We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.
ABSTRACT. We provide a-priori L ∞ bounds for positive solutions to a class of subcritical elliptic problems in bounded C 2 domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution u, the maximum of any positive solution in the domain Ω, is bounded above by a constant multiplied by the maximum on an open subset strongly contained in Ω. The constant and the open subset depend only on geometric properties of Ω, and are independent of the non-linearity and on the solution u. Our analysis answers a longstanding open problem.
We consider the Dirichlet problem for positive solutions of the equation −∆ p (u) = f (u) in a convex, bounded, smooth domain Ω ⊂ R N , with f locally Lipschitz continuous.We provide sufficient conditions guarantying L ∞ a priori bounds for positive solutions of some elliptic equations involving the p-Laplacian and extend the class of known nonlinearities for which the solutions are L ∞ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains. 2010 Mathematics Subject Classification. 35B45,35J92, 35B09, 35B33, 35J62. Key words and phrases. A priori estimates, quasilinear elliptic equations with p-Laplacian, critical Sobolev esponent, moving planes method, Pohozaev identity, Picone identity, positive solutions.
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