## 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

Zbl 1343.35081
Full Text:

### References:

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