Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

Abstract

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities $ - {\Delta _{\mathbb{B}}}u + \lambda u = a\left( z \right)f\left( u \right),\,\,\,\,\,\,u \ge 0\,\,{\rm{in}}\,\,_ + ^N,$

where N = n + 1 ≥ 3, λ > 0, z = (t, x ₁, ⋯, x _n), and ${\Delta _{\mathbb{B}}} = {\left( {t{\partial _t}} \right)^2} + \partial _{{x_1}}^2 + \cdots + \partial _{{x_n}}^2$. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.

Keywords

Asymptotically linear / Pohozaev identity / Cone degenerate elliptic operators

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