Least energy solution for a scalar field equation with a singular nonlinearity. (English) Zbl 1459.35209

Summary: We are concerned with a nonnegative solution to the scalar field equation \[ \Delta u+f(u)=0\text{ in }\mathbb{R}^N,\quad \lim\limits_{|x|\to\infty}u(x)=0. \] A classical existence result by H. Berestycki and P.-L. Lions [Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] considers only the case when \(f\) is continuous. In this paper, we are interested in the existence of a solution when \(f\) is singular. For a singular nonlinearity \(f\), F. Gazzola et al. [Adv. Differ. Equ. 5, No. 1–3, 1–30 (2000; Zbl 0987.35064)] proved an existence result when \(f\in L^1_{\mathrm{loc}}(\mathbb{R})\cap\text{Lip}_{\mathrm{loc}}(0,\infty)\) with \(\int_0^uf(s)\,\mathrm{d}s<0\) for small \(u>0\). Since they use a shooting argument for their proof, they require the property that \(f\in\text{Lip}_{\mathrm{loc}}(0,\infty)\). In this paper, using a purely variational method, we extend the previous existence results for \(f\in L^1_{\mathrm{loc}}(\mathbb{R})\cap C(0,\infty)\). We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.


35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
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