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Multiple solutions for a class of nonlinear Schrödinger equations. (English) Zbl 1072.35166

The authors of this very interesting paper consider the nonlinear Schrödinger equation having the form

-u+V(x)u=g(x,u)

for x N , where u(x)0 as |x|. This equation arises in numerous applications in physics and chemistry. It turns out that there exist infinitely many homoclinic type solutions under the following assumptions:

(1) VC 1 ( N ,), V(x)>0 and is 1-periodic in x j (j=1,,N);

(2) gC 2 ( N ×,) and g(x,u) is 1-periodic in x j (j=1,,N);

(3) g(x,0)=0=g u (x,0) and G(x,u) 0 u g(x,s)ds, 2 * 2N/(N-2) for N3;

(4) G(x,u)0, g u (x,u)V (x) as |u| uniformly in x, where V (x) is 1-periodic in x j (j=1,,N) and inf V ( N )>λ ¯;

(5) G ˜(x,u)(1/2)g(x,u)u-G(x,u)0 and there is ρ(0,λ ̲) such that G ˜(x,u)ρ whenever g(x,u)/uλ ̲-ρ.

Then there exist infinitely many N -distinct solutions. Here two solutions u 1 ,u 2 of the equation under consideration are said to be geometrically distinct (or N -distinct) if u 1 τ k u 2 for all k=(k 1 ,,k N ) N , where (τ k u)(x)=u(x 1 +k 1 ,,x N +k N ). The second result concerns the case under assumptions (1)-(3) and a modification of (4) and (5). Then the authors prove again that there exist infinitely many N -distinct solutions.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35B38Critical points in solutions of PDE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods