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Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball. (English) Zbl 0852.34021
The nonlinear Dirichlet problem under consideration has the form $$- \Delta_p u= f(u)+ h(|x|),\quad x\in B,\quad u|_{\partial B}= 0,\tag1$$ where $B$ denotes an open ball in $\bbfR^n$, $\Delta_p$ is the $p$-Laplacian operator for $p> 1$, $f: \bbfR\to \bbfR$ is locally Lipschitzian and superlinear, and $h\in L^\infty(B, \bbfR)$. Let $y(t, a)$ denote a solution of the polar form of (1) satisfying $y(0)= a\ne 0$, $y'(0)= 0$ $(t\ge 0)$. 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 {\it M. Struwe} [Arch. Math. 39, 233-240 (1982; Zbl 0496.35034)], {\it A. Castro} and {\it A. Kurepa} [Proc. Am. Math. Soc. 101, 57-64 (1987; Zbl 0656.35048)].

34B15Nonlinear boundary value problems for ODE
35J65Nonlinear boundary value problems for linear elliptic equations
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