Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci; Roberto Livrea

Annales Polonici Mathematici, Tome 81 (2003), p. 77-85 / Harvested from The Polish Digital Mathematics Library

Résumé

We deal with a bifurcation result for the Dirichlet problem ⎧ $- Δ_{p} u = μ / {| x |}^{p} {| u |}^{p - 2} u + λ f (x, u)$ a.e. in Ω, ⎨ ⎩ $u_{| \partial Ω} = 0$ . 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 μ close to zero, there exists a positive number $λ *_{μ}$ such that for every $λ \in] 0, λ *_{μ} [$ the above problem admits a nonzero weak solution $u_{λ}$ in $W ₀^{1, p} (Ω)$ satisfying $l i m_{λ \to 0 ⁺} | | u_{λ} | | = 0$ .

Publié le : 2003-01-01

Zbl 1273.35039

EUDML-ID : urn:eudml:doc:280183

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     title = {Bifurcation theorems for nonlinear problems with lack of compactness},
     journal = {Annales Polonici Mathematici},
     volume = {81},
     year = {2003},
     pages = {77-85},
     zbl = {1273.35039},
     language = {en},
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Francesca Faraci; Roberto Livrea. Bifurcation theorems for nonlinear problems with lack of compactness. Annales Polonici Mathematici, Tome 81 (2003) pp. 77-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap82-1-9/