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In this paper, we achieve a ground state to a Hartree–Fock type system having an external potential and 3-lower nonlinearity. An interesting finding is that 3-order power-type nonlinearity could be controlled by the Coulomb interaction. [ABSTRACT FROM AUTHOR]

In this work, we consider the initial-value problem associated with a coupled system of generalized Korteweg–de Vries equations. We present a relationship between the best constant for a Gagliardo–Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary wave solutions with minimal mass, the so-called ground state solutions. A characterization of the ground states and the orbital instability of the solitary waves are also established. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac equation -i∑k=13αk∂ku+aβu+V(x)u=f(x,|u|)u,x∈R3.The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground-state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground-state solutions, such as compactness of set of ground-state solutions and exponential decay of ground-state solutions. [ABSTRACT FROM AUTHOR]