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Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms. (English) Zbl 1288.35234

Calc. Var. Partial Differ. Equ. 49, No. 1-2, 271-286 (2014); erratum ibid. 49, No. 1-2, 287-290 (2014).
The authors of this interesting paper investigate a class of resonant cooperative elliptic systems having the form \[ -\Delta u=\lambda u+\delta v+f(x,u,v) , \quad -\Delta v=\delta u+\gamma v+g(x,u,v) \text{ in }\Omega, \] and \(u=v=0\) in \(\partial\Omega\) with sublinear and superlinear terms, where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) (\(N\geq 3\)), and \(\lambda,\delta,\gamma\in\mathbb{R}\). It is assumed that there exists a function \(F\in C^1(\overline{\Omega }\times\mathbb{R}^2,\mathbb{R})\) such that \(\nabla F=(f,g)\), which is the cooperative case. If the condition \(\sigma (A^{\ast })\cap \sigma (-\Delta )\neq \emptyset\) holds, then the elliptic system is resonant. \(A^{\ast }\) is the matrix constructed by the coefficients \(\lambda \), \(\delta \) (first row), and \(\delta \), \(\gamma \) (second row) at the linear terms of the above stated elliptic equations, \(\sigma (A^{\ast })\) is the spectrum of \(A^{\ast }\), \[ \sigma (A^{\ast })=\{\xi ,\zeta \}= \biggl\{ \frac{\lambda +\gamma }{2}\pm \sqrt{\biggl(\frac{\lambda - \gamma }{2}\biggr)^2+\delta^2} \;\biggr\}, \] \(\sigma (-\Delta )=\{\lambda_k:k=1,2,\dots; 0<\lambda_1<\lambda_2<\dots \}\), \(\lambda_i\) – eigenvalues of the Laplacian on \(\Omega \) with zero boundary condition. The subquadratic case is considered provided that three conditions hold: (a) \(F(x,U)\geq 0\), \((x,U)\in\Omega\times\mathbb{R}^2\), and \((\nabla F,U)\leq\mu F\), \(x\in\Omega \), \(|U|\geq R_1\) (\(\mu\in [1;2]\), \(R_1>0\)); (b) \(\lim_{|U|\to 0}\frac{F}{|U|^2}=\infty \) uniformly for \(x\in\Omega \), \(F(x,U)\leq c_2|U|\), \(x\in\Omega \), \(|U|\leq R_2\) (\(c_2,U>0\)); (c) \(\liminf_{|U|\to\infty }\frac{F}{|U|}\geq d>0\) uniformly for \(x\in\Omega \).
The main statement here is that under the above stated conditions (a)–(c) and if \(F(x,U)\) is even in \(U\), then the system under consideration possesses infinitely many nontrivial solutions.
The second result is stated under the following requirements:
(i)
\(|\nabla F|\leq a_1(1+|U|^{\nu -1})\), \((x,U)\in \Omega\times\mathbb{R}^2\), \(2<\nu < 2^{\ast }=2N(N-2)^{-1}\);
(ii)
\(\lim_{|U|\to\infty }\frac{F}{|U|^2}=\infty\) uniformly for \(x\in\Omega \), \(F\geq 0\) for all \((x,U)\in\Omega \times\mathbb{R}^2\);
(iii)
\(\liminf_{|U|\to\infty } \frac{(\nabla F,U)-2F}{|U|^{\nu }}\geq b>0\) uniformly for \(x\in\Omega \).
Here the authors also prove existence of infinitely many nontrivial solutions under the conditions (i)–(iii), and \(F\) is even in \(U\) as well.

MSC:

35J50 Variational methods for elliptic systems
35P05 General topics in linear spectral theory for PDEs
35J47 Second-order elliptic systems
35B34 Resonance in context of PDEs
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References:

[1] Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241-273 (1979) · Zbl 0465.49006 · doi:10.1007/BF01389883
[2] Costa, D.G., Magalhães, C.A.: A variational approach to subquadratic perturbations of elliptic systems. J. Differ. Equ. 111, 103-122 (1994) · Zbl 0803.35052 · doi:10.1006/jdeq.1994.1077
[3] Costa, D.G., Magalhães, C.A.: A unified approach to a class of strongly indefinite functionals. J. Differ. Equ. 125, 521-547 (1996) · Zbl 0890.47038 · doi:10.1006/jdeq.1996.0039
[4] Chen, G., Ma, S.: Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0. J. Math. Anal. Appl. 379, 842-851 (2011) · Zbl 1218.37079 · doi:10.1016/j.jmaa.2011.02.013
[5] Chen, G., Ma, S.: Homoclinic orbits of superlinear Hamiltonian systems. Proc. Am. Math. Soc. 139(11), 3973-3983 (2011) · Zbl 1242.37041 · doi:10.1090/S0002-9939-2011-11185-7
[6] Chen, G., Ma, S.: Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput. 218, 5496-5507 (2012) · Zbl 1254.39006 · doi:10.1016/j.amc.2011.11.038
[7] Chen, G., Ma, S.: Ground state periodic solutions of second order Hamiltonian systems without spectrum 0. Isr. J. Math. (in press). · Zbl 1280.37053
[8] Chen, G., Ma, S.: Asymptotically or super linear cooperative elliptic systems in the whole space. Preprinted. · Zbl 1279.35037
[9] Fei, G.: Multiple solutions of some nonlinear strongly resonant elliptic equations without the (PS) condition. J. Math. Anal. Appl. 193, 659-670 (1995) · Zbl 0836.35053 · doi:10.1006/jmaa.1995.1259
[10] Lazzo, M.: Nonlinear differential problems and Morse theory. Nonlinear Anal. TMA 30, 169-176 (1997) · Zbl 0898.34043 · doi:10.1016/S0362-546X(96)00220-9
[11] Li, S., Liu, J.Q.: Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance. Houston J. Math. 25, 563-582 (1999) · Zbl 0981.58011
[12] Li, S., Zou, W.: The Computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue. J. Math. Anal. Appl. 235, 237-259 (1999) · Zbl 0935.35055 · doi:10.1006/jmaa.1999.6396
[13] Ma, S.: Infinitely many solutions for cooperative elliptic systems with odd nonlinearity. Nonlinear Anal. 71, 1445-1461 (2009) · Zbl 1170.35388 · doi:10.1016/j.na.2008.12.012
[14] Ma, S.: Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups. Nonlinear Anal. 73(12), 3856-3872 (2010) · Zbl 1202.35086 · doi:10.1016/j.na.2010.08.013
[15] Pomponio, A.: Asymptotically linear cooperative elliptic system: existence and multiplicity. Nonlinear Anal. 52, 989-1003 (2003) · Zbl 1022.35014 · doi:10.1016/S0362-546X(02)00148-7
[16] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1986) · Zbl 0609.58002
[17] Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802-3822 (2009) · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013
[18] Tang, C.-L., Gao, Q.-J.: Elliptic resonant problems at higher eigenvalues with an unbounded nonlinear term. J. Differ. Equ. 146, 56-66 (1998) · Zbl 0908.35043 · doi:10.1006/jdeq.1998.3411
[19] Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1
[20] Zou, W.: Solutions of resonant elliptic systems with nonodd or odd nonlinearities. J. Math. Anal. Appl. 223, 397-417 (1998) · Zbl 0921.35062 · doi:10.1006/jmaa.1998.5938
[21] Zou, W.: Multiple solutions for asymptotically linear elliptic systems. J. Math. Anal. Appl. 255, 213-229 (2001) · Zbl 0989.35049 · doi:10.1006/jmaa.2000.7236
[22] Zou, W.: Variant fountain theorems and their applications. Manuscr. Math. 104, 343-358 (2001) · Zbl 0976.35026 · doi:10.1007/s002290170032
[23] Zou, W., Li, S.: Nontrivial Solutions for resonant cooperative elliptic systems via computations of critical groups. Nonlinear Anal. 38, 229-247 (1999) · Zbl 0940.35074 · doi:10.1016/S0362-546X(98)00191-6
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