## 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.

### MSC:

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

### Citations:

Zbl 0533.35029; Zbl 0987.35064
Full Text:

### References:

 [1] Acciaio, B., and Pucci, P.. Existence of radial solutions for quasilinear elliptic equations with singular nonlinearities. Adv. Nonlinear Stud.3 (2003), 511-539. · Zbl 1080.35019 [2] Berestycki, H., Gallouet, T. and Kavian, O.. Équations de champs scalaires euclidiens non linaires dans le plan. Compt. Rend. Acad. Sci.297 (1983), 307-310. · Zbl 0544.35042 [3] Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rational Mech. Anal.82 (1983), 313-345. · Zbl 0533.35029 [4] Byeon, J., Jeanjean, L. and Maris, M.. Symmetry and monotonicity of least energy solutions. Calc. Var. Partial Differ. Equ.36 (2009), 481-492. · Zbl 1226.35041 [5] Chung, J., Kim, Y.-J., Kwon, O. and Pan, X., Discontinuous nonlinearity and finite time extinction, submitted. · Zbl 1431.35073 [6] Cortázar, C., Elgueta, M. and Felmer, P.. Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equatio. Comm. Partial Differ. Equ.21 (1996), 507-520. · Zbl 0854.35033 [7] Davila, J. and Montenegro, M.. Concentration for an elliptic equation with singular nonlinearity. J. Math. Pures Appl.97 (2012), 545-578. · Zbl 1246.35085 [8] Gazzola, F., Serrin, J. and Tang, M.. Existence of ground states and free boundary problems for quasilinear elliptic operators. Adv. Differ. Equ.5 (2000), 1-30. · Zbl 0987.35064 [9] Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd ed., 224 (Berlin: Springer-Verlag, 1983). · Zbl 0562.35001 [10] Gui, C.. Symmetry of the blow-up set of a porous medium type equation. Comm. Pure Appl. Math.48 (1995), 471-500. · Zbl 0827.35014 [11] Kaper, H. G., Kwong, M. K. and Li, Y.. Symmetry results for reaction-diffusion equations. Differ. Int. Equ.6 (1993), 1045-1056. · Zbl 0799.35083 [12] Pohožaev, S. I.. On the eigenfunctions of the equation $$\Delta u + \lambda f(u) = 0$$. Dokl. Akad. Nauk SSSR61 (1965), 36-39. [13] Pucci, P., Garcia-Huidobro, M., Manasevich, R. and Serrin, J.. Qualitative properties of ground states for singular elliptic equations with weights. Ann. Mat. Pura Appl.185 (2006), S205-S243. · Zbl 1115.35050 [14] Pucci, P., Serrin, J. and Zou, H.. A strong maximum principle and a compact support principle for singular elliptic inequalities. J. Math. Pures Appl.78 (1999), 769-789. · Zbl 0952.35045 [15] Redheffer, R.. Nonlinear differential inequalities and functions of compact support. Trans. Amer. Math. Soc.220 (1976), 133-157. · Zbl 0361.35029 [16] Serrin, J. and Tang, M.. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J.49 (2000), 897-923. · Zbl 0979.35049 [17] Vázquez, J. L.. A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim.3 (1984), 191-202. · Zbl 0561.35003
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