani
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Results ◽  
Order Equation ◽  
Fourth Order Equation

‎The existence of at least three weak solutions for a class of perturbed‎‎fourth-order problems with a perturbed nonlinear term is investigated‎. ‎Our‎‎approach is based on variational methods and critical point theory‎.

Multiple solutions for a class of oscillatory discrete problems

2015 ◽  
Vol 4 (3) ◽  
pp. 221-233 ◽  
Author(s):  
Maria Mălin
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Term ◽  
Nonlinear Boundary Value Problem ◽  
Sufficient Condition ◽  
Power Type ◽  
Discrete Problems

AbstractIn this paper, we study a discrete nonlinear boundary value problem that involves a nonlinear term oscillating at infinity and a power-type nonlinearity up. By using variational methods, we establish the existence of a sequence of non-negative weak solutions that converges to +∞ if 0 < p ≤ 1. In the superlinear case, we establish a sufficient condition for the existence of at least n solutions.

Qualitative Analysis of Solutions for a Class of Anisotropic Elliptic Equations with Variable Exponent

2015 ◽  
Vol 59 (3) ◽  
pp. 541-557 ◽  
Author(s):  
G. A. Afrouzi ◽  
M. Mirzapour ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Special Form ◽  
Variable Exponent ◽  
Nonlinear Term ◽  
Fountain Theorem ◽  
Image Position ◽  
Anisotropic Elliptic Equations

AbstractWe are concerned with the degenerate anisotropic problemWe first establish the existence of an unbounded sequence of weak solutions. We also obtain the existence of a non-trivial weak solution if the nonlinear termfhas a special form. The proofs rely on the fountain theorem and Ekeland's variational principle.

scholarly journals Multiplicity Results for a Perturbed Elliptic Neumann Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì
Keyword(s):  
Variational Methods ◽  
Neumann Problem ◽  
Nonlinear Term ◽  
Three Solutions ◽  
Multiplicity Results ◽  
Neumann Problems

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods.

scholarly journals The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ge Dong ◽  
Xiaochun Fang
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Nonlinear Term ◽  
Extremal Solutions ◽  
Quasilinear Elliptic Equations ◽  
Quasilinear Elliptic Problem ◽  
Analysis Theory ◽  
Carathéodory Function ◽  
Quasilinear Elliptic ◽  
Differential Part

We prove the existence of extremal solutions of the following quasilinear elliptic problem -∑i=1N∂/∂xiai(x,u(x),Du(x))+g(x,u(x),Du(x))=0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W01LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth

2017 ◽  
Vol 8 (1) ◽  
pp. 339-371 ◽  
Author(s):  
Kaimin Teng ◽  
Xiaofeng Yang
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Elliptic Equations ◽  
Nonnegative Function ◽  
Quasilinear Elliptic Equations ◽  
Sobolev Critical Exponent ◽  
Concentration Behavior ◽  
Quasilinear Elliptic

Abstract In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent -\varepsilon^{p}\Delta_{p}u-\varepsilon^{p}\Delta_{p}(u^{2})u+V(x)\lvert u% \rvert^{p-2}u=h(u)+\lvert u\rvert^{2p^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplace operator, {p^{*}=\frac{Np}{N-p}} ( {N\geq 3} , {N>p\geq 2} ) is the usual Sobolev critical exponent, the potential {V(x)} is a continuous function, and the nonlinearity {h(u)} is a nonnegative function of {C^{1}} class. Under some suitable assumptions on V and h, we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category.

Semilinear elliptic equations in unbounded domains of Rn

1981 ◽  
Vol 88 (1-2) ◽  
pp. 109-119 ◽  
Author(s):  
Patrizia Donato ◽  
Lucia Migliaccio ◽  
Rosanna Schianchi
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Nonlinear Term ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Quadratic Growth ◽  
Multiplicity Results ◽  
Semilinear Elliptic ◽  
Least Solution ◽  
Semilinear Problem

SynopsisWe study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.

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