Existence of \(H\)-bubbles in a perturbative setting
Let \(H: \mathbb R^3 \to \mathbb R\) be a \(C^1\) function and consider the problem of finding smooth non-constant maps \(\omega :\mathbb R^2 \to \mathbb R^3\) satisfying \[ \Delta \omega = 2 H(\omega) \omega_x \wedge \omega_y \quad\text{in } \mathbb R^2,\quad \text{and}\quad \int_{\mathbb R^2} | \nabla \omega | ^2 < + \infty.\tag{1} \] It is known that such a solution must be conformal and, at each regular point \(p=\omega (z)\), \(H(p)\) is the mean curvature of the surface parameterized by \(\omega\) at the point \(p\). Moreover, if \(\sigma : S^2 \to \mathbb R^2\) denotes the stereographic projection, the map \(\omega \circ \sigma : S^2 \to \mathbb R^3\) defines an \(S^2\)-type parametric surface in \(\mathbb R^3\) having prescribed mean curvature, simply called an \(H\)-bubble. The problem has also a variational description in which \(H\)-bubbles can be found as critical points of an energy functional. When the prescribed mean curvature is a non-zero constant function, \(H(u)\equiv H_0\), \textit{H. Brezis} and \textit{J.-M. Coron} [Arch. Ration. Mech. Anal. 89, 21--56 (1985; Zbl 0584.49024)] have proved that the only non-zero solutions to (1) are spheres of radius \(| H_0| ^{-1}\) placed anywhere in \(\mathbb R^3\). The present paper deals with the case \[ H(u)=H_0(u)+\varepsilon H_1(u)=:H_{\varepsilon}(u),\tag{2} \] where \(H_0 \in C^1(\mathbb R^3)\) satisfies some particular conditions, \(| \varepsilon| \) is small, and \(H_1 : \mathbb R^3 \to \mathbb R\) is any \(C^1\) function. The conditions for \(H_0\) are motivated by a previous result of the authors [Commun. Contemp. Math. 4, No. 2, 177--209 (2002; Zbl 1009.53008)] regarding the existence of solutions to (1) when \(H\) is nonconstant. For such an \(H_0\), the authors have proved the existence of \(H_0\) bubbles which are minimal in the sense of having minimal energy in the variational description mentioned above. The main result of the paper states that for any \(H_{\varepsilon}\) as in (2) there exists an \(\widehat{\varepsilon} >0\), such that \(\forall \varepsilon \in (-\widehat{\varepsilon}, \widehat{\varepsilon})\), there exists an \(H_{\varepsilon}\)-bubble. Furthermore, as \(\varepsilon \to 0\), \(\omega^{\varepsilon}\) converges to some minimal \(H_0\)-bubble \(\omega\), as \(\omega^{\varepsilon} \circ \sigma \to \omega \circ \sigma\) in \(C^1(S^2, \mathbb R^3)\).
scientific article; zbMATH DE number 2110200