A non-smooth three critical points theorem with applications in differential inclusions

We extend a recent result of Ricceri concerning the existence of three critical points of certain non-smooth functionals. Two applications are given, both in the theory of differential inclusions; the first one concerns a non-homogeneous Neumann boundary value problem, the second one treats a quasilinear elliptic inclusion problem in the whole \mathbb R^N{\mathbb R^N}.     

On the continuity of minimizers for quasilinear functionals

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)⁻¹. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.     

Boundary blow-up solutions and their applications in quasilinear elliptic equations

Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions, for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain or over the entireR ^N .     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems

Summary. A new numerical algorithm for solving semilinear elliptic problems is presented. A variational formulation is used and critical points of a C¹-functional subject to a constraint given by a level set of another C¹-functional (or an intersection of such level sets of finitely many functionals) are sought. First, constrained local minima are looked for, then constrained mountain pass points. The approach is based on the deformation lemma and the mountain pass theorem in a constrained setting. Several examples are given showing new numerical solutions in various applications.Mathematics Subject Classification (2000):35J20, 65N99The author would like to thank the referee for helpful comments in particular on Section 4.     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ? ⁿ . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.     

Necessary and Sufficient Conditions of Existence for Positive Solution to a Class of Quasilinear Elliptic Systems in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we establish the necessary and sufficient conditions of existence for a positive solution to a class of non-variational quasilinear elliptic systems in R ^N . The sufficient condition of existence result bases on the Mountain Pass Lemma and the sub-super solution methods, and the necessary condition is a consequence of a Picone’s identity. The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology and so on.     

Cauchy problem for quasilinear hyperbolic systems with higher order dissipative terms

In this paper the author first investigates the global existence, singularites and life span of C ¹ solutions to the Cauchy problem for quasilinear hyperbolic systems with higher order dissipative terms and give some applications with physical interest. Received April 4, 1996     

Quasilinear asymptotically periodic Schrödinger equations with critical growth

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in ${\mathbb{R}^N}$ with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in ${H^1(\mathbb{R}^N)}$ and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.     

Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.     

Generalized solutions for a class of non-uniformaly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence from with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV ^loc and are C ^1,α almost everywhere.     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

The mountain pass method in the problem on a nontrivial solution of a quasilinear equation with a parameter in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In the present paper, we consider a quasilinear elliptic equation in ℝ ^N with a parameter whose values lie in a neighborhood of an eigenvalue of the linear problem. To prove the existence of a nontrivial solution, we use a modification of the conditional mountain pass method. The difficulties related to the lack of compactness of the Sobolev operator in the case of an unbounded domain are eliminated with the use of the Lions concentration-compactness method.     

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted L^s space, the existence of nontrivial solutions is obtained via variational methods.     

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δ _p u = a(x)u ^m −b(x)f(u) with p >  1 and 0 <  m <  p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

An overdetermined problem in nonlinear elliptic equations of second order

We define some functionals involvingu(x) andx'u _i , whereu(x) is a classical solution of the equation (q ^p-2 u _i ) _i +k(u)q ^p =0,p > 1, and prove that such functionals satisfy a second order elliptic differential equation. By a suitable choice of such functionals we investigate an overdetermined problem.Partially supported by Regione Autonoma della Sardegna.     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .