Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0864.35036
Troestler, C.; Willem, M.
Nontrivial solution of a semilinear Schrödinger equation. (English)
[J] Commun. Partial Differ. Equations 21, No.9-10, 1431-1449 (1996). ISSN 0360-5302; ISSN 1532-4133/e

The authors consider the nonlinear, stationary Schrödinger equation $-\Delta u+V(\cdot)u= f(\cdot,u)$ in $\bbfR^n$. Assume that $V\in C(\bbfR^n)$ and $f\in C^1(\bbfR^n\times \bbfR)$ are 1-periodic in $x_k$, $1\leq k\leq n$, and $D:H^2(\bbfR^n)\to L^2(\bbfR^n)$, $u\mapsto-\Delta u+V(\cdot)u$ is invertible. Moreover, let $f$ satisfy $|f_u(x,u)|\le\text{const}(|u|^{q-2}+|u|^{p-2})$ for $2<q\leq p<2^*$ $(={{2n}\over{n-2}}$ for $n\geq 3$, $=\infty$ otherwise). Then the equation has a nontrivial solution $u\in H^1(\bbfR^n)$, provided that $0<\alpha F(x,u)\leq f(x,u)u$, $u\ne 0$, with some constant $\alpha>2$ and $F(x,u):= \int^u_0 f(x,t)dt$. \par To obtain this solution, the existence of a Palais-Smale sequence to a suitable level of the functional $${\cal E}(u):= \int_{\bbfR^n} {\textstyle{1\over2}} |\nabla u(x)|^2+ {\textstyle{1\over2}} V(x)u^2(x)- F(x,u(x))dx, \qquad u\in H^1(\bbfR^n),$$ is proved, i.e., there is a $c\in(0,\infty)$ and $\{u_n\}\subset H^1(\bbfR^n)$ such that ${\cal E}(u_n)\to c$ and $\nabla{\cal E}(u_n)\to 0$ in $H^1(\bbfR^n)$. The difficulty here is that ${\cal E}$ 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 $D$.
[R.Beyerstedt (Aachen)]

MSC 2000:: *35J60 Nonlinear elliptic equations
49J35 Minimax problems (existence)
35J20 Second order elliptic equations, variational methods

Keywords: indefinite functionals; periodic coefficients; Palais-Smale condition

Cited in: Zbl 1117.58007 Zbl 1059.35506

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]