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A reduction method for proving the existence of solutions to elliptic equations involving the \(p\)-Laplacian. (English) Zbl 1038.58022
The authors use a reduction method in proving existence of solutions to certain elliptic equations involving the \(p\)-Laplace operator such as \[ \Delta_p u+ku^{p-1}-Ku^q=0,\qquad q>p-1, \] where \(K\geq0\) and \(k\leq K\) are smooth functions on the complete non-compact Riemannian manifold \((M,g)\) and \(\Delta_pu=\text{div\,}(| \nabla u| ^{p-2}\nabla u).\)
Precisely, the existence is implied by existence of a positive essentially weak subsolution on a manifold and the existence of a positive supersolution on each compact domain of this manifold. The existence/nonexistence of positive supersolutions is expressed in terms of the sign of the first eigenvalue of the nonlinear operator under consideration.
MSC:
58J05 Elliptic equations on manifolds, general theory
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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