Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci¹ ; Roberto Livrea²

¹ Dipartimento di Matematica e Informatica Università di Catania Viale A. Doria 6 95125 Catania, Italy
² Dipartimento di Matematica Università di Messina Salita Sperone 31 98166 Sant'Agata, Messina, Italy

Annales Polonici Mathematici, Tome 82 (2003) no. 1, pp. 77-85

Résumé

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 $\lim_{\lambda\to 0^+}\|u_\lambda\|=0$.

Détail

DOI : 10.4064/ap82-1-9

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