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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.
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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