A Schechter type critical point result in annular conical domains of a Banach space and applications. (English) Zbl 1338.47117
Summary: Using Ekeland’s variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii’s fixed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to \(p\)-Laplace equations, where the geometric condition on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or infinitely many solutions for certain classes of quasilinear equations.
MSC:
| 47J30 | Variational methods involving nonlinear operators |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
critical point; extremum point; Palais-Smale condition; Ekeland’s variational principle; \({p}\)-Laplacian; weak Harnack inequalityReferences:
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