We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: [View the MathML source Turn MathJax on] where k∈C(R3) changes sign in R3, lim∣x∣→∞k(x)=k∞<0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that μ>μ1 and near μ1, where μ1 is the first eigenvalue of −Δ+id in H1(R3) with weight function h, whose corresponding positive eigenfunction is denoted by e1. An interesting phenomenon here is that we do not need the condition View the MathML source, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Costa and Tehrani, 2001).

This article concerns the existence of ground state and bound states, and the study of their bifurcation properties for the Schrödinger-Poisson system(Forumala Presented). Under suitable assumptions on the coefficient h(x), we prove that the ground state must bifurcate from zero, and that another bound state bifurcates from a solution, when µ = µ 1 is the first eigenvalue of −∆u + u = µh(x)u in H 1 (R 3 )

This paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on R involving the p-Laplacian, with a non-autonomous perturbation. The first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain-pass argument. The special features of the problem here is the “complex” structure of the linear part which, in particular, oblige to work into the space W 1,p (R). Then one faces problems in the convergence of the sequences of derivatives un → u.

In this work a review of the supersymmetric quantum mechanics formalism combined with the variational method is done. This approach is useful in order to obtain numerical values for the energy eigenvalues from Schrödinger equation. As an example, the energy eigenvalues from the Lennard-Jones (12,6) potential are determined and the results are compared with other ones obtained from different methods. Copyright by the Sociedade Brasileira de Física.