Bifurcation theorems for nonlinear problems with lack of compactness
Volume 82 / 2003
Annales Polonici Mathematici 82 (2003), 77-85
MSC: 35B32, 35J60.
DOI: 10.4064/ap82-1-9
Abstract
We deal with a bifurcation result for the Dirichlet problem $$ \cases{ \displaystyle -{\mit\Delta}_pu=\frac{\mu}{|x|^p}\,|u|^{p-2}u +\lambda f(x,u) &\hbox{a.e. in }{\mit\Omega},\cr u_{|\partial{\mit\Omega}}=0.\cr} $$ Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for $\mu$ close to zero, there exists a positive number $\lambda^*_\mu$ such that for every $\lambda\in \mathopen{]}0,\lambda^*_\mu\mathclose{[}$ the above problem admits a nonzero weak solution $u_\lambda$ in $W_0^{1,p}({\mit\Omega})$ satisfying $\lim_{\lambda\to 0^+}\|u_\lambda\|=0$.
