A multiplicity theorem for problems with the p-Laplacian

Abstract

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ ∈ R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ 2 = the second eigenvalue of (− p , W 1,p 0 (Z)), then the problem has at least three nontrivial solutions. Our approach combines the method of upper-lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p = 2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].

Anisotropic nonlinear Neumann problems

“…Recently there have been three solutions theorems for Dirichlet problems driven by the p-Laplacian ( p = constant). We mention the works of Bartsch-Liu [6], Carl-Perera [8], Dancer-Perera [12], Filippakis-KristalyPapageorgiou [20], Gasiński-Papageorgiou [23], Liu-Liu [30], Papageorgiou-Papageorgiou [34,35] and Zhang-Chen-Li [38]. From the aforementioned works, the p-superlinear case was investigated by Bartsch-Liu [6] and Filippakis-Kristaly-Papageorgiou [20].…”

Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

“…We mention that three solutions theorems for coercive equations were proved by Ambrosetti-Lupo [2], Ambrosetti-Mancini [3], Iannizzotto [24], Struwe [34] for certain parametric semilinear equations (Iannizzotto [24] deals with hemivariational inequalities, while the other consider "smooth" problems) and by Averna-Marano-Motreanu [4], Liu-Liu [27], Liu [28], Papageorgiou-Papageorgiou [33] for problems driven by p-Laplacian (Averna-Marano-Motreanu [4] deal with parametric hemivariational inequalities, while the others examine "smooth" potentials). Our work here is closer to those of Liu-Liu [27] and Liu [28], since no parameter appears in (1.1) and our theorems extend the results of [27] and [28] in many different ways.…”

Constant sign and nodal solutions for a class of nonlinear Dirichlet problems

“…p = 2) ones by Ambrosetti and Lupo [3], Ambrosetti and Mancini [4] and Struwe [19,20] and the nonlinear work of Papageorgiou and Papageorgiou [18].…”

“…In fact, in all the aforementioned works (with the exception of Papageorgiou and Papageorgiou [18]) the problem is semilinear and parametric and the authors produce three nontrivial solutions for certain values of the parameter. The hypotheses on the reaction are more restrictive and they do not prove the existence of nodal solutions.…”