Multiple solutions of quasilinear periodic-parabolic inclusions

The existence of multiple solutions for a class of quasilinear periodic-parabolic boundary value problems is proved. Our proof is based on the one hand on the sub-supersolution method for general quasilinear periodic-parabolic inclusions. On the other hand it relies on the construction of non-overlapping ordered intervals of sub-supersolutions by the use of appropriately designed quasilinear elliptic problems and the maximum and anti-maximum principles. Our approach also allows us to provide an elementary existence proof for semilinear periodic-parabolic problems which have been studied otherwise by rather involved and elaborated abstract topological tools.     

General comparison principle for quasilinear elliptic inclusions

The main goal of this paper is to prove existence and comparison results for elliptic differential inclusions governed by a quasilinear elliptic operator and a multivalued function given by Clarke’s generalized gradient of some locally Lipschitz function. These kinds of problems have been treated in the past by various authors including the authors of this paper. However, in all the works we are aware of, additional assumptions on the structure of the elliptic operator and/or the generalized Clarke’s gradient are needed to get comparison results in terms of sub-supersolutions. Comparison principles were obtained recently, e.g., in the case where the elliptic operator is of potential type, or Clarke’s gradient is required to satisfy some one-sided growth condition, or the sub-supersolutions are supposed to satisfy additional properties. The novelty of this paper is that we are able to obtain a comparison principle without assuming any of the above restrictions. To the best of our knowledge this is the first mathematical treatment of the considered elliptic inclusion in its full generality. The obtained results of this paper complement the development of the sub-supersolution method for nonsmooth problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu.     

Nonexistence of solutions for singular quasilinear differential inequalities with a gradient nonlinearity

This paper is devoted to prove some new nonexistence theorems for the singular quasilinear differential inequalities with a gradient nonlinearity in bounded and unbounded domains. The proofs are based on the test function method developed by Mitidieri and Pohozaev.     

Existence results for quasilinear parabolic hemivariational inequalities

This paper is devoted to the periodic problem for quasilinear parabolic hemivariational inequalities at resonance as well as at nonresonance. By use of the theory of multi-valued pseudomonotone operators, the notion of generalized gradient of Clarke and the property of the first eigenfunction, we build a Landesman-Lazer theory in the nonsmooth framework of quasilinear parabolic hemivariational inequalities.     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations

In this article, we introduce a new approach for proving Maximum Principle type results for viscosity solutions of second-order, fully nonlinear possibly degenerate elliptic equations. This approach leads, in particular, to a better understanding of the conditions on the equation which are necessary to obtain such results. It allows us to provide new comparison results for semilinear and quasilinear equations.     

Isoperimetric inequalities,Wulff shape and related questions for strongly nonlinear elliptic operators

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p--Laplace operator $\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert\nabla u \vert^{p-2}{{\partial u}\over{\partial x_i}})$, the pseudo-p-Laplace operator $\tilde\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert {{\partial u}\over{\partial x_i}} \vert^{p-2} {{\partial u}\over{\partial x_i}})$ and others. We derive the positivity of the first eingefunction, simlicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H ^o. They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls.     

Viscosity solutions to second order partial differential equations on Riemannian manifolds

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x,u,du,d²u)=0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d²u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.     

A comparison principle for weak solutions of the Cauchy problem for quasilinear parabolic inequalities

We obtain a new comparison principle for weak solutions of the Cauchy problem for a wide class of quasilinear parabolic inequalities. This is a nonlinear result with no analogue in linear theory. Received: 13 January 2005     

On some degenerate quasilinear equations involving variable exponents

In the present paper, we are concerned with some degenerate quasilinear equations involving variable exponents. Using various (variational and nonvariational) techniques, we prove existence, nonexistence and multiplicity results.     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)$p(x)$-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.     

Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory

A nonsmooth version of a three critical point theorem of Ricceri (due to Iannizzotto) is used to obtain three anti-periodic solutions for a second-order impulsive differential inclusions with a perturbed nonlinearity and two parameters.     

Structure of least-energy solutions for quasilinear elliptic equations on convex domains

In this paper we study the existence and structure of the least-energy solutions for a class of singularly perturbed quasilinear elliptic equations. Using the moving plane method and a geometric lemma we show that any least-energy solution develops to a single spike-layer solution on convex domains.