## On De Giorgi’s conjecture in dimensions 4 and 5.(English)Zbl 1165.35367

The authors develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago.
Conjecture: Suppose that $$u$$ is an entire solution of the equation
$\Delta u+u-u^3=0, \quad |u|=1, \qquad x=(x',x_n)\in\mathbb R^n$
satisfying $$\frac{\partial u}{\partial x_n}>0$$, $$x\in\mathbb R^n$$. Then, at least for $$n\leq 8$$, the level sets of $$u$$ must be hyperplanes. The main message of the authors is that De Giorgi’s conjecture is true in dimensions $$n=4,5$$ provided that the solutions are also assumed to satisfy an anti-symmetry condition.

### MSC:

 35J60 Nonlinear elliptic equations 35J15 Second-order elliptic equations 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

entire solution; level sets; anti-symmetry condition
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