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A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity. (English) Zbl 07534195

Summary: By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form \[ -\left( a+ b\int_{\mathbb{R}^3}|\nabla u|^2\right) \Delta{u}+V(x)u=f(u),\,\,x\in \mathbb{R}^3, \] where \(a,b>0\) are constants, \(V\in C(\mathbb{R}^3,\mathbb{R}), f\in C(\mathbb{R},\mathbb{R})\). The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on \(f\) nor the coercivity condition on \(V\) is required. Our result improves the study made by Y. Deng et al. [J. Funct. Anal. 269, No. 11, 3500–3527 (2015; Zbl 1343.35081)] in the sense that, in the present paper, the nonlinearities include the power-type case \(f(u)=|u|^{p-2}u\) for \(p\in (2,4)\), in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, energy doubling is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small \(b>0\).

MSC:

35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence

Citations:

Zbl 1343.35081
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