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 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.

Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Qing-Mei Zhou ◽  
Ke-Qi Wang
Keyword(s):  
Eigenvalue Problem ◽  
Critical Points ◽  
Fractional Laplacian ◽  
Nonlinear Eigenvalue Problem ◽  
Elliptic Problems ◽  
Nonlinear Elliptic Problems ◽  
Three Solutions ◽  
Multiplicity Of Solutions ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

AbstractIn this paper we consider a nonlinear eigenvalue problem driven by the fractional Laplacian. By applying a version of the three-critical-points theorem we obtain the existence of three solutions of the problem in

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

2016 ◽  
Vol 16 (1) ◽  
pp. 51-65 ◽  
Author(s):  
Salvatore A. Marano ◽  
Sunra J. N. Mosconi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Multiple Solutions ◽  
First Eigenvalue ◽  
Reaction Term ◽  
The First Eigenvalue

AbstractThe existence of multiple solutions to a Dirichlet problem involving the ${(p,q)}$-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of ${-\Delta_{p}}$ in ${W^{1,p}_{0}(\Omega)}$ and exhibits a concave behavior near zero.

scholarly journals Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yong-Yi Lan ◽  
Xian Hu ◽  
Bi-Yun Tang
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

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 Nonresonance conditions on the potential for a nonlinear nonautonomous Neumann problem

2019 ◽  
Vol 38 (3) ◽  
pp. 79-96 ◽  
Author(s):  
Ahmed Sanhaji ◽  
A. Dakkak
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Problems ◽  
Neumann Boundary ◽  
Existence Results ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
The First Eigenvalue

The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.

scholarly journals Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problems

2014 ◽  
Vol 33 (2) ◽  
pp. 203-217 ◽  
Author(s):  
El Miloud Hssini ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Technical Approach ◽  
Multiplicity Of Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problems

This paper is concerned with the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type equations with Neumann boundary condition. Our technical approach is based on variational methods.

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