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In this paper, ground-state solutions to a Hartree–Fock type system with a critical growth are studied. Firstly, instead of establishing the local Palais–Smale (P.-S.) condition and estimating the mountain-pass critical level, a perturbation method is used to recover compactness obtain the existence of ground-state solutions. To achieve this, an important step is to get the right continuity of the mountain-pass level on the coefficient in front of perturbing terms. Subsequently, depending on the internal parameters of coupled nonlinearities, whether the ground state is semi-trivial or vectorial is proved.

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]

This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: (−Δ)su+V(x)u=∫RNA(εy)∣u(y)∣p∣x−y∣μdyA(εx)∣u(x)∣p−2u(x),x∈RN,{\left(-\Delta )}^{s}u+V\left(x)u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{A\left(\varepsilon y)| u(y){| }^{p}}{| x-y{| }^{\mu }}{\rm{d}}y\right)A\left(\varepsilon x)| u\left(x){| }^{p-2}u\left(x),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where s∈(0,1)s\in \left(0,1), N>2sN\gt 2s, 0

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 consider the existence of ground-state solutions to nonlinear Kirchhoff-type transmission problems by using the methods from (Silvia Cingolani and Tobias Weth, On the planar Schrödinger– Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, 169–197). Here, we avoid the conditions under which the nonlinear terms of f and g are forms of C1 . In particular, when N = 2, the existence of ground-state solutions is established to the Kirchhoff-type transmission problem with exponent-type nonlinearity. [ABSTRACT FROM AUTHOR]

The paper deals with the following problem: where is a parameter, and , . Under some certain assumptions on V(x), we establish the existence of ground-state solutions for quasi-linear elliptic equations with Sobolev exponent via the method of Pohožaev manifold, the monotonic trick and global compactness lemma. Some recent results are extended. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equations: [ABSTRACT FROM AUTHOR]

In this article, we deal with the following Schrödinger systems in ℝN:where λ > 0 is a parameter, α > 1, β > 1, α + β < 2* and, forN ≥ 3, 2* = +∞, forN = 1, 2 is the Sobolev critical exponent. By exploiting the Nehari manifold method and the concentration compactness principle, we establish the existence of ground-state solutions when the parameter λ is sufficiently large. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we study the following Hamiltonian elliptic system with gradient terms: for , where , , and are -periodic in . Under weak superquadratic assumptions for the nonlinearities, we use much more direct methods to prove that all the Cerami sequences for the energy functional are bounded and establish the existence of ground-state solutions for this system. [Copyright &y& Elsevier]