Abstract
This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation \( - \left({a + b\,\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}} \right)\Delta u + V\left(x \right)u = f\left(u \right)\,\,{\rm{in}}\,\,{\mathbb{R}^3}\), with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.
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This work was partially supported by NSFC (11701108).
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Pi, H., Zeng, Y. Existence Results for the Kirchhoff Type Equation with a General Nonlinear Term. Acta Math Sci 42, 2063–2077 (2022). https://doi.org/10.1007/s10473-022-0519-8
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DOI: https://doi.org/10.1007/s10473-022-0519-8