scholarly journals The Multiplicity of Nontrivial Solutions for a New p x -Kirchhoff-Type Elliptic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chang-Mu Chu ◽  
Yu-Xia Xiao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Existence Of Weak Solutions ◽  
Kirchhoff Problem

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

scholarly journals p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

2018 ◽  
Vol 8 (1) ◽  
pp. 1111-1131 ◽  
Author(s):  
Alessio Fiscella ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

scholarly journals Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 83 ◽  
Author(s):  
Mohammed Massar ◽  
EL Miloud Hssini ◽  
Najib Tsouli
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Saddle Point Theorem ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

MOUNTAIN PASS TYPE SOLUTIONS FOR QUASILINEAR ELLIPTIC INCLUSIONS

2002 ◽  
Vol 04 (04) ◽  
pp. 607-637 ◽  
Author(s):  
PH. CLÉMENT ◽  
M. GARCÍA-HUIDOBRO ◽  
R. MANÁSEVICH
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Maximal Monotone ◽  
Mountain Pass ◽  
Inclusion Problem ◽  
Existence Of Weak Solutions ◽  
Smooth Case ◽  
Quasilinear Elliptic

We establish the existence of weak solutions to the inclusion problem [Formula: see text] where Ω is a bounded domain in ℝN, [Formula: see text], and ψ ∊ ℝ × ℝ is a maximal monotone odd graph. Under suitable conditions on ψ, g (which reduce to subcritical and superlinear conditions in the case of powers) we obtain the existence of non-trivial solutions which are of mountain pass type in an appropriate not necessarily reflexive Orlicz Sobolev space. The proof is based on a version of the Mountain Pass Theorem for a non-smooth case.

scholarly journals Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Boundary Problem ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Ekeland’S Variational Principle ◽  
Laplacian Operator ◽  
Ekeland's Variational Principle ◽  
Existence Of Weak Solutions

We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator:  -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in  Ω,∂u/∂ν=0,on  ∂Ω. Under some appropriate conditions on the functionsp,  e,  a, and  f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.

Existence results for a Kirchhoff type equation in Orlicz–Sobolev spaces

2017 ◽  
Vol 8 (3) ◽  
Author(s):  
EL Miloud Hssini ◽  
Najib Tsouli ◽  
Mustapha Haddaoui
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Existence Results ◽  
Mountain Pass ◽  
Nonlocal Problems ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation

AbstractIn this paper, based on the mountain pass theorem and Ekeland’s variational principle, we show the existence of solutions for a class of non-homogeneous and nonlocal problems in Orlicz–Sobolev spaces.

scholarly journals Multiple solutions for a problem with resonance involving thep-Laplacian

1998 ◽  
Vol 3 (1-2) ◽  
pp. 191-201 ◽  
Author(s):  
C. O. Alves ◽  
P. C. Carrião ◽  
O. H. Miyagaki
Keyword(s):  
Variational Principle ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Three Solutions ◽  
Bounded Functions

In this paper we will investigate the existence of multiple solutions for the problem(P)                                                         −Δpu+g(x,u)=λ1h(x)|u|p−2u,     in     Ω,    u∈H01,p(Ω)whereΔpu=div(|∇u|p−2∇u)is thep-Laplacian operator,Ω⫅ℝNis a bounded domain with smooth boundary,handgare bounded functions,N≥1and1<p<∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).

Existence theorems for fractional p-Laplacian problems

2016 ◽  
Vol 15 (05) ◽  
pp. 607-640 ◽  
Author(s):  
Paolo Piersanti ◽  
Patrizia Pucci
Keyword(s):  
Weak Solutions ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Positive Function ◽  
Existence Results ◽  
Laplacian Operator ◽  
Bounded Domains ◽  
Lipschitz Boundary ◽  
Nontrivial Solutions ◽  
Existence Of Weak Solutions

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.

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.

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

scholarly journals Existence results for some nonlinear degenerate problems in the anisotropic spaces

2021 ◽  
Vol 39 (6) ◽  
pp. 53-66
Author(s):  
Mohamed Boukhrij ◽  
Benali Aharrouch ◽  
Jaouad Bennouna ◽  
Ahmed Aberqi
Keyword(s):  
Elliptic Problem ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Sign Condition ◽  
Degenerate Elliptic ◽  
Anisotropic Spaces ◽  
Nonlinear Degenerate

Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.

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