The elliptic Kirchhoff equation in \(\mathbb {R}^{N}\) perturbed by a local nonlinearity. (English) Zbl 1265.35069

Azzollini, Antonio

Differ. Integral Equ. 25, No. 5-6, 543-554 (2012).

The author proves the existence of a positive solution on \(C^2(\mathbb {R}^N)\) to the equation \[ -M(\int _{\mathbb {R}^N}| \nabla u| ^2)\Delta u=g(u) \] in \(\mathbb {R}^N\), (\(N\geq 3\)) for zero-mass Berestycki-Lions nonlinearity. In the second part of the paper the existence of a ground-state solution of the above equation with \(M(s)=a+bs\), \(g(u)\) being a zero-mass Berestycki-Lions nonlinearity and \(N=3,4\), is studied.

Reviewer: Marie Kopáčková (Praha)

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
35Q74	PDEs in connection with mechanics of deformable solids

Keywords:

elliptic Kirchhoff equation; ground-state solution; Berestycki-Lions assumptions