Elliptic equations with diffusion parameterized by the range of nonlocal interactions
We consider quasilinear elliptic equations where the diffusion at each point depends on all the values of the solution in a neighborhood of this point. The size of this neighborhood is parameterized by some non-negative number which represents the range of nonlocal interactions. For fixed values of the parameter, the issue of the existence and local uniqueness of the solution is addressed. In a radial symmetric setting, we give pointwise estimates of the solutions and prove the existence of multiple solutions. Regarding bifurcation theory, we show that many local branches of solutions may exist while, among them, only one is global and has no bifurcation point.
Mathematics Subject Classification (2000)35J60 35B30
KeywordsNonlocal elliptic equations Branches of solutions Parameter Nonlocal diffusion
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