Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.     

Landesman-Lazer type conditions for a system of p-Laplacian like operators

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems.     

Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation

Using theory of global attractors for multi-valued semiflows, we prove the existence of a global attractor for the m-semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain.     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

A priori estimates and existence of positive solutions for strongly nonlinear problems

In this paper we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-Laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Liouville-type theorem to obtain a priori estimates. Some Harnack-type inequalities which are needed in our reasonings are also proved.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.     

Solvability of nonlinear variational-hemivariational inequalities

In this paper we study nonlinear elliptic differential equations driven by the p-Laplacian with unilateral constraints produced by the combined effects of a monotone term and of a nonmonotone term (variational-hemivariational inequality). Our approach is variational and uses the subdifferential theory of nonsmooth functions and the theory of accretive and monotone operators. Also using these ideas and a special choice of the monotone term, we prove the existence of a strictly positive smooth solution for a class of nonlinear equations with nonsmooth potential (hemivariational inequality).     

On a new system of nonlinear A-monotone multivalued variational inclusions

In this paper, we introduce and study a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces. By using the concept and properties of A-monotone mappings, and the resolvent operator technique associated with A-monotone mappings due to Verma, we construct a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with A-monotone mappings in Hilbert spaces. We also prove the existence of solutions for the nonlinear multivalued variational inclusions and the convergence of iterative sequences generated by the algorithm. Our results improve and generalize many known corresponding results.     

Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first p-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain L ^∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L ^p norms.     

Gradient estimates via linear and nonlinear potentials

We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by −Δpu=μ. In particular, no matter the nonlinearity of the equations considered, we show that in the case p?2 a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case p?2. In the case p>2 we prove a new gradient estimate employing nonlinear potentials of Wolff type.     

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.     

On a class of boundary value problems involving the p-biharmonic operator

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p>1. It describes various problems in the theory of elasticity, e.g., the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p−1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

Increasing variational solutions for a nonlinear p-laplace equation without growth conditions

By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the p-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.     

Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs

In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher-dimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, via a contraction argument, the existence and local uniqueness of solutions for a rather large class of nonlinear problems. We apply this technique to prove existence and local uniqueness of equilibrium solutions for the Cahn-Hilliard and the Swift-Hohenberg equations defined on two- and three-dimensional spatial domains.     

Coefficient bounds for biholomorphic mappings which have a parametric representation

The main purpose of this paper is to prove a collection of new fixed point theorems for so-called weakly F-contractive mappings. By analogy, we introduce also a class of strongly F-expansive mappings and we prove fixed point theorems for such mappings. We provide a few examples, which illustrate these results and, as an application, we prove an existence and uniqueness theorem for the generalized Fredholm integral equation of the second kind. Finally, in Appendix A, we apply the Mönch fixed point theorem to prove two results on the existence of approximate fixed points of some continuous mappings.     

Nonlinear quasistatic problems of gradient type in inelastic deformations theory

In this paper we study nonlinear quasistatic problems from inelastic deformations theory. Only strictly monotone, gradient-type constitutive equations are considered. We prove existence for both coercive and non-coercive models, using energy estimates and Young measures. For non-coercive models we use the L² self-controlling property.     

Large Time Behavior via the Method of ?-Trajectories

The method of ?-trajectories is presented in a general setting as an alternative approach to the study of the large-time behavior of nonlinear evolutionary systems. It can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear. Here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics. In both cases we prove the existence of a finite-dimensional global attractor and the existence of an exponential attractor.     

Ground state periodic solutions for difference equations with discrete p-Laplacian

We use the critical point theory to establish existence results for periodic solutions of some nonlinear boundary value problems involving the discrete p-Laplacian operator. As an application we give an alternative proof to the upper and lower solutions theorem.