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A note on the existence of infinitely many radially symmetric solutions of a quasilinear elliptic problem. (English) Zbl 0901.35033

Summary: The existence of infinitely many classical radially symmetric solutions of the quasilinear elliptic problem in a ball \[ -\text{div} \biggl(a\bigl( |\nabla u|^2 \bigr) \nabla u \biggr) =f(u)+ h\bigl(| x| \bigr) \text{ in } B_R, \quad u=0 \text{ on } \partial B_R \] is established assuming that the quotient \(F(s)/A(s)\), where \(F(s)= \int^s_0 f(\xi)d\xi\) and \(A(s^2) =2 \int^s_0 \xi a(\xi^2) d\xi\), exhibits an oscillatory behaviour at infinity. The case in which the domain is an annulus is also discussed.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
35J20 Variational methods for second-order elliptic equations
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