Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the \(p\)-Laplacian. (English) Zbl 1064.35053

Anello, Giovanni; Cordaro, Giuseppe

Proc. R. Soc. Edinb., Sect. A, Math. 132, No. 3, 511-519 (2002); corrigendum 133, No. 1, 1 (2003).

The authors present a result of existence of infinitely many arbitrarily small positive solutions to the Dirichlet problem involving the \(p\)-Laplacian: \[ -\Delta_pu=\lambda f(x,u)\text{ in }\Omega\qquad u=0\text{ on } \partial \Omega, \] where \(\Omega\in {\mathbb R}^N\) is a bounded set with sufficiently smooth boundary \(\partial \Omega\), \(p>1\), \(\lambda>0\), and \(f:\Omega\times \mathbb R\to \mathbb R\) is a Carathéodory function satisfying the condition: there exists \(\bar{t}>0\) such that \[ \sup_{t\in[0,\bar t\, ]} f(\cdot,t)\in L^{\infty}(\Omega). \] {In the corrigendum several errors are corrected, esp. “a.e. positive” has to be changed into “nonzero and nonnegative”}.

Reviewer: Josef Diblík (Brno)

MSC:

35J60	Nonlinear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35D05	Existence of generalized solutions of PDE (MSC2000)
35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Dirichlet problem; small positive solutions; \(p\)-Laplacian