Patrizia Pucci

A mountain pass theorem

143 In other words, the mountain separating 0 and e may even be assumed of “zero” altitude and the conclusion is still true; moreover (as we shall show) if c = a the critical point can be chosen with llx,,ll = R. It is interesting to ask whether this extension of the Ambrosetti- Rabinowitz theorem remains true in the infinite-dimensional case. We prove here that if (1) is strengthened a little, to the form (1’) there exist real numbers u, r, R such that 0 a for every x E A := {X E X: Y a. Moreover, if c = a the critical point can be chosen with r < I/x0/I < R. Roughly speaking, in brief, the mountain pass theorem continues to hold for a mountain of zero altitude, provided it also has non-zero thickness; in addition, if c = a, the “pass” itself occurs precisely on the mountain-i.e., satisfying r < llroll < R. There are two interesting and immediate corollaries. The first one says that a C’ function which has IWO local minimum points also has a third critical point. The second states that a u-periodic C’ function with a local minimum e has a critical point x,, # e + ku, k = 0, + 1, +2 ,.... The precise statements of the above results will be given in the next sec- tions. The proof depends upon a lemma of Clark [3], formulated here in a version suitable to our purpose. In Section 3 some applications are presen- ted for the forced pendulum equation. 2.

Extensions of the mountain pass theorem

On considere une fonctionnelle C 1 :I:X→R ou X est un espace de Banach reel et ou I satisfait la condition de compacite de Palais-Smale. On presente une serie de theoremes qui generalisent le theoreme du col

A strong maximum principle and a compact support principle for singular elliptic inequalities

Abstract Vazquez in 1984 established a strong maximum principle for the classical m-Laplace differential inequality Δ m u−f(u)≤0, where Δ m u =div(| Du | m −2 Du ) and f ( u ) is a non-decreasing continuous function with f (0)=0. We extend this principle to a wide class of singular inequalities involving quasilinear divergence structure elliptic operators, and also consider the converse problem of compact support solutions in exterior domains.

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations

Abstract The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the nonlinearities.

Asymptotic stability for intermittently controlled nonlinear oscillators

The authors prove a number of asymptotic stability theorems for intermittently damped quasi-variational systems, extending and generalizing previous work on the subject.

On the existence of stationary solutions for higher-order p-Kirchhoff problems

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space , endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ Const. and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1 of the Rayleigh quotient for the p(x)-polyharmonic operator .

A note on the strong maximum principle for elliptic differential inequalities

We consider the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. This allows us to give necessary and sufficient conditions for the validity of both principles.

A Representation Theorem for Aumann Integrals

In recent years the study of set-valued functions has been developed extensively by many authors, with applications to mathematical economics and control theory; see Refs. [5, 11, 15, 161. In those papers, three approaches can be distinguished according to whether the range space (values of setvalued functions) is .Z”, a Banach space, or a locally convex topological space. The purpose of this paper is to establish properties of Aumann’s integrals of set-valued functions, F: T+ 2’, whose values are nonempty subsets of a real separable reflexive Banach space X, and to continue the work due to Aumann [2] and Datko [7-81. While previous analysis has always treated the case of special finite nonatomic measure spaces, we focus here on the case of general a-finite nonatomic measure spaces. In this last situation, moreover, the analogous results we establish hold under less stringent hypotheses. More precisely, all through the paper we consider a measure space (T, C, p), where p is supposed to be positive, nonatomic and u-finite, and we give the following statements.

LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC KIRCHHOFF SYSTEMS

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.