scholarly journals Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-33
Author(s):  
Siegfried Carl ◽  
Dumitru Motreanu
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Point Theory ◽  
Energy Functional ◽  
Constant Sign ◽  
Nodal Solutions ◽  
Variational Characterization ◽  
Spectrum Of The Laplacian ◽  
Sign Changing Solutions

The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

Existence of nodal solutions for quasilinear elliptic problems in ℝN

2015 ◽  
Vol 145 (5) ◽  
pp. 937-957
Author(s):  
Ann Derlet ◽  
François Genoud
Keyword(s):  
Elliptic Equations ◽  
Weighted Sobolev Space ◽  
Point Theory ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solutions ◽  
Laplacian Equation ◽  
Quasilinear Elliptic Problems ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Constraint Minimization

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

Multiple Solutions of Sublinear Quasilinear Schrödinger Equations with Small Perturbations

2018 ◽  
Vol 62 (2) ◽  
pp. 471-488
Author(s):  
Liang Zhang ◽  
X. H. Tang ◽  
Yi Chen
Keyword(s):  
Critical Point Theory ◽  
Multiple Solutions ◽  
Point Theory ◽  
Energy Functional ◽  
Negative Energy ◽  
Quasilinear Schrödinger Equation ◽  
Small Perturbations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

AbstractIn this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation $$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$ where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.

scholarly journals Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zuoshi Xie ◽  
Yuanfeng Jin ◽  
Chengmin Hou
Keyword(s):  
Boundary Value Problem ◽  
Critical Point Theory ◽  
Variational Approach ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Point Theory ◽  
Linking Theorem ◽  
Fractional Difference ◽  
Variational Framework

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

scholarly journals Multiple solutions of the Kirchhoff-type problem in RN

2016 ◽  
Vol 1 (1) ◽  
pp. 229-238 ◽  
Author(s):  
Jiahua Jin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Variational Principle ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Nonlinear Partial Differential Equation ◽  
Point Theory ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

AbstractIn this paper, we concern with a class of quasilinear Kirchhoff-type problem. By using the Ekeland’s Variational Principle and Mountain Pass Theorem, the existence of multiple solutions is obtained. Besides, we also take this problem as an example to give the main frame of using critical point theory to find the weak solutions of nonlinear partial differential equation.

Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights

2021 ◽  
pp. 207-218
Author(s):  
Safia Benmansour ◽  
Atika Matallah ◽  
Mustapha Meghnafi
Keyword(s):  
Multiple Solutions ◽  
Elliptic Problems

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

scholarly journals Constant sign and nodal solutions for parametric anisotropic (p, 2) -equations

2021 ◽  
pp. 1-18
Author(s):  
Nikolaos S. Papageorgiou ◽  
Dušan D. Repovš ◽  
Calogero Vetro
Keyword(s):  
Constant Sign ◽  
Nodal Solutions
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