scholarly journals The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaoyao Jia ◽  
Zhenluo Lou
Keyword(s):  
Sobolev Space ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Quasilinear Equation ◽  
Minimax Theorem ◽  
Nontrivial Solutions ◽  
Quasilinear Equations

In this paper, we study the following quasilinear equation: − div ϕ ∇ u ∇ u + ϕ u u = f u   in   ℝ N , where ϕ ∈ C 1 ℝ + , ℝ + and Φ t = ∫ 0 t s ϕ ∣ s ∣ d s . In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.

scholarly journals NONTRIVIAL SOLUTIONS OF p-SUPERLINEAR p-LAPLACIAN PROBLEMS VIA A COHOMOLOGICAL LOCAL SPLITTING

2010 ◽  
Vol 12 (03) ◽  
pp. 475-486 ◽  
Author(s):  
MARCO DEGIOVANNI ◽  
SERGIO LANCELOTTI ◽  
KANISHKA PERERA
Keyword(s):  
Laplace Operator ◽  
Nontrivial Solution ◽  
Morse Theory ◽  
Quasilinear Equation ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Superlinear Nonlinearity ◽  
Sign Condition ◽  
Mountain Pass Geometry

We consider a quasilinear equation, involving the p-Laplace operator, with a p-superlinear nonlinearity. We prove the existence of a nontrivial solution, also when there is no mountain pass geometry, without imposing a global sign condition. Techniques of Morse theory are employed.

scholarly journals A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality

2018 ◽  
Vol 20 (04) ◽  
pp. 1750037 ◽  
Author(s):  
Fashun Gao ◽  
Minbo Yang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Nontrivial Solution ◽  
Sobolev Inequality ◽  
Schrodinger Equation ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Semilinear Equation ◽  
Nonlinear Choquard Equation

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

scholarly journals Multiplicity solutions of a class fractional Schrödinger equations

2017 ◽  
Vol 15 (1) ◽  
pp. 1010-1023
Author(s):  
Li-Jiang Jia ◽  
Bin Ge ◽  
Ying-Xin Cui ◽  
Liang-Liang Sun
Keyword(s):  
Sobolev Space ◽  
Variational Methods ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Operator ◽  
Nontrivial Solutions ◽  
Fractional Sobolev Space ◽  
Fractional Schrödinger Equations

Abstract In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations $$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$ where $ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

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 On a p x -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hafid Lebrimchi ◽  
Mohamed Talbi ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Problem

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Existence of Two Solutions for a Second-Order Discrete Boundary Value Problem

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Pasquale Candito ◽  
Giovanni Molica Bisci
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Nontrivial Solutions ◽  
Discrete Boundary Value Problem

AbstractThe existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

scholarly journals A Hopf-type Boundary Point Lemma for Pairs of Solutions to Quasilinear Equations

2019 ◽  
Vol 62 (3) ◽  
pp. 607-621 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Weak Solutions ◽  
Boundary Point ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Quasilinear Equations ◽  
Differentiable Functions ◽  
The Difference ◽  
Type Boundary

AbstractWe present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

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 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.

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