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A multiplicity result for asymptotically linear Kirchhoff equations. (English) Zbl 1419.35037

Summary: In this paper, we study the following Kirchhoff type equation: \[ -\left(1+b\int_{\mathbb R^N}\vert\nabla u\vert^2\,dx\right)\Delta u+u=a(x)f(u)\quad\text{in }\mathbb R^N,\qquad u\in H^1(\mathbb R^N),\] where \(N\geq 3\), \(b>0\) and \(f(s)\) is asymptotically linear at infinity, that is, \(f(s)\sim O(s)\) as \(s\rightarrow+\infty\). By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on \(a(x)\).

MSC:

35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
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