Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case

Strongly nonlinear second order differential inclusions with generalized boundary conditions

AbstractIn this paper, we study second order differential inclusions in RN with a maximal monotone term and generalized boundary conditions. The nonlinear differential operator need not be necessary homogeneous and incorporates as a special case the one-dimensional p-Laplacian. The generalized boundary conditions incorporate as special cases well-known problems such as the Dirichlet (Picard), Neumann and periodic problems. As application to the proven results we obtain existence theorems for both “convex” and “nonconvex” problems when the maximal monotone term A is defined everywhere and when not (case of variational inequalities)

Three Nontrivial Solutions for a Quasilinear Elliptic Differential Equation at Resonance with Discontinuous Right Hand Side

We study a quasilinear elliptic equation at resonance with discontinuous right hand side. To have an existence theory, we pass to a multivalued version of the problem by filling in the gaps at the discontinuity points. Using the nonsmooth critical point theory of Chang for lically Lipschitz functionals and the Ekeland variational principle, we show that the resulting elliptic inclusion has three distinct nontrivial solutions

Nonlinear Elliptic Eigenvalue Problems with Discontinuities

AbstractIn this paper we study the existence of solution for two different eigenvalue problems. The first is nonlinear and the second is semilinear. Our approach is based on results from the nonsmooth critical point theory. In the first theorem we prove the existence of at least two nontrivial solutions when λ is in a half-axis. In the second theorem (based on a nonsmooth variant of the generalized mountain pass theorem), we prove the existence of at least one nontrivial solution for every λ∈R