Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket
Abstract
UDC 517.9
We study the existence of a weak (strong) solution of the nonlinear elliptic problem\begin{gather*} -\Delta u- \lambda ug_1 +h(u)g_2=f \quad\text{in}\quad V\setminus V_0,\\u=0 \quad\text{on}\quad V_0,\end{gather*} where $V$ is a Sierpi\'nski gasket in $\mathbb{R}^{N-1},$ $N\geq 2,$ $V_0$ is its boundary (consisting of $N$ its corners), and $\lambda$ is a real parameter. Here, $f,g_1,g_2\colon V\to\mathbb{R}$ and $h\colon \mathbb{R}\to\mathbb{R}$ are functions satisfying suitable hypotheses.
References
A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349 – 381 (1973). DOI: https://doi.org/10.1016/0022-1236(73)90051-7
M. T. Barlow, R. F. Bass, Transition densities for Brownian motion on the Sierpi'nski carpet, Probab. Theory and Related Fields, 91, 307 – 330 (1992). DOI: https://doi.org/10.1007/BF01192060
M. T. Barlow, R. F. Bass, Brownian motion and harmonic analysis on Sierpi'nski carpet, Canad. J. Math., 51, 673 – 744 (1999). DOI: https://doi.org/10.4153/CJM-1999-031-4
G. M. Bisci, V. Ru adulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc., 143, 2959 – 2968 (2015). DOI: https://doi.org/10.1090/S0002-9939-2015-12475-6
G. M. Bisci, D. Repovv s, R. Servadei, Nonlinear problems on the Sierpi'nski gasket, J. Math. Anal. and Appl., 452, 883 – 895 (2017). DOI: https://doi.org/10.1016/j.jmaa.2017.03.032
B. E. Breckner, Real-valued functions of finite energy on the Sierpi'nski gasket, Mathematica, 55(78), 142 – 158 (2013).
B. E. Breckner, A short note on harmonic functions and zero divisors on the Sierpi'nski fractal, Arch. Math. (Basel), 106, 183 – 188 (2016). DOI: https://doi.org/10.1007/s00013-015-0838-2
B. E. Breckner, V. Ru adulescu, C. Varga, Infinitely many solutions for the Dirichlet problem on the Sierpi'nski gasket, Anal. and Appl., 9, 235 – 248 (2011). DOI: https://doi.org/10.1142/S0219530511001844
B. E. Breckner, D. Repovv s, C. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpi'nski gasket, Nonlinear Anal., 73, 2980 – 2990 (2010). DOI: https://doi.org/10.1016/j.na.2010.06.064
B. E. Breckner, C. Varga, A note on gradient-type systems on fractals, Nonlinear Anal. Real World Appl., 21, 142 – 152 (2015). DOI: https://doi.org/10.1016/j.nonrwa.2014.07.004
B. E. Breckner, C. Varga, Multiple solutions of Dirichlet problems on the Sierpi'nski gasket, J. Optim. Theory and Appl., 167, 842 – 861 (2015). DOI: https://doi.org/10.1007/s10957-013-0368-7
P. Hess, On the Fredholm alternative for nonlinear functional equations in Banach spaces, Proc. Amer. Math. Soc., 33, 55 – 61 (1972). DOI: https://doi.org/10.1090/S0002-9939-1972-0301585-9
K. J. Falconer, Semilinear PDEs on self-similar fractals, Comm. Math. Phys., 206, 235 – 245 (1999). DOI: https://doi.org/10.1007/s002200050703
K. J. Falconer, Fractal geometry: mathematical foundations and applications, 2nd ed., John Wiley & Sons (2003). DOI: https://doi.org/10.1002/0470013850
K. J. Falconer, J. Hu, Nonlinear elliptic equations on the Sierpi'nski gasket, J. Math. Anal. and Appl., 240, 552 – 573 (1999). DOI: https://doi.org/10.1006/jmaa.1999.6617
F. Faraci, A. Krist'aly, One-dimensional scalar field equations involving an oscillatory nonlinear term, Discrete and Contin. Dyn. Syst., 18, № 1, 107 – 120 (2007). DOI: https://doi.org/10.3934/dcds.2007.18.107
M. Fukushima, T. Shima, On a spectral analysis for the Sierpi'nski gasket, Potential Anal., 1, 1 – 35 (1992). DOI: https://doi.org/10.1007/BF00249784
Z. He, Sublinear elliptic equation on fractal domains, J. Partial Different. Equat., 24, 97 – 113 (2011). DOI: https://doi.org/10.4208/jpde.v24.n2.1
J. Hu, Multiple solutions for a class of nonlinear elliptic equations on the Sierpi'nski gasket, Sci. China Ser. A, 47, 772 – 786 (2004). DOI: https://doi.org/10.1360/02ys0366
J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335, 721 – 755 (1993). DOI: https://doi.org/10.1090/S0002-9947-1993-1076617-1
S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153, 339 – 357 (1993). DOI: https://doi.org/10.1007/BF02096647
M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, GITTL, Moscow (1956).
A. Kufner, O. John, S. Fuv cik, Functions spaces, Noordhoff, Leyden (1977).
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI (1986). DOI: https://doi.org/10.1090/cbms/065
V. Raghavendra, R. Kar, Existence of a weak solution for a class of fractional Laplacian equation, J. Aust. Math. Soc., 102, № 3, 392 – 404 (2016). DOI: https://doi.org/10.1017/S144678871600032X
B. Ricceri, On a classical existence theorem for nonlinear elliptic equations, M. Th' era (Ed.), Esperimental, Constructive and Nonlinar Analysis, CMS Conf. Proc., vol. 27, Canad. Math. Soc. (2000), p.~275 – 278.
R. S. Strichartz, Some properties of Laplacian on fractals, J. Funct. Anal., 164, 181 – 208 (1999). DOI: https://doi.org/10.1006/jfan.1999.3400
R. S. Strichartz, Solvability for differential equations on fractals, J. Anal. Math., 96, 247 – 267 (2005). DOI: https://doi.org/10.1007/BF02787830
M. Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin, Heidelberg (1990).
E. Zeidler, Nonlinear functional analysis and its applications, Pt II/A, Springer-Verlag, New York (1990).
E. Zeidler, Nonlinear functional analysis and its applications, Pt II/B, Springer-Verlag, New York (1990). DOI: https://doi.org/10.1007/978-1-4612-0981-2
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