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By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$-norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

In the first part of this paper, we study the best constant involving the L2 norm in Wente's inequality. We prove that this best constantis universal for any Riemannian surface with boundary, or respectively, for any Riemannian surface without boundary. The secondpart concerns the study of critical points of the associate energy functional, whose Euler equation corresponds to H-surfaces. We willestablish the existence of a non-trivial critical point for a plan domain with small holes.

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Positive solutions for a degenerate Kirchhoff problem

Estimations of the best constant involving the L2 norm in Wente's inequality andcompact H-surfaces in Euclidean space

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2 results

Positive solutions for a degenerate Kirchhoff problem

Estimations of the best constant involving the L2 norm in Wente's inequality andcompact H-surfaces in Euclidean space