On problems with weighted elliptic operator and general growth nonlinearities

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This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form −div(|x| aDu) = f(x, u), u > 0, in Ω, where N ≥ 3, Ω is an open domain in R N containing the origin, N−2+a > 0 and f satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided f exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for f exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in Ω = R N exists provided the growth of f is subcritical. The results are then extended to systems of the form −div(|x| aDu1)=f1(x, u1, u2), −div(|x| aDu2)=f2(x, u1, u2), u1, u2 >0, in Ω, but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems.


Original published source available at http://dx.doi.org/10.3934/cpaa.2021023

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Communications on Pure and Applied Analysis