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Discrete & Continuous Dynamical Systems - S

2012 , Volume 5 , Issue 4

Issue on Variational Methods in
Nonlinear Elliptic Equations

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Preface
Siegfried Carl, Salvatore A. Marano and Dumitru Motreanu
2012, 5(4): i-i doi: 10.3934/dcdss.2012.5.4i +[Abstract](192) +[PDF](88.0KB)
Abstract:
The present issue intends to provide an exposition of very recent topics and results in the qualitative study of nonlinear elliptic equations or systems such as, e.g., existence, multiplicity, and comparison principles. Emphasis is put on variational techniques, combined with topological arguments and sub-super-solution methods, in both a smooth and non-smooth framework.
    The collected papers investigate a wide range of questions. Let us mention for instance multiple solutions to elliptic equations and systems in bounded or unbounded domains, sub-super-solutions of elliptic problems whose relevant energy functionals can be non-differentiable, singular elliptic equations, asymptotically critical problems on higher dimensional spheres, local $C^1$-minimizers versus local $W^{1,p}$-minimizers.
    Each contribution is original and thoroughly reviewed.
On some nonlocal eigenvalue problems
Ravi P. Agarwal, Kanishka Perera and Zhitao Zhang
2012, 5(4): 707-714 doi: 10.3934/dcdss.2012.5.707 +[Abstract](182) +[PDF](319.0KB)
Abstract:
We study a class of nonlocal eigenvalue problems related to certain boundary value problems that arise in many application areas. We construct a nondecreasing and unbounded sequence of eigenvalues that yields nontrivial critical groups for the associated variational functional using a nonstandard minimax scheme that involves the $\mathbb{Z}_2$-cohomological index. As an application we prove a multiplicity result for a class of nonlocal boundary value problems using Morse theory.
Multiplicity results to elliptic problems in $\mathbb{R}^N$
Giuseppina Barletta and Gabriele Bonanno
2012, 5(4): 715-727 doi: 10.3934/dcdss.2012.5.715 +[Abstract](122) +[PDF](371.7KB)
Abstract:
The aim of this paper is to investigate elliptic variational-hemivariational inequalities on unbounded domains. In particular, by using a recent critical point theorem, existence results of at least two nontrivial solutions are established.
Fourth-order hemivariational inequalities
Gabriele Bonanno and Beatrice Di Bella
2012, 5(4): 729-739 doi: 10.3934/dcdss.2012.5.729 +[Abstract](158) +[PDF](178.5KB)
Abstract:
The aim of this paper is to investigate an ordinary fourth-order hemivariational inequality. By using non-smooth variational methods, infinitely many solutions satisfying this type of inequality, whenever the potential of the nonlinear term has a suitable growth condition or convenient oscillatory assumptions at zero or at infinity, are guaranteed. As a consequence, a multiplicity result for non-smooth fourth-order boundary value problems is pointed out.
Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator
Pasquale Candito and Giovanni Molica Bisci
2012, 5(4): 741-751 doi: 10.3934/dcdss.2012.5.741 +[Abstract](167) +[PDF](353.7KB)
Abstract:
The existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the $p$--biharmonic operator is investigated. Our approach is chiefly based on critical point theory.
Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian
Antonia Chinnì and Roberto Livrea
2012, 5(4): 753-764 doi: 10.3934/dcdss.2012.5.753 +[Abstract](215) +[PDF](393.0KB)
Abstract:
Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian.
Multiple solutions to a Neumann problem with equi-diffusive reaction term
Giuseppina D’Aguì, Salvatore A. Marano and Nikolaos S. Papageorgiou
2012, 5(4): 765-777 doi: 10.3934/dcdss.2012.5.765 +[Abstract](191) +[PDF](363.4KB)
Abstract:
The existence of four solutions, one negative, one positive, and two sign-changing (namely, nodal), for a Neumann boundary-value problem with right-hand side depending on a positive parameter is established. Proofs make use of sub- and super-solution techniques as well as Morse theory.
Three nonzero periodic solutions for a differential inclusion
Francesca Faraci and Antonio Iannizzotto
2012, 5(4): 779-788 doi: 10.3934/dcdss.2012.5.779 +[Abstract](213) +[PDF](359.2KB)
Abstract:
We prove the existence of three non-zero periodic solutions for an ordinary differential inclusion. Our approach is variational and based on a multiplicity theorem for the critical points of a nonsmooth functional, which extends a recent result of Ricceri.
Multiple solutions for a perturbed system on strip-like domains
Alexandru Kristály and Ildikó-Ilona Mezei
2012, 5(4): 789-796 doi: 10.3934/dcdss.2012.5.789 +[Abstract](154) +[PDF](333.1KB)
Abstract:
We prove a multiplicity result for a perturbed gradient-type system defined on strip-like domains. The approach is based on a recent Ricceri-type three critical point theorem.
Stable and unstable initial configuration in the theory wave fronts
Ruediger Landes
2012, 5(4): 797-808 doi: 10.3934/dcdss.2012.5.797 +[Abstract](164) +[PDF](213.7KB)
Abstract:
In this paper we study the wavefront like phase transition of solutions of a parabolic nonlinear boundary value problem used to model phase transitions in the theory of boiling liquids. Using weak supersolutions we provide bounds for the propagation speed of such a phase transition. Also we construct stable supersolutions to initial configurations which have locally supercritical values.
On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces
Vy Khoi Le
2012, 5(4): 809-818 doi: 10.3934/dcdss.2012.5.809 +[Abstract](136) +[PDF](362.1KB)
Abstract:
This paper is about an alternate variational inequality formulation for the boundary value problem $$ \begin{array}{l} -{\rm div} (a(|\nabla u|) \nabla u) + \partial_u G(x,u) \ni 0 \;\mbox{ in } \;\Omega , \\ u=0 \;\mbox{ on } \;\partial\Omega , \end{array} $$ where the principal part may have non-polynomial or very slow growth. As a consequence of this formulation, we can apply abstract nonsmooth linking theorems to study the existence and multiplicity of nontrivial solutions to the above problem.
A variational approach to a class of quasilinear elliptic equations not in divergence form
M. Matzeu and Raffaella Servadei
2012, 5(4): 819-830 doi: 10.3934/dcdss.2012.5.819 +[Abstract](156) +[PDF](196.5KB)
Abstract:
The aim of this paper is to use a variational approach in order to obtain the existence of non-trivial weak solutions of a quasilinear elliptic equation not in divergence form, in dimension $N=3$. Moreover, we prove that our solution is $C^{1, \alpha}(\overline\Omega)$ and also locally $C^{2, \alpha}(\overline\Omega)$ for a suitable $\alpha\in (0,1)$.
Three solutions with precise sign properties for systems of quasilinear elliptic equations
Dumitru Motreanu
2012, 5(4): 831-843 doi: 10.3934/dcdss.2012.5.831 +[Abstract](159) +[PDF](199.8KB)
Abstract:
For a quasilinear elliptic system, the existence of two extremal solutions with components of opposite constant sign is established. If the system has a variational structure, the existence of a third nontrivial solution is shown.
Multiplicity of solutions for variable exponent Dirichlet problem with concave term
V. V. Motreanu
2012, 5(4): 845-855 doi: 10.3934/dcdss.2012.5.845 +[Abstract](157) +[PDF](363.0KB)
Abstract:
We consider a nonlinear Dirichlet boundary value problem involving the $p(x)$-Laplacian and a concave term. Our main result shows the existence of at least three nontrivial solutions. We use truncation techniques and the method of sub- and supersolutions.
Noncoercive elliptic equations with subcritical growth
Vicenţiu D. Rădulescu
2012, 5(4): 857-864 doi: 10.3934/dcdss.2012.5.857 +[Abstract](162) +[PDF](330.3KB)
Abstract:
We study a class of nonlinear elliptic equations with subcritical growth and Dirichlet boundary condition. Our purpose in the present paper is threefold: (i) to establish the effect of a small perturbation in a nonlinear coercive problem; (ii) to study a Dirichlet elliptic problem with lack of coercivity; and (iii) to consider the case of a monotone nonlinear term with subcritical growth. This last feature enables us to use a dual variational method introduced by Clarke and Ekeland in the framework of Hamiltonian systems associated with a convex Hamiltonian and applied by Brezis to the qualitative analysis of large classes of nonlinear partial differential equations. Connections with the mountain pass theorem are also made in the present paper.
A priori bounds for weak solutions to elliptic equations with nonstandard growth
Patrick Winkert and Rico Zacher
2012, 5(4): 865-878 doi: 10.3934/dcdss.2012.5.865 +[Abstract](266) +[PDF](409.7KB)
Abstract:
In this paper we study elliptic equations with a nonlinear conormal derivative boundary condition involving nonstandard growth terms. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems.

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