scholarly journals Multiple Solutions for Double Phase Problems with Hardy Type Potential

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 376
Author(s):  
Chun-Bo Lian ◽  
Bei-Lei Zhang ◽  
Bin Ge
Keyword(s):  
Variational Principle ◽  
Variational Approach ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Phase Operator ◽  
Nontrivial Solutions ◽  
Hardy Type ◽  
Double Phase ◽  
Type Potential

In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.

Multiple solutions for Dirichlet impulsive equations

2014 ◽  
Vol 3 (S1) ◽  
pp. s89-s98 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Shapour Heidarkhani ◽  
Pasquale F. Pizzimenti
Keyword(s):  
Banach Space ◽  
Boundary Conditions ◽  
Critical Points ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Reflexive Banach Space ◽  
Dirichlet Boundary Conditions ◽  
Nontrivial Solutions ◽  
Three Solutions ◽  
Impulsive Equations

AbstractIn this paper we are interested to ensure the existence of multiple nontrivial solutions for some classes of problems under Dirichlet boundary conditions with impulsive effects. More precisely, by using a suitable analytical setting, the existence of at least three solutions is proved exploiting a recent three-critical points result for smooth functionals defined in a reflexive Banach space. Our approach generalizes some well-known results in the classical framework.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

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.

An existence result for multiple solutions to a Dirichlet problem

2017 ◽  
Vol 24 (1) ◽  
pp. 55-62
Author(s):  
Saeid Shokooh ◽  
Ghasem A. Afrouzi ◽  
John R. Graef
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Existence Result ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Three Solutions ◽  
Quasilinear Elliptic Problem ◽  
Critical Point Result ◽  
Quasilinear Elliptic

AbstractThe authors establish the existence of at least three solutions to a quasilinear elliptic problem subject to Dirichlet boundary conditions in a bounded domain in ${\mathbb{R}^{N}}$. A critical point result for differentiable functionals is used to prove the results.

Multiple Solutions for Perturbed Elliptic Equations in Unbounded Domains

2003 ◽  
Vol 3 (1) ◽  
pp. 1-23 ◽  
Author(s):  
A. Salvatore
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Unbounded Domains ◽  
Dirichlet Boundary Conditions

AbstractWe look for solutions of a nonlinear perturbed Schrödinger equation with nonhomogeneous Dirichlet boundary conditions. By using a perturbation method introduced by Bolle, we prove the existence of multiple solutions in spite of the lack of the symmetry of the problem.

scholarly journals Existence results for a sublinear second order Dirichlet boundary value problem on the half-line

2020 ◽  
Vol 40 (5) ◽  
pp. 537-548
Author(s):  
Dahmane Bouafia ◽  
Toufik Moussaoui
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Half Line ◽  
Dirichlet Boundary Value Problem ◽  
Nontrivial Solutions

In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by using Ekeland's variational principle and critical point theory.

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