Existence and multiplicity of solutions for discontinuous elliptic problems in ℝ N

2020 ◽  
pp. 1-25
Author(s):  
Claudianor O. Alves ◽  
Ziqing Yuan ◽  
Lihong Huang
Keyword(s):  
Variational Principle ◽  
Variational Methods ◽  
Multiple Solutions ◽  
Elliptic Problems ◽  
Ekeland’S Variational Principle ◽  
Discontinuous Nonlinearity ◽  
Nontrivial Solutions ◽  
Ekeland's Variational Principle ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

Abstract This paper concerns with the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity. By using dual variational methods, properties of the Nehari manifolds and Ekeland's variational principle, we show how the ‘shape’ of the graph of the function A affects the number of nontrivial solutions.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals SUPERLINEAR ELLIPTIC PROBLEMS WITH SIGN CHANGING COEFFICIENTS

2012 ◽  
Vol 14 (01) ◽  
pp. 1250001 ◽  
Author(s):  
EUGENIO MASSA ◽  
PEDRO UBILLA
Keyword(s):  
Variational Methods ◽  
Critical Case ◽  
Elliptic Problems ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Suitable Space

Via variational methods, we study multiplicity of solutions for the problem [Formula: see text] where a simple example for g(x, u) is |u|p-2u; here a, λ are real parameters, 1 < q < 2 < p ≤ 2* and b(x) is a function in a suitable space Lσ. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any λ > 0, and a total of five nontrivial solutions are obtained when λ is small and a ≥ λ1. Note that this type of results are valid even in the critical case.

scholarly journals Multiple Solutions for a Nonlocal Elliptic Problem Involving p x , q x -Biharmonic Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Qi Zhang ◽  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Sufficient Conditions ◽  
Biharmonic Operator ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

scholarly journals Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Mohammed El Mokhtar Ould El Mokhtar
Keyword(s):  
Variational Principle ◽  
Elliptic Equation ◽  
Critical Exponent ◽  
Elliptic Equations ◽  
Local Minimizer ◽  
Ekeland’S Variational Principle ◽  
Nontrivial Solutions ◽  
Ekeland's Variational Principle

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.

Small perturbations of elliptic problems with variable growth in RN

2021 ◽  
Vol 477 (2249) ◽  
Author(s):  
Bin Ge ◽  
Hai-Cheng Liu ◽  
Bei-Lei Zhang
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Ekeland’S Variational Principle ◽  
Mountain Pass ◽  
Ekeland's Variational Principle ◽  
Small Perturbations ◽  
Whole Space ◽  
Laplacian Equations

In this paper, we study the existence of at least two non-trivial solutions for a class of p ( x )-Laplacian equations with perturbation in the whole space. Using Ekeland’s variational principle and the mountain pass theorem, under appropriate assumptions, we prove the existence of two solutions for the equations.

Existence of nontrivial solution for elliptic systems involving the p(x)-Laplacian

2014 ◽  
Vol 51 (2) ◽  
pp. 213-230
Author(s):  
Ali Taghavi ◽  
Ghasem Afrouzi ◽  
Horieh Ghorbani
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Variational Methods ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Ekeland’S Variational Principle ◽  
Ekeland's Variational Principle

In this paper, we consider the system \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\begin{array}{*{20}c} {\left\{ { - \Delta _{p\left( x \right)} u = \lambda a\left( x \right)\left| u \right|} \right.^{r_1 \left( x \right) - 2} u - \mu b\left( x \right)\left| u \right|^{\alpha \left( x \right) - 2} u\;x \in \Omega } \\ {\left\{ { - \Delta _{q\left( x \right)} \nu = \lambda c\left( x \right)\left| \nu \right|} \right.^{r_2 \left( x \right) - 2} \nu - \mu d\left( x \right)\left| \nu \right|^{\beta \left( x \right) - 2} \nu \;x \in \Omega } \\ {u = \nu = 0\;x \in \partial \Omega } \\ \end{array} } \right.$$ \end{document} where Ω is a bounded domain in ℝN with smooth boundary, λ, μ > 0, p, q, r1, r2, α and β are continuous functions on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\bar \Omega$$ \end{document} satisfying appropriate conditions. We prove that for any μ > 0, there exists λ* sufficiently small, and λ* large enough such that for any λ ∈ (0; λ*) ∪ (λ*, ∞), the above system has a nontrivial weak solution. The proof relies on some variational arguments based on the Ekeland’s variational principle and some adequate variational methods.

Resonant-Superlinear Elliptic Problems Using Variational Methods

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Edcarlos Domingos da Silva ◽  
Bruno Ribeiro
Keyword(s):  
Variational Methods ◽  
Linear Problem ◽  
Elliptic Problems ◽  
First Eigenvalue ◽  
Multiplicity Of Solutions ◽  
The First Eigenvalue ◽  
Existence And Multiplicity ◽  
Resonance Phenomena

AbstractIn this work we establish existence and multiplicity of solutions for resonant-superlinear elliptic problems using appropriate variational methods. The nonlinearity is resonant at −∞ and superlinear at +∞ and the resonance phenomena occurs precisely in the first eigenvalue of the corresponding linear problem. Our main theorems are stated without the well known Ambrosetti-Rabinowitz condition.

Existence and multiplicity of solutions for Kirchhoff–Schrödinger–Poisson system with critical growth

2021 ◽  
Author(s):  
Guofeng Che ◽  
Haibo Chen
Keyword(s):  
Variational Methods ◽  
Critical Growth ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

This paper is concerned with the following Kirchhoff–Schrödinger–Poisson system: [Formula: see text] where constants [Formula: see text], [Formula: see text] and [Formula: see text] are the parameters. Under some appropriate assumptions on [Formula: see text], [Formula: see text] and [Formula: see text], we prove the existence and multiplicity of nontrivial solutions for the above system via variational methods. Some recent results from the literature are greatly improved and extended.

scholarly journals Multiple Solutions for Nonlinear Navier Boundary Systems Involving(p1(x),…,pn(x))-Biharmonic Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Multiplicity Of Solutions ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Variable Exponent Sobolev Spaces

We improve some results on the existence and multiplicity of solutions for the(p1(x),…,pn(x))-biharmonic system. Our main results are new. Our approach is based on general variational principle and the theory of the variable exponent Sobolev spaces.

scholarly journals Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz–Sobolev spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. Heidari ◽  
A. Razani
Keyword(s):  
Variational Methods ◽  
Sobolev Spaces ◽  
Multiple Solutions ◽  
Elliptic Systems ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Quasilinear Elliptic

AbstractIn this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

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