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An elliptic resonance problem with multiple solutions. (English) Zbl 1155.35358

Let \(\Omega\) be a bounded open domain with smooth boundary. In this paper it is studied the semilinear elliptic equation \(-\Delta u=f(x,u)\) in \(\Omega\), under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The mapping \(f\) is a smooth nonlinearity such that \(\lim_{| t| \rightarrow\infty}f(x,t)/t=\lambda_k\) for any \(x\in\Omega\), where \(\lambda_k\) is the \(k\)th eigenvalue of \((-\Delta)\) in \(H^1_0(\Omega )\). The main results of the present paper establish sufficient conditions for the existence of one or several solutions to the above resonant problem. The proofs combine techniques involving the Morse critical point theory, critical groups computation, and minimax methods (mountain pass and local linking). The reviewer believes that the techniques developed in this paper can be extended for the study of other classes of nonlinear boundary value problems, including quasilinear elliptic equations or equations involving singular potentials.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Ahmad, S., Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems, Proc. amer. math. soc., 96, 405-409, (1986) · Zbl 0634.35029
[2] Amann, H.; Zehnder, E., Nontrivial solutions for a class of nonresonance problem and applications to nonlinear differential equations, Ann. scuola norm. sup. Pisa cl. sci. (4), 7, 539-603, (1980) · Zbl 0452.47077
[3] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to nonlinear problems with “strong” resonance at infinity, Nonlinear anal., 7, 981-1012, (1983) · Zbl 0522.58012
[4] Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018
[5] Cerami, G., Un criterio di esistenza per i punti critici su varietà illimitate, Rend. acad. sci. lett. ist. lombardo, 112, 332-336, (1978) · Zbl 0436.58006
[6] Chang, K.C., Solutions of asymptotically linear operator via Morse theory, Comm. pure appl. math., 34, 693-712, (1981) · Zbl 0444.58008
[7] Chang, K.C., Infinite dimensional Morse theory and multiple solutions problems, (1993), Birkhäuser Boston, MA
[8] Chang, K.C.; Li, S.J.; Liu, J.Q., Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. methods nonlinear anal., 3, 179-187, (1994) · Zbl 0812.35031
[9] Hirano, N.; Nishimura, T., Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities, J. math. anal. appl., 180, 566-586, (1993) · Zbl 0835.35052
[10] Landesman, E.M.; Lazer, A.C., Nonlinear perturbations of linear eigenvalues problem at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203
[11] Landesman, E.; Robinson, S.; Rumbos, A., Multiple solutions of semilinear elliptic problems at resonance, Nonlinear anal., 24, 1049-1059, (1995) · Zbl 0832.35048
[12] Li, S.; Liu, J.Q., Some existence theorems on multiple critical points and their applications, Kexue tongbao, 17, 1025-1027, (1984)
[13] Li, S.J.; Liu, J.Q., Nontrivial critical point for asymptotically quadratic functions, J. math. anal. appl., 165, 333-345, (1992) · Zbl 0767.35025
[14] Li, S.J.; Liu, J.Q., Computations of critical groups at degenerate critical point and application to nonlinear differential equations with resonance, Houston J. math., 25, 563-582, (1999) · Zbl 0981.58011
[15] Li, S.J.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 6-32, (1995) · Zbl 0820.58012
[16] Li, S.J.; Willem, M., Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue, Nodea nonlinear differential equations appl., 5, 4, 479-490, (1998) · Zbl 0933.35066
[17] Liu, J.Q., A Morse index for a saddle point, Syst. sci. math. sci., 2, 32-39, (1989) · Zbl 0732.58011
[18] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer Berlin · Zbl 0676.58017
[19] Mizoguchi, N., Asymptotically linear elliptic equation without nonresonance conditions, J. differential equations, 113, 150-165, (1994) · Zbl 0806.35040
[20] Rabinowitz, P., Minimax methods in critical point theory with application to differential equations, Cbms, vol. 65, (1986), Amer. Math. Soc. Providence, RI
[21] Robinson, S., Multiple solutions for semilinear elliptic boundary value problems at resonance, Electron. J. differential equations, 1995, 01, 1-14, (1995)
[22] Su, J.B., Semilinear elliptic resonant problems at higher eigenvalues with unbounded nonlinear terms, Acta. math. sinica (N.S.), 14, 3, 411-419, (1998)
[23] Su, J.B., Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear anal., 48, 6, 881-895, (2002) · Zbl 1018.35037
[24] Su, J.B., Multiplicity results for asymptotically linear elliptic problems at resonance, J. math. anal. appl., 278, 2, 397-408, (2003) · Zbl 1290.35109
[25] Su, J.B.; Tang, C.L., Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear anal., 44, 311-321, (2001) · Zbl 1153.35336
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