Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Giuseppina D’Aguí
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Asymptotic Condition ◽  
Multiplicity Results ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
Finite Dimensional ◽  
Superlinear Growth ◽  
Discrete Problems ◽  
Minimum Theorem

AbstractIn this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

A variational approach to multiplicity results for boundary-value problems on the real line

2015 ◽  
Vol 145 (1) ◽  
pp. 13-29 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina Barletta ◽  
Donal O’Regan
Keyword(s):  
Real Line ◽  
Nonlinear Term ◽  
Laplacian Operator ◽  
Multiplicity Result ◽  
Multiplicity Results ◽  
The Real ◽  
Existence And Multiplicity ◽  
Superlinear Growth ◽  
Unbounded Intervals ◽  
Special Case

We study the existence and multiplicity of solutions for a parametric equation driven by the p-Laplacian operator on unbounded intervals. Precisely, by using a recent local minimum theorem we prove the existence of a non-trivial non-negative solution to an equation on the real line, without assuming any asymptotic condition either at 0 or at ∞ on the nonlinear term. As a special case, we note the existence of a non-trivial solution for the problem when the nonlinear term is sublinear at 0. Moreover, under a suitable superlinear growth at ∞ on the nonlinearity we prove a multiplicity result for such a problem.

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 Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

scholarly journals Multiple solutions for Grushin operator without odd nonlinearity

2019 ◽  
Vol 13 (07) ◽  
pp. 2050131 ◽  
Author(s):  
Mohamed Karim Hamdani
Keyword(s):  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Grushin Operator ◽  
Clark’S Theorem ◽  
Superlinear Growth ◽  
Degenerate Operator ◽  
Strongly Degenerate ◽  
Odd Nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

scholarly journals EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL HARDY–SOBOLEV EXPONENTS AND SIGN-CHANGING WEIGHT FUNCTION

2012 ◽  
Vol 17 (3) ◽  
pp. 330-350 ◽  
Author(s):  
Nemat Nyamoradi
Keyword(s):  
Weight Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Sobolev Exponent ◽  
Quasilinear Elliptic

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiple positive solutions for a critical elliptic system with concave—convex nonlinearities

2009 ◽  
Vol 139 (6) ◽  
pp. 1163-1177 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiple solutions for a class of oscillatory discrete problems

2015 ◽  
Vol 4 (3) ◽  
pp. 221-233 ◽  
Author(s):  
Maria Mălin
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Term ◽  
Nonlinear Boundary Value Problem ◽  
Sufficient Condition ◽  
Power Type ◽  
Discrete Problems

AbstractIn this paper, we study a discrete nonlinear boundary value problem that involves a nonlinear term oscillating at infinity and a power-type nonlinearity up. By using variational methods, we establish the existence of a sequence of non-negative weak solutions that converges to +∞ if 0 < p ≤ 1. In the superlinear case, we establish a sufficient condition for the existence of at least n solutions.

scholarly journals AN EXISTENCE RESULT OF ONE NONTRIVIAL SOLUTION FOR TWO POINT BOUNDARY VALUE PROBLEMS

2011 ◽  
Vol 84 (2) ◽  
pp. 288-299 ◽  
Author(s):  
GABRIELE BONANNO ◽  
ANGELA SCIAMMETTA
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Classical Result ◽  
Existence Result ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Existence Results ◽  
Asymptotic Condition ◽  
Point Boundary

AbstractExistence results of positive solutions for a two point boundary value problem are established. No asymptotic condition on the nonlinear term either at zero or at infinity is required. A classical result of Erbe and Wang is improved. The approach is based on variational methods.

scholarly journals Many closed K-magnetic geodesics on $${\mathbb {S}}^2$$

2021 ◽  
Author(s):  
Roberta Musina ◽  
Fabio Zuddas
Keyword(s):  
Dimensional Reduction ◽  
Variational Formulation ◽  
Analytical Approach ◽  
Scalar Function ◽  
Multiplicity Results ◽  
Finite Dimensional Reduction ◽  
Existence And Multiplicity ◽  
Finite Dimensional

AbstractIn this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere $${\mathbb {S}}^2$$ S 2 , where $$K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}$$ K : S 2 → R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem [21] and Bottkoll results [7].

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