an:01790828
Zbl 1064.35053
Anello, Giovanni; Cordaro, Giuseppe
Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the \(p\)-Laplacian
EN
Proc. R. Soc. Edinb., Sect. A, Math. 132, No. 3, 511-519 (2002); corrigendum 133, No. 1, 1 (2003).
00085245
2003
j
35J60 35B05 35D05 35J20 35J25
Dirichlet problem; small positive solutions; \(p\)-Laplacian
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 Carath??odory 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''\}.
Josef Dibl??k (Brno)