El Hachimi, A.; de Thélin, F. Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball. (English) Zbl 0852.34021 J. Differ. Equations 128, No. 1, 78-102 (1996). The nonlinear Dirichlet problem under consideration has the form \[ - \Delta_p u= f(u)+ h(|x|),\quad x\in B,\quad u|_{\partial B}= 0,\tag{1} \] where \(B\) denotes an open ball in \(\mathbb{R}^n\), \(\Delta_p\) is the \(p\)-Laplacian operator for \(p> 1\), \(f: \mathbb{R}\to \mathbb{R}\) is locally Lipschitzian and superlinear, and \(h\in L^\infty(B, \mathbb{R})\). Let \(y(t, a)\) denote a solution of the polar form of (1) satisfying \(y(0)= a\neq 0\), \(y'(0)= 0\) \((t\geq 0)\). The energy functional for \(y(t, a)\) is \[ E(t, a)= {p- 1\over p} |y'(t, a)|^p+ \int^{y(t, a)}_0 f(s) ds. \] The main theorem states that (1) has infinitely many radial solutions \(u\) with \(u(0)> 0\) \((u(0)< 0)\) if \(E(t, a)\to + \infty\) as \(a\to +\infty\) (\(a\to - \infty\), respectively) uniformly in \(t\). The proof is based on phase plane analysis of the polar form of (1). More than half of the paper is devoted to the development of sufficient conditions, via a shooting method, for the energy hypothesis of the theorem to hold. Results of this type for \(p= 2\) were obtained by M. Struwe [Arch. Math. 39, 233-240 (1982; Zbl 0496.35034)], A. Castro and A. Kurepa [Proc. Am. Math. Soc. 101, 57-64 (1987; Zbl 0656.35048)]. Reviewer: Charles A.Swanson (Vancouver) Cited in 1 ReviewCited in 15 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:nonlinear Dirichlet problem; energy functional; infinitely many radial solutions; shooting method Citations:Zbl 0496.35034; Zbl 0656.35048 PDFBibTeX XMLCite \textit{A. El Hachimi} and \textit{F. de Thélin}, J. Differ. Equations 128, No. 1, 78--102 (1996; Zbl 0852.34021) Full Text: DOI Link