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Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient

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  • In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.

    Mathematics Subject Classification: Primary: 35B09, 35B32; Secondary: 35J60, 35P30.


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  • Figure 1.  Bifurcation diagrams when $\lambda _0<\lambda _\infty<\infty$ and $\lambda _\infty<\lambda _0<\infty$, respectively

    Figure 2.  Bifurcation diagrams when $\lambda _\infty = 0$ and $\lambda_\infty = \infty$, respectively. For example, this last diagram appears when $b\geq b_0>0$

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