DCDS
Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials
Norihisa Ikoma
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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