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

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.

### MSC:

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