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

Abstract

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 J λ , λ ∈ I ⊂ R. Under suitable assumptions, we locate an open subinterval of values λ in I for which J λ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.

MULTIPLE SOLUTIONS FOR EQUATIONS OF p(x)-LAPLACE TYPE WITH NONLINEAR NEUMANN BOUNDARY CONDITION

“…The next result represents the differentiable version of the Arcoya and Carmona; see Theorem 3.10 in [3].…”

“…Motivated by the pioneer work of A. Ambrosetti and P. Rabinowitz in [1], J. Yao [28] showed the existence of nontrivial solutions for the inhomogeneous and nonlinear Neumann boundary value problems involving the p(x)-Laplacian; see [7] for p(x)-Laplace type operator. The purpose of this paper is to establish the existence of at least three solutions for problem (N) as applications of an abstract three critical points theorem [3] which is the extension of the famous result of B. Ricceri [23]. The study about the existence of at least three solutions for elliptic equations has been an interesting topic; see [2,6,4,9,18,21,23,24].…”

“…It is well known that B. Ricceri's theorems in [21,22,23] gave no accurate information on the location and size of an interval of the parameter λ in R for the existence of at least three critical points. The authors in [9] localized the interval for the existence of three solutions for equations of p-Laplace type with various boundary conditions (for example, homogeneous Dirichlet and inhomogeneous Robin problems) which were motivated by the study of D. Arcoya and J. Carmona [3]. Recently, J.-H. Bae, Y.-H. Kim, and C. Zhang [4] established the existence of at least three solutions for equations of p(x)-Laplace type and localized a three critical points interval for this problem which was based on the works of [6,9].…”

“…Recently, J.-H. Bae, Y.-H. Kim, and C. Zhang [4] established the existence of at least three solutions for equations of p(x)-Laplace type and localized a three critical points interval for this problem which was based on the works of [6,9]. In this respect, we determine precisely the intervals of λ's for which problem (N) admits only the trivial solution and for which problem (N) admits at least two nontrivial solutions by applying the three critical points theorems given in [3]. This paper is organized as follows.…”

A non-smooth three critical points theorem for general hemivariational inequality on bounded domains

“…It is well known that many problems in mathematics and physics that comes from the real world by some authors have investigated (see cf. [1], [2], [29], [30], [31]). The applications to nonsmooth variational problems have been seen in (cf.…”