Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems
2008 (English)In: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 15, no 4-5, 581-598 p.Article in journal (Refereed) Published
Let c be the usual mountain pass level for the semilinear elliptic functional G(u) = 1/2 integral(Omega) (vertical bar del u vertical bar(2) +lambda u(2))dx -integral(Omega) F(x, u(x))dx. (0.1) In general, c <= c(#), where c# is the analogous mountain pass level of the asymptotic functional G(#) defined with respect to unbounded shifts or dilations. We show under general conditions that whenever the strict inequality c <= c(#) holds, the functional G satisfies the Palais-Smale condition at the level c and, consequently, has a critical point at this level. This sets a solvability framework for unconstrained mountain pass similar to that of P.-L. Lions set for constrained minimization. The nonlinearity F is allowed to have critical growth with asymptotically selfsimilar oscillations about the critical "stem" vertical bar u vertical bar(2)* and not only "stem" asymptotics. For example, the main existence result, Theorem 3.2, holds for F(x, s) = vertical bar 8 vertical bar 2 e(sigma Nx2/e1+x2sin(2log(vertical bar s vertical bar))). Since the unconstrained minimax is studied, the convexity-type conditions that arise with the use of Nehari constraint (G' (u), u) = 0 are not required.
Place, publisher, year, edition, pages
2008. Vol. 15, no 4-5, 581-598 p.
Semilinear elliptic equations, concentration compactness, mountain pass, positive solutions, variational problems
IdentifiersURN: urn:nbn:se:uu:diva-129271DOI: 10.1007/s00030-008-7046-8ISI: 000261985600009OAI: oai:DiVA.org:uu-129271DiVA: diva2:339014