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Three solutions with precise sign properties for systems of quasilinear elliptic equations
Dumitru Motreanu
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.
keywords: constant sign solutions Quasilinear elliptic system sub-supersolution variational methods.
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Preface
Siegfried Carl Salvatore A. Marano Dumitru Motreanu
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.
keywords:
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Periodic and homoclinic solutions for a class of unilateral problems
Samir Adly Daniel Goeleven Dumitru Motreanu
This paper contains some existence and multiplicity results for periodic solutions of second order nonautonomous and nonsmooth Hamiltonian systems involving nonconvex superpotentials. This study is achieved by proving the existence of homoclinic solutions. These solutions are constructed as critical points of the corresponding nonconvex and nonsmooth energy functional.
keywords: nonconvex energy functional Hamiltonian system. homoclinic orbit Critical point theory calculus of variation for nondifferentiable functions
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Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method
Dumitru Motreanu Calogero Vetro Francesca Vetro

For the homogeneous Dirichlet problem involving a system of equations driven by $(p,q)$-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic equations. Positive and negative solutions are obtained.

keywords: Dirichlet problem system of elliptic equations (p, q)-Laplacian subsolution-supersolution and gradient dependence
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Location of Nodal solutions for quasilinear elliptic equations with gradient dependence
Dumitru Motreanu Viorica V. Motreanu Abdelkrim Moussaoui

Existence and regularity results for quasilinear elliptic equations driven by $(p, q)$-Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign-changing) solutions is obtained by means of constant-sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are finally established.

keywords: Quasilinear elliptic equations (p, q)-Laplacian gradient dependence nodal solutions constant-sign solutions location sub-supersolutions extremal solutions

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