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Fractional equations with indefinite nonlinearities. (English) Zbl 1408.35037

Summary: In this paper, we consider a fractional equation with indefinite nonlinearities
\[ (-\Delta)^{\alpha/2} u = a(x_1) f(u) \] for \(0<\alpha<2\), where \(a\) and \(f\) are nondecreasing functions. We prove that there is no positive bounded solution. In particular, in the case \(a(x_1) = x_1\) and \(f(u) = u^p\), this remarkably improves the result in [the authors, J. Differ. Equations 260, No. 5, 4758–4785 (2016; Zbl 1336.35089)] by extending the range of \(\alpha\) from \([1,2)\) to \((0,2)\), due to the introduction of new ideas, which may be applied to solve many other similar problems.

MSC:

35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
35B09 Positive solutions to PDEs

Citations:

Zbl 1336.35089
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References:

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