## Nontrivial solution of a semilinear Schrödinger equation.(English)Zbl 0864.35036

The authors consider the nonlinear, stationary Schrödinger equation $$-\Delta u+V(\cdot)u= f(\cdot,u)$$ in $$\mathbb{R}^n$$. Assume that $$V\in C(\mathbb{R}^n)$$ and $$f\in C^1(\mathbb{R}^n\times \mathbb{R})$$ are 1-periodic in $$x_k$$, $$1\leq k\leq n$$, and $$D:H^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$$, $$u\mapsto-\Delta u+V(\cdot)u$$ is invertible. Moreover, let $$f$$ satisfy $$|f_u(x,u)|\leq\text{const}(|u|^{q-2}+|u|^{p-2})$$ for $$2<q\leq p<2^*$$ $$(={{2n}\over{n-2}}$$ for $$n\geq 3$$, $$=\infty$$ otherwise). Then the equation has a nontrivial solution $$u\in H^1(\mathbb{R}^n)$$, provided that $$0<\alpha F(x,u)\leq f(x,u)u$$, $$u\neq 0$$, with some constant $$\alpha>2$$ and $$F(x,u):= \int^u_0 f(x,t)dt$$.
To obtain this solution, the existence of a Palais-Smale sequence to a suitable level of the functional ${\mathcal E}(u):= \int_{\mathbb{R}^n} {\textstyle{1\over2}} |\nabla u(x)|^2+ {\textstyle{1\over2}} V(x)u^2(x)- F(x,u(x))dx, \qquad u\in H^1(\mathbb{R}^n),$ is proved, i.e., there is a $$c\in(0,\infty)$$ and $$\{u_n\}\subset H^1(\mathbb{R}^n)$$ such that $${\mathcal E}(u_n)\to c$$ and $$\nabla{\mathcal E}(u_n)\to 0$$ in $$H^1(\mathbb{R}^n)$$. The difficulty here is that $${\mathcal E}$$ satisfies no Palais-Smale condition because of the periodicity. Moreover, the functional is strongly indefinite since 0 lies in a spectral gap of the linear operator $$D$$.

### MSC:

 35J60 Nonlinear elliptic equations 49J35 Existence of solutions for minimax problems 35J20 Variational methods for second-order elliptic equations
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### References:

 [1] Alama S., J.Diff.Equ. 96 pp 89– (1992) · Zbl 0766.35009 [2] Benci V., Inventiones Math. 52 pp 241– (1979) · Zbl 0465.49006 [3] Coste J., The continuous model. Physical review B 39 (18) pp 13096– (1989) [4] Buffoni B., Proc. Amer. Math. Soc. 119 (18) pp 179– (1993) [5] Esteban M.J., Commun.Math.Phys. 171 (18) pp 323– (1995) · Zbl 0843.35114 [6] Gohberg, I., Golberg, S. and Kaashoek, M. 1991. ”Classes of linear operators”. Vol. 1, Birkhäuser. [7] Heinz H.P., J.Diff.Equ. 100 pp 341– (1992) · Zbl 0767.35006 [8] Höfer H., Mathematische Annalen pp 483– [9] Höfer H., Inventiones Math. 90 pp 1– (1987) · Zbl 0631.58022 [10] Jeanjean I., J.Diff.Equ. 112 pp 53– (1994) · Zbl 0804.35033 [11] Lions P.L., Part 1 and 2. Ann. Inst. Henri Poincaré 1 pp 109– (1984) [12] Moulis M., Ann. Inst. Inst.Fourier 21 4 pp 293– (1971) · Zbl 0271.41025 [13] Nabiev R.F., Optic letters 18 (19) pp 1612– (1993) [14] Séré E., commun. Math. Phys. 172 (19) pp 293– (1995) · Zbl 0840.34046 [15] Smale S., American Journ. of Math. 87 (19) pp 861– (1965) · Zbl 0143.35301 [16] Tanaka K., J.Diff. Equat. 94 (19) pp 315– (1991) · Zbl 0787.34041 [17] Willem M., Progress in Diff. Eqns, to appear
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