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This paper proves that any nonregular nonparametric saddle surface in a three-dimensional space of nonzero constant curvature , which is bounded by a rectifiable curve, is a space of curvature not greater than in the sense of Aleksandrov. This generalizes a classical theorem by Shefel′ on saddle surfaces in .
By means of Morse theory we prove the existence of a nontrivial solution to a superlinear -harmonic elliptic problem with Navier boundary conditions having a linking structure around the origin. Moreover, in case of both resonance near zero and nonresonance at the existence of two nontrivial solutions is shown.
We study minimal solutions for one-dimensional variational problems on a torus. We show that, for a generic integrand and any rational number , there exists a unique (up to translations) periodic minimal solution with rotation number .
This work is devoted to the study of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many solutions of a related nonlinear eigenvalue problem. Applying an abstract minimax theorem, we obtain a solution of the quasilinear system , under conditions involving the first and the second eigenvalues.