ISSN:
1937-1632
eISSN:
1937-1179
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Discrete and Continuous Dynamical Systems - S
August 2012 , Volume 5 , Issue 4
Issue on Variational Methods in
Nonlinear Elliptic Equations
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2012, 5(4): i-i
doi: 10.3934/dcdss.2012.5.4i
+[Abstract](2620)
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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.
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.
2012, 5(4): 707-714
doi: 10.3934/dcdss.2012.5.707
+[Abstract](3216)
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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.
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.
2012, 5(4): 715-727
doi: 10.3934/dcdss.2012.5.715
+[Abstract](2273)
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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.
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.
2012, 5(4): 729-739
doi: 10.3934/dcdss.2012.5.729
+[Abstract](2992)
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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.
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.
2012, 5(4): 741-751
doi: 10.3934/dcdss.2012.5.741
+[Abstract](3078)
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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.
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.
2012, 5(4): 753-764
doi: 10.3934/dcdss.2012.5.753
+[Abstract](2806)
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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.
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.
2012, 5(4): 765-777
doi: 10.3934/dcdss.2012.5.765
+[Abstract](2731)
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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.
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.
2012, 5(4): 779-788
doi: 10.3934/dcdss.2012.5.779
+[Abstract](2658)
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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.
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.
2012, 5(4): 789-796
doi: 10.3934/dcdss.2012.5.789
+[Abstract](2295)
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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.
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](2455)
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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.
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](2395)
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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.
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](2561)
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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 $\alpha\in (0,1)$.
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](2377)
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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.
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](2745)
+[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.
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](2478)
+[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.
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.
2012, 5(4): 865-878
doi: 10.3934/dcdss.2012.5.865
+[Abstract](3415)
+[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.
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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