Salvatore A. Marano

On the structure of the critical set of non-differentiable functions with a weak compactness condition

The nature of critical points generated by a Ghoussoubs type min–max principle for locally Lipschitz continuous functionals fulfilling a weak Palais–Smale assumption, which contains the so-called non-smooth Cerami condition, is investigated. Two meaningful special cases are then pointed out; see Theorems 3.5 and 3.6.

MULTIPLE SOLUTIONS FOR A DIRICHLET PROBLEM WITH p -LAPLACIAN AND SET-VALUED NONLINEARITY

The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via suband supersolution methods as well as variational techniques for nonsmooth functions.

Elliptic boundary-value problems with discontinuous nonlinearities

For a class of elliptic boundary value problems with discontinuous nonlinearities, the existence of strong solutions is established. Two applications are then developed. In particular, one of them is devoted to implicit elliptic equations of the form ψ(−Δu)=ϕ(u), where ψ is a continuous function and ϕ has a set of discontinuity points of Lebesgue measure zero. The abstract framework where these problems are studied is that of set-valued analysis.

A deformation theorem and some critical point results for non-differentiable functions

A deformation lemma for functionals which are the sum of a locally Lipschitz continuous function and of a concave, proper and upper semicontinuous function is established. Some critical point theorems are then deduced and an application to a class of elliptic variational-hemivariational inequalities is presented.

Fixed points of contractive multivalued maps

For a class of contractive multivalued maps defined on a complete absolute retract and with closed bounded values, the set of fixed points is proved to be an absolute retract. This result unifies and extends to arbitrary absolute retracts both Theorem 1 by Ricceri [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81 (1987), 283–286] and Theorem 1 by Bressan, Cellina, and Fryszkowski [Proc. Amer. Math. Soc. 112 (1991), 413–418].

SOME REMARKS ON CRITICAL POINT THEORY FOR LOCALLY LIPSCHITZ FUNCTIONS

In this paper, a dual version of the Mountain Pass Theorem and the Generalized Mountain Pass Theorem are extended to functions that are locally Lipschitz only. An application involving elliptic hemivariational inequalities is next examined.

Elliptic eigenvalue problems with highly discontinuous nonlinearities

For a family of elliptic eigenvalue problems with highly discontinuous nonlinearities, the existence of unbounded continua of positive solutions containing (0,0) is established by using techniques and results from set-valued analysis. Some special cases are then presented and discussed.

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

Abstract The existence of multiple solutions to a Dirichlet problem involving the ( p , q )

Implicit elliptic differential equations

{(p,q)}

Space homogeneous solutions to the Cauchy problem for semiconductor Boltzmann equations

-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of - Δ p