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Sign-changing solutions for the boundary value problem involving the fractional \(p\)-Laplacian. (English) Zbl 1479.35497

Summary: In the paper, we consider the following boundary value problem involving the fractional \(p\)-Laplacian: \[ \begin{cases} (-\triangle)_p^s u(x)=f(x,u) &\text{in } \Omega,\\ u(x)=0 &\text{in } \mathbb{R}^N \setminus\Omega. \end{cases} \tag{\(\mathcal{P}\)} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) with \(N\geq 1\), \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian with \(s\in (0,1)\), \(p\in (1,N/s)\), \(f(x,u)\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R} \). Under the improved subcritical polynomial growth condition and other conditions, the existences of a least-energy sign-changing solution for the problem \((\mathcal{P})\) has been established.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35R11 Fractional partial differential equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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