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littlelittlelittle
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Solvability of quasilinear elliptic equations on closed manifolds
Yes, I think it is necessary and sufficient. You can check Barta's Lemma in this paper (Lemma 3.1): arxiv.org/abs/1211.2705
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A question on Schrodinger operator
$u=1$ requires either $\phi>0$ or $\psi>0$.