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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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