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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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