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On a semilinear Schrödinger equation with periodic potential. (English) Zbl 0952.35047

The existence of a nontrivial solution for the Schrödinger equation \(-\Delta u+V(x)u=f(x,u)\), \(x\in \mathbb{R}^N\) is studied for indefinite periodic potentials \(V\) and superlinear, subcritical nonlinearities \(f\) in \(u\) and with the same period of \(f\) in \(x\) as \(V.\) Under the assumption that \(0\) is in a gap of the spectrum of \(-\Delta+V\) it is proved the existence of nontrivial solutions having finite energy. The proof is based on the approximation of the considered problem by periodic problems in cubes.

MSC:

35J60 Nonlinear elliptic equations
35B10 Periodic solutions to PDEs
35A35 Theoretical approximation in context of PDEs
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[1] Alama, S.; Li, Y.Y., Existence of solutions for semilinear elliptic equations with indefinite linear part, J. differential equations, 96, 89-115, (1992) · Zbl 0766.35009
[2] Buffoni, B.; Jeanjean, L.; Stuart, C.A., Existence of nontrivial solutions to a strongly indefinite semilinear equation, Proc. am. math. soc., 119, 179-186, (1993) · Zbl 0789.35052
[3] V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on\(R\^{}\{n\}\), Comm. Pure Appl. Math. 45 (1992) 1217-1269. · Zbl 0785.35029
[4] Heinz, H.P.; Küpper, T.; Stuart, C.A., Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation, J. differential equations, 100, 341-354, (1992) · Zbl 0767.35006
[5] Jeanjean, L., Solutions in spectral gaps for a nonlinear equation of Schrödinger type, J. differential equations, 112, 53-80, (1994) · Zbl 0804.35033
[6] W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, preprint, Stockholm. · Zbl 0947.35061
[7] P.L. Lions, The concentration-compactness method in the calculus of variations, The locally compact case, I, II, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984) 109 -145, 223-283.
[8] A.A. Pankov, Semilinear elliptic equations in\(R\^{}\{n\}\) with nonstabilizing coefficients, Ukraine Math. J. 41 (9) (1989) 1075-1078. Transl. from Ukr. Mat. Zh. 41 (9) (1987) 1247-1251. · Zbl 0729.35043
[9] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, AMS Reg. Conf. Ser. Math. 65 (1986). · Zbl 0609.58002
[10] P.H. Rabinowitz, A note on a semilinear elliptic equation on\(R\^{}\{n\}\), Nonlinear Analysis (a tribute in honour of G. Prodi, Q. Sc. Norm. Sup. Pisa) (1991) 307-318.
[11] Reed, M.; Simon, B., Methods of mathematical physics IV, (1978), Academic Press New York
[12] C. Troestler, M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, to appear. · Zbl 0864.35036
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