Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

Abstract

This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term$$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

“…For more mathematical and physical background on Kirchhoff type problems, we refer the readers to previous works. [2][3][4][5][6][7][8][9] After Lions 10 proposed an abstract functional analysis framework to Kirchhoff Equation (1.2), based on variational methods, a number of important results of the existence and multiplicity of solutions for problem (1.2) have been established when 𝑓 satisfies various conditions, here we just cite previous studies for the subcritical growth case [11][12][13][14][15][16][17][18][19][20] :…”

“…A basic motivation for the study of this equation comes from the fact that this problem is related to the stationary analogue of the Kirchhoff equation $$$$ {u}_{tt}-\left(a+b\underset{{\mathbb{R}}^3}{\int }{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u=g\left(x,t\right), $$$$ which is proposed by Kirchhoff 1 as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. For more mathematical and physical background on Kirchhoff type problems, we refer the readers to previous works 2–9 …”