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Some remarks on a conjecture of De Giorgi. (English) Zbl 0938.35057
This article concerns one-dimensional solutions to the $$p$$-Laplacian equation $\text{div} \bigl(|\nabla u|^{p-2}\nabla u\bigr)= f'(u)\text{ in }\mathbb{R}^n,\tag{1}$ where $$n\geq 2$$, $$p\geq 2$$, $$f\in C^2(\mathbb{R})$$, $$f$$ is nonnegative with (at least) two zeros $$z_\pm$$, $$z_-<z_+$$, $$f'(t)$$ has exactly one zero in $$(z_-,z_+)$$, and $$f(t)=O (|t-z|^p)$$ near any zero $$z$$ of $$f$$. For a bounded domain $$\Omega$$ in $$\mathbb{R}^{n-1}$$, define $$S_\Omega$$ to be the set of all functions $$v\in W^{1,p}_{\text{loc}}(\mathbb{R} \times\Omega)$$ such that $$v(x)\to z_{\pm}$$ as $$x_1\to \pm\infty$$ uniformly. The main theorem states that a solution $$u\in W^{1,p}_{\text{loc}}(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n)$$ to (1) obtained as a minimizer for an energy functional associated with (1) over $$S_\Omega$$ must be a one-dimensional function of type $$u(x)=g(x_1+a)$$ for some constant $$a\in\mathbb{R}$$ and a unique function $$g\in C^3(\mathbb{R})$$ with $$g(x_1)\in(z_-,z_+)$$, $$g'(x_1)>0$$ for all $$x_1\in\mathbb{R}$$. The proof employs results/techniques of L. Caffarelli, N. Garofalo, and F. Segala [Commun. Pure Appl. Math. 47, 1457-1473 (1994; Zbl 0819.35016)] and G. Carbou [Ann. Inst. H. Poincaré, Anal. Non Linéaire 12, 305-318 (1995; Zbl 0835.35045)]. The first theorem of the present type was obtained by Modica and Mortola in the case $$n=2$$, $$p=2$$, $$4f(t)= (t^2-1)^2$$ (i.e., scalar Ginzburg-Landau equation) [L. Modica, St. Mortola, Boll. Unione Mat. Ital., IV. Ser. B17, 614-622 (1980; Zbl 0448.35044)], confirming a conjecture of De Giorgi in 1978.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations
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