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On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities. (English. Russian original) Zbl 1298.35069
Differ. Equ. 50, No. 6, 765-776 (2014); translation from Differ. Uravn. 50, No. 6, 768-779 (2014).
Summary: We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions \(u_{\lambda}\) for \(\lambda \in (-\infty, \lambda ^*_{0})\). The critical value \(\lambda^*_{0} >0\) is found by a spectral analysis procedure according to Pokhozhaev’s fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter \(\lambda\). Moreover, we prove the existence of an interval \((-\infty ,\tilde \lambda )\), where \(\tilde \lambda > 0\), on which this branch consists of solutions with exactly two nodal domains.
35J57 Boundary value problems for second-order elliptic systems
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[1] Ambrosetti, A; Brezis, H; Cerami, G, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519-543, (1994) · Zbl 0805.35028
[2] Radulescu, V; Repovs, D, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75, 1524-1530, (2012) · Zbl 1237.35043
[3] Callegari, A; Nachman, A, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38, 275-281, (1980) · Zbl 0453.76002
[4] Gierer, A; Meinhardt, H, A theory of biological pattern formation, Kybernetik, 12, 30-39, (1972)
[5] Keller, EF; Segel, LA, The initiation of slime mold aggregation viewed As an instability, J. Theoret. Biol., 26, 399-415, (1970) · Zbl 1170.92306
[6] Il’yasov, Ya, On nonlocal existence results for elliptic equations with convex-concave nonlinearities, Nonlinear Anal., 61, 211-236, (2005) · Zbl 1190.35112
[7] Bartsch, T; Willem, M, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123, 3555-3561, (1995) · Zbl 0848.35039
[8] Lubyshev, VF, Multiple positive solutions of an elliptic equation with a convex-concave nonlinearity containing a sign-changing term, Tr. Mat. Inst. Steklova, 269, 167-180, (2010)
[9] Castro, A; Cossio, J; Neuberger, JM, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27, 1041-1053, (1997) · Zbl 0907.35050
[10] Bartsch, T; Weth, T, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22, 1-14, (2003) · Zbl 1094.35041
[11] Clapp, M; Weth, T, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var. Partial Differential Equations, 21, 1-14, (2004) · Zbl 1097.35048
[12] Liu, Z; Wang, Z-Q, Sign-changing solutions of nonlinear elliptic equations, Front. Math. in China, 3, 221-238, (2008) · Zbl 1158.35370
[13] Cepicka, J; Drabek, P; Girg, P, Open problems related to the \(p\)-Laplacian, 13-34, (2009)
[14] Courant, R. and Hilbert, D., Metody matematicheskoi fiziki (Methods ofMathematical Physics), Moscow, 1933, 1945, vols. 1,2.
[15] Drabek, P; Robinson, SB, On the generalization of the Courant nodal domain theorem, J. Differential Equations, 181, 58-71, (2002) · Zbl 1163.35449
[16] Helffer, B; Hoffmann-Ostenhof, T; Terracini, S, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 101-138, (2009) · Zbl 1171.35083
[17] Nehari, Z, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95, 101-123, (1960) · Zbl 0097.29501
[18] Szulkin, A. and Weth, T., The Method of Nehari Manifold: Handbook of Nonconvex Analysis and Applications, Boston, 2010. · Zbl 1218.58010
[19] Pokhozhaev, SI, An approach to nonlinear equations, Dokl. Akad. Nauk SSSR, 247, 1327-1331, (1979)
[20] Pokhozhaev, SI, The fibering method for solving nonlinear boundary value problems, Tr. Mat. Inst. Steklova, 192, 146-163, (1990) · Zbl 0734.35036
[21] Il’yasov, Ya, On a procedure of projective fibration of functionals on Banach spaces, Proc. Steklov Inst. Math., 232, 150-156, (2001) · Zbl 1032.46095
[22] Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, New York, 1980. · Zbl 0457.35001
[23] Dunford, N. and Schwartz, J., Linear Operators. General Theory, New York, 1958. Translated under the title Lineinye operatory. T. 1. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962.
[24] Ambrosetti, A. and Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge, 2007. · Zbl 1125.47052
[25] Willem, M., Minimax Theorems, Boston, 1996. · Zbl 0856.49001
[26] Vrahatis, MN, A short proof and a generalization of miranda’s existence theorem, Proc. Amer. Math. Soc., 107, 701-703, (1989) · Zbl 0695.55001
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