An area condition on f and g is necessary and sufficient for nonnegative C^1 solutions with maximum in (α,β] to the p-Laplacian problem with gradient term when f(0)≥0; sufficiently negative g ensures existence for arbitrary f.
Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems , Comm
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Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities
An area condition on f and g is necessary and sufficient for nonnegative C^1 solutions with maximum in (α,β] to the p-Laplacian problem with gradient term when f(0)≥0; sufficiently negative g ensures existence for arbitrary f.