DCDS
Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials
Norihisa Ikoma
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 943-966 doi: 10.3934/dcds.2015.35.943
In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations \[ - m \left( \| \nabla u \|_{L^2(\mathbf{R}^N)}^{2} \right) \Delta u + V(x) u = |u|^{p-1} u \quad {\rm in}\ \mathbf{R}^N, \ u \in H^1(\mathbf{R}^N), \ N \geq 1 \] where $ 1 < p < \infty$ when $N=1,2$, $1 < p < (N+2)/(N-2)$ when $N \geq 3$, $m: [0,\infty) \to (0,\infty)$ is a continuous function and $V:\mathbf{R}^N \to \mathbf{R}$ a smooth function. Under suitable conditions on $m(s)$ and $V$, it is shown that a ground state solution to the above equation exists.
keywords: ground state solutions the Pohozaev identity. Kirchhoff type equations Variational methods monotonicity trick

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