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A remark on the uniqueness of positive solutions to semilinear elliptic equations with double-power nonlinearities. (English) Zbl 1240.35200

The paper studies the existence and uniqueness of positive solutions of the problem \[ \Delta u -\omega u +u^p-u^{2p-1} =0\quad\text{ in }\mathbb R^n,\quad u(x)\to 0\quad\text{as}\;| x| \to \infty. \] It is shown that the problem has exactly one positive solution for \[ p(7p-5)/[4(p+1)(2p-1)^2] \leq \omega <p/(p+1)^2. \]

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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