We study the existence of ground state sign-changing solutions for the fractional Kirchhoff problem. Under mild assumptions on the nonlinearity, by using some new analytical skills and the non-Nehari manifold method, we prove that the fractional Kirchhoff problem possesses a ground state sign-changing solution ub. Moreover, we show that the energy of ub is strictly larger than twice that of the ground state solutions of Nehari-type. Finally, we establish the convergence property of ub as the parameter b ↘ 0. Our results generalize some results obtained by Shuai [J. Differ. Equations 259, 1256 (2015)] and Tang and Cheng [J. Differ. Equations 261, 2384 (2016)].
In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditionwhere Ω ⊂ R n (n ≥ 2) is a bounded, smooth domain and f (x, u) is asymptotically linear at infinity with respect to u. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).
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