Multiple positive solutions for equations involving critical Sobolev exponent in \(\mathbb{R}^ N\). (English) Zbl 0886.35056

Alves, C. O.

Electron. J. Differ. Equ. 1997, Paper 13, 10 p. (1997).

Summary: This article deals with the problem \[ -\text{div}(|\nabla u|^{m-2}\nabla u) = \lambda h u^q+u^{m^*-1}\quad\text{in}\quad {\mathbb R}^N . \] Using the Ekeland variational principle and the mountain pass theorem, we show the existence of \(\lambda ^*>0\) such that there are at least two nonnegative solutions for each \(\lambda\) in \((0,\lambda ^*)\).

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations

Keywords:

mountain pass theorem; Ekeland variational principle; \(p\)-Laplacian