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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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