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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.

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