Ground state solutions for some indefinite variational problems

Abstract

We consider the nonlinear stationary Schrödinger equation − u + V (x)u = f (x, u) in R N . Here f is a superlinear, subcritical nonlinearity, and we mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of − + V . Inspired by previous work of Li et al. (2006) [11] andPankov (2005) [13], we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed. For odd f , we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain.

Generalized Fountain Theorem and applications to strongly indefinite semilinear problems

“…Both Y and Z are infinite-dimensional, so the functional ϕ is strongly indefinite. This case has been of large interest in the last two decades, existence and multiplicity results have been obtained by different methods, see for instance [2,13,14,15,16,17,18]. We prove in this paper that, under the above assumptions, pPq has infinitely many large energy solutions.…”

Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

“…Remark 2.3 In [16], A. Mai and Z. Zhou treated the discrete nonlinear Schrödinger equation with superquadratic nonlinearity, they required the condition (f 4 ) s → f n (s)/|s| is strictly increasing on (-∞, 0) and (0, ∞) for all n ∈ Z, and obtained ground state solutions by using the generalized Nehari manifold approach developed by Szulkin and Weth [27]. In [15], they considered the following DNLS equation in M dimensional lattices:…”

Homoclinic orbits of superlinear Hamiltonian systems

“…We should mention that some results concerning the existence of ground state solutions for Schrödinger equations have been obtained by [16,23]. Our main idea lies in the application of a variant generalized weak linking theorem for a strongly indefinite problem developed by Schechter and Zou [15].…”

“…(2) The proof for the case λ = 1 is contained in [16]; we outline it here for the completeness of the paper. Suppose by contradiction that there exist…”