# American Institue of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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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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Siegfried Carl , Salvatore A. Marano and  Dumitru Motreanu
2012, 5(4): i-i doi: 10.3934/dcdss.2012.5.4i +[Abstract](34) +[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.
Ravi P. Agarwal , Kanishka Perera and  Zhitao Zhang
2012, 5(4): 707-714 doi: 10.3934/dcdss.2012.5.707 +[Abstract](33) +[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.
Giuseppina Barletta and  Gabriele Bonanno
2012, 5(4): 715-727 doi: 10.3934/dcdss.2012.5.715 +[Abstract](22) +[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.
Gabriele Bonanno and  Beatrice Di Bella
2012, 5(4): 729-739 doi: 10.3934/dcdss.2012.5.729 +[Abstract](38) +[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.
Pasquale Candito and  Giovanni Molica Bisci
2012, 5(4): 741-751 doi: 10.3934/dcdss.2012.5.741 +[Abstract](40) +[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.
Antonia Chinnì and  Roberto Livrea
2012, 5(4): 753-764 doi: 10.3934/dcdss.2012.5.753 +[Abstract](37) +[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.
Giuseppina D’Aguì , Salvatore A. Marano and  Nikolaos S. Papageorgiou
2012, 5(4): 765-777 doi: 10.3934/dcdss.2012.5.765 +[Abstract](32) +[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.
Francesca Faraci and  Antonio Iannizzotto
2012, 5(4): 779-788 doi: 10.3934/dcdss.2012.5.779 +[Abstract](42) +[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.
Alexandru Kristály and  Ildikó-Ilona Mezei
2012, 5(4): 789-796 doi: 10.3934/dcdss.2012.5.789 +[Abstract](30) +[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.
2012, 5(4): 797-808 doi: 10.3934/dcdss.2012.5.797 +[Abstract](32) +[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.
2012, 5(4): 809-818 doi: 10.3934/dcdss.2012.5.809 +[Abstract](29) +[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.
2012, 5(4): 819-830 doi: 10.3934/dcdss.2012.5.819 +[Abstract](25) +[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 $\alpha\in (0,1)$.
2012, 5(4): 831-843 doi: 10.3934/dcdss.2012.5.831 +[Abstract](31) +[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.
2012, 5(4): 845-855 doi: 10.3934/dcdss.2012.5.845 +[Abstract](31) +[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.
2012, 5(4): 857-864 doi: 10.3934/dcdss.2012.5.857 +[Abstract](27) +[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.
Patrick Winkert and  Rico Zacher
2012, 5(4): 865-878 doi: 10.3934/dcdss.2012.5.865 +[Abstract](40) +[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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