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Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials. (English) Zbl 1432.35192

Summary: Based on the strategy introduced by the authors [Adv. Nonlinear Anal. 9, 496–515 (2020; Zbl 1422.35023)] and some new tricks, we prove that the nonlinear problem of Kirchhoff-type \(- \left(a + b \int_{\mathbb{R}^3} | \nabla u |^2 \operatorname{d}x\right) \triangle u + V(x) u = f(u), x \in R^3\) in \(H^1(\mathbb{R}^3)\) admits two classes of ground state solutions under the general “Berestycki-Lions assumptions” on the nonlinearity \(f\), which are almost necessary conditions, as well as some weak assumptions on the potential \(V\). Moreover, we also give a simple minimax characterization of the ground state energy. Our results improve and extend recent theorems in several directions.
©2019 American Institute of Physics

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J10 Schrödinger operator, Schrödinger equation
35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Citations:

Zbl 1422.35023
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References:

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