scholarly journals Infinitely many solutions for anisotropic elliptic equations with variable exponent

2021 ◽  
Vol 40 (5) ◽  
pp. 1071-1096
Author(s):  
Abdelrachid El Amrouss ◽  
Ali El Mahraoui
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Infinitely Many Solutions ◽  
Anisotropic Space ◽  
Fountain Theorem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Dual Fountain Theorem ◽  
Anisotropic Elliptic Equations

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak 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 Existence and Multiplicity of Solutions for Sublinear Schrödinger Equations with Coercive Potentials

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Dong-Lun Wu
Keyword(s):  
Growth Conditions ◽  
Schrodinger Equations ◽  
Linking Theorem ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Fountain Theorem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Working Space ◽  
Variant Fountain Theorem

In this study, we consider the following sublinear Schrödinger equations −Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞, where fx,u satisfies some sublinear growth conditions with respect to u and is not required to be integrable with respect to x. Moreover, V is assumed to be coercive to guarantee the compactness of the embedding from working space to LpℝN for all p∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions when fx,u is odd in u by using the variant fountain theorem.

scholarly journals Existence Results for apx-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Libo Wang ◽  
Minghe Pei
Keyword(s):  
Existence Of Solutions ◽  
Type Equation ◽  
Existence Results ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Multiplicity Of Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity

We consider the existence and multiplicity of solutions for thepx-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.

scholarly journals Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

2017 ◽  
Vol 148 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Eun Bee Choi ◽  
Jae-Myoung Kim ◽  
Yun-Ho Kim
Keyword(s):  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Neumann Problems ◽  
Neumann Boundary Value Problem ◽  
Cerami Condition ◽  
Image Position ◽  
Laplace Type

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

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