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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References:

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