Abstract
We consider the problem of finding a harmonic function u in a bounded domain \({\Omega \subset {\mathbb R}^N, N \ge 2,}\) satisfying a nonlinear boundary condition of the form \({\partial_{\nu}u(x)=\lambda\,(\eta(x)u(x)+\mu(x)h(u(x)), x\in \partial \Omega}\) where μ and η are bounded functions and h is a \({\mathcal{C}^1}\) odd function with subcritical growth at infinity and such that lim s→∞ h′(s) = +∞. By using variants of the mountain pass lemma based on index theory, we discuss existence and multiplicity of non trivial solutions to the problem for every value of λ.
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References
Escobar J.: Conformal metrics with prescribed mean curvature on the boundary. Calc. Var. 4, 559–592 (1996)
Zhu M.: On elliptic problems with indefinite superlinear boundary conditions. J. Differ. Equ. 193(1), 180–195 (2003)
Vogelius M., Xu J.-M.: A nonlinear elliptic boundary value problem related to corrosion modeling. Q. Appl. Math. 56, 479–505 (1998)
Pagani C.D., Pierotti D.: Variational methods for nonlinear Steklov eigenvalues problems with an indefinite weight function. Calc. Var. 39(1), 35–58 (2010)
Kavian, O., Vogelius, M.: On the existence and ‘blow-up’ of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modeling. Proc. Royal Soc. Edinb. Sect. A 133, 119–149 (2003). Corrigendum to the same, Proc. Royal Soc. Edinb. Sect. A 133, 729–730 (2003)
Struwe M.: Variational Methods and Applications to Nonlinear Partial Differential Equations and Hamiltonian systems. Springer, Berlin (1990)
Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Bartolo P., Benci V., Fortunato D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. Theory Methods Appl. 7(9), 981–1012 (1983)
Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA (1985)
Torné O.: Steklov problem with an indefinite weight for the p−Laplacian. Electron. J. Differ. Equ. 87, 1–8 (2005)
Jerison D.S., Kenig C.E.: The Neumann problem on Lipschitz domains. Bull. AMS 4(2), 203–207 (1981)
Berestycki H., Capuzzo Dolcetta I., Nirenberg L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA 2, 553–572 (1995)
Caffarelli L., Friedman A.: Partial regularity of the zero set of linear elliptic equations. J. Differ. Equ. 60, 420–439 (1985)
Hardt R., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.: Critical sets of solutions to elliptic equations. J. Differ. Geom. 51, 359–373 (1999)
Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, vol. 65, AMS, Providence (1986)
Cherrier P.: Problèmes de Neumann nonlinéaires sur les variétés riemanniennes. J. Funct. Anal. 57, 154–207 (1984)
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Pagani, C.D., Pierotti, D. Multiple variational solutions to nonlinear Steklov problems. Nonlinear Differ. Equ. Appl. 19, 417–436 (2012). https://doi.org/10.1007/s00030-011-0136-z
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DOI: https://doi.org/10.1007/s00030-011-0136-z