Separating solutions of nonlinear problems using nonlinear generalized Rayleigh quotients
DOI:
https://doi.org/10.12775/TMNA.2020.075Keywords
Nonlinear generalized Rayleigh quotient, multiplicity result, Nehari manifold, ground state solutionAbstract
This paper deals with nonlinear elliptic boundary value problems with complicated geometry of nonlinearities. A new method for obtaining multiple solutions based on a recursive use of the nonlinear generalized Rayleigh quotients to the split Nehari manifold into subsets without degeneracies is introduced. The method is applied to prove the multiplicity result for nonnegative solutions, as well as to find a ground state of elliptic boundary value problems with nonlinearities of polynomial type.References
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543.
V.I. Arnol’d, Catastrophe Theory, Springer Science & Business Media, 2003.
A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32.
V. Bobkov, P. Drábek and Y. Il’yasov, On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities, Appl. Math. Lett. 95 (2019), 23–28.
L. Cherfils and Y. Il’yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22.
C. Cortázar, M. Elgueta and P. Felmer, On a semi-linear elliptic problem in RN with a non-Lipschitzian non-linearity, Adv. Difference Equ. 1 (1996), 199–218.
J.I. Dı́az, J. Hernández and Y. Sh. Il’yasov, On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets, Adv. Nonlinear Anal. 9 (2019), no. 1, 1046–1065.
J. Dı́az, J. Hernández and Y. Il’yasov, Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3, Chinese Ann. Math. Ser. B 38 (2017), no. 1, 345–378.
Y. Il’yasov, On nonlocal existence results for elliptic equations with convex-concave nonlinearities, Nonlinear Anal. 61 (2005), no. 1–2, 211–236.
Y. Il’yasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 683–714.
Y. Il’yasov and Y. Egorov, Hopf boundary maximum principle violation for semilinear elliptic equations, Nonlinear Anal. 72 (2010), no. 7–8, 3346–3355.
H. Kaper and M. Kwong, Free boundary problems for Emden–Fowler equation, Differential Integral Equations 3 (1990), 353–362.
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259–287.
N.S. Papageorgiou, V.D. Rădulescu and D. Repovš, Nonlinear Analysis Theory and Methods, Springer International Publishing, 2019.
S.I. Pokhozhaev, On the method of fibering a solution in nonlinear boundary value problems, Proc. Steklov Inst. Math. 192 (1992), 157–173.
L.E. Payne and D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 , no. 3–4, 273–303.
M. Struwe, Variational Methods, vol. 31999, Springer, Berlin, Heildelberg, 1990.
A. Szulkin and T. Weth, The Method of Nehari Manifold, Handbook of Nonconvex Analysis and Applications 597632 (2010).
M. Willem, Minimax Theorems, vol. 24, Springer Science & Business Media, 1997.
E. Zeidler, Nonlinear Functional Analysis and its Applications: III. Variational Methods and Optimization, Springer Science & Business Media, 2013.
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Copyright (c) 2021 Marcos Leandro Carvalho, Yavdat Il'yasov, Carlos Alberto Santos
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
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