EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

doi:10.1007/s10473-022-0519-8

Abstract

Abstract: This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation $-\big(a + b \int_{\mathbb{R}^{3}}|\nabla u|^2 \big)\Delta u + V(x) u = f(u)$ 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.

Key words: Kirchhoff type equation, general nonlinearity, variational methods, Pohozaev identity

CLC Number:

35J20

Huirong PI, Yong ZENG. EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM[J].Acta mathematica scientia,Series B, 2022, 42(5): 2063-2077.

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