Multiplicity and concentration for Kirchhoff type equations around topologically critical points in potential
Keywords
Kirchhoff type equations, topologically critical points, multiplicityAbstract
We consider the multiplicity and concentration of solutions for the Kirchhoff Type Equation \[ -\varepsilon^2 M\bigg( \varepsilon^{2-N}\int_{\mathbb{R}^N} |\nabla v|^2dx \bigg) \Delta v+V(x)v=f(v), \quad \mathrm{in }\ \mathbb{R}^N. \] Under suitable conditions on functions $M$, $V$ and $f$, we obtain the existence of positive solutions concentrating around the local maximum points of $V$, which gives an affirmative answer to the problem raised in \cite{fij}. Moreover, we also obtain multiplicity of solutions which are affected by the topology of critical points set of potential $V$.References
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