No abstract

In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.where λ < λ 1 , λ 1 the first eigenvalue of −∆ p and f a subcritical power-type nonlinearity. Also, they considered elliptic problems in R N likeunder some conditions on V and f . These two problems (and other two also considered in [17]) have a common feature which allows the use of the standard Nehari method described by Szulkin and Weth. In fact, the energy functionals associated to them has the "principal part" p−homogenous. This condition, by its side, has been dropped out in the paper of Figueiredo and Ramos [11],

Some superlinear fourth order elliptic equations are considered. A family of solutions is proved to exist and to concentrate at a point in the limit. The proof relies on variational methods and makes use of a weak version of the Ambrosetti-Rabinowitz condition. The existence and concentration of solutions are related to a suitable truncated equation.

In this work it is studied a quasilinear elliptic problem in the whole space R N involving the 1−Laplacian operator, with potentials which can vanish at infinity. The Euler-Lagrange functional is defined in a space whose definition resembles BV (R N ) and, in order to avoid working with extensions of it to some Lebesgue space, we state and prove a version of the Mountain Pass Theorem without the Palais-Smale condition to Lipschitz continuous functionals.

In this work we use variational methods to prove results on existence and concentration of solutions to a problem in R N involving the 1−Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation BV (R N ) is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.