Exactly two entire positive solutions for a class of nonhomogeneous elliptic equations. (English) Zbl 1164.35302

Chen, Kuan-Ju

Differ. Integral Equ. 17, No. 1-2, 1-16 (2004).

Summary: We study the nonhomogeneous elliptic equation \(-\Delta u(x)+u(x)=\lambda (f(x,u)+h(x))\) in \(\mathbb R^N\), where \(f(x,u)\) and \(h(x)\) satisfy some assumptions. We use variational methods to obtain existence results and give a concise argument to establish the exact number of solutions. In addition to a lack of compactness the main difficulty to overcome is the degenerated structure of the set of possible critical points.

MSC:

35A15	Variational methods applied to PDEs
35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

variational methods; existence results