##
**Existence and uniqueness of nonnegative solutions of quasilinear equations in \(\mathbb{R}^ n\).**
*(English)*
Zbl 0853.35035

The main theorems establish existence and uniqueness of radially symmetric ground states \(u\) to nonlinear elliptic equations of the type
\[
- \text{div} [g(|\nabla u|) \nabla u]= f(u)\quad \text{in } \mathbb{R}^n,\quad n\geq 2\tag{1}
\]
for functions \(f\in C(\overline{\mathbb{R}}_+, \mathbb{R})\), \(g\in C(\mathbb{R}_+, \mathbb{R}_+)\) such that \(f(0)= 0\), \(f(t)< 0\) for small \(t> 0\), \(f(t)\) changes sign at some \(t_0> 0\), \(tg(t)\to 0\) as \(t\to 0+\), \(tg(t)\) is strictly increasing in \(\mathbb{R}_+\), and \(f\), \(g\) satisfy a variety of additional technical conditions. A ground state is defined to be a nonnegative nontrivial classical solution to (1) with limit zero at \(\infty\). In particular, the results apply to \(p\)-Laplacian equations (i.e., \(g(t)= t^{p- 2}\), \(p> 1\)) and mean curvature equations (i.e., \(g(t)= (1+ t^2)^{- 1/2}\)).

The lucid fascinating introduction contains a historical review, connections with earlier work, a summary of the results, and an outline of the procedures here and elsewhere. Section 1 contains 13 preliminary lemmas including some a priori identities for nonnegative classical radial solutions to (1), necessary conditions for the existence of such solutions, and sufficient or necessary conditions for solutions to have compact support. The existence theorems are proved in §2 by shooting methods applied to the polar form of (1). Uniqueness is proved in §3 via the monotone separation theorem of L. A. Peletier and the third author [Arch. Ration. Mech. Anal. 81, 181-197 (1983; Zbl 0516.35031); J. Differ. Equations 61, 380-397 (1986; Zbl 0577.35035)] and a remarkable new identity for nonnegative solutions \(u(r)\) to the polar form of (1) satisfying \(u'(0)= 0= u(\infty)\).

A short review precludes a detailed description of hypotheses, conclusions, techniques, or illustrations, nor can it reveal the depth of results, ingenuity of procedures, and perspicuity of presentation. The flavor is indicated by the following specialization: Suppose there exist constants \(b\), \(c\) with \(c> b> 0\) such that (i) \(F(t)< 0\) for \(0< t< b\), \(F(b)= 0\), where \(F(t)= \int^t_0 f(s) ds\); (ii) \(f(t)> 0\) for \(b\leq t< c\), \(f(c)= 0\) if \(c< \infty\); and (iii) \((t- b)^{1- p} f(t)\) is nonincreasing for \(b< t< c\). Then the \(p\)-Laplacian equation (1) \((p> 1)\) has a radial ground state \(u(r)\) such that \(u(0)\in (b, \infty)\) if \(c= \infty\) or \(u(0)\in (b, c]\) if \(c< \infty\). The solution is unique if, in addition, \(f\) is Lipschitzian in \((b, c)\) and either \(c= \infty\) or \(u(0)< c< \infty\). Even if the solution is not unique, \(u(0)\) is still unique.

The lucid fascinating introduction contains a historical review, connections with earlier work, a summary of the results, and an outline of the procedures here and elsewhere. Section 1 contains 13 preliminary lemmas including some a priori identities for nonnegative classical radial solutions to (1), necessary conditions for the existence of such solutions, and sufficient or necessary conditions for solutions to have compact support. The existence theorems are proved in §2 by shooting methods applied to the polar form of (1). Uniqueness is proved in §3 via the monotone separation theorem of L. A. Peletier and the third author [Arch. Ration. Mech. Anal. 81, 181-197 (1983; Zbl 0516.35031); J. Differ. Equations 61, 380-397 (1986; Zbl 0577.35035)] and a remarkable new identity for nonnegative solutions \(u(r)\) to the polar form of (1) satisfying \(u'(0)= 0= u(\infty)\).

A short review precludes a detailed description of hypotheses, conclusions, techniques, or illustrations, nor can it reveal the depth of results, ingenuity of procedures, and perspicuity of presentation. The flavor is indicated by the following specialization: Suppose there exist constants \(b\), \(c\) with \(c> b> 0\) such that (i) \(F(t)< 0\) for \(0< t< b\), \(F(b)= 0\), where \(F(t)= \int^t_0 f(s) ds\); (ii) \(f(t)> 0\) for \(b\leq t< c\), \(f(c)= 0\) if \(c< \infty\); and (iii) \((t- b)^{1- p} f(t)\) is nonincreasing for \(b< t< c\). Then the \(p\)-Laplacian equation (1) \((p> 1)\) has a radial ground state \(u(r)\) such that \(u(0)\in (b, \infty)\) if \(c= \infty\) or \(u(0)\in (b, c]\) if \(c< \infty\). The solution is unique if, in addition, \(f\) is Lipschitzian in \((b, c)\) and either \(c= \infty\) or \(u(0)< c< \infty\). Even if the solution is not unique, \(u(0)\) is still unique.

Reviewer: Charles A.Swanson (Vancouver)

### MSC:

35J60 | Nonlinear elliptic equations |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

34B15 | Nonlinear boundary value problems for ordinary differential equations |