A critical point theorem for perturbed functionals and localization of critical point in bounded convex
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Date
2023-06-10
Journal Title
Journal ISSN
Volume Title
Publisher
University of M'sila
Abstract
In this work, we have studied a new critical point theorem in the following two cases:
In the first case, we have established a new critical point theorem for a class of perturbed
functionals without satisfying the Palais-Smale condition, which asserts the existence of critical
point of functionals of the type I = I1 + I2, provided that I1 has at least one critical point. The
main abstract result is applied to the following nonhomogeneous Problem:
( −div(|∇u|p−2 .∇uu) = = 0|u|q−2 u + λg(x, u) on in ΩΓ,.
Where λ ∈ R, Ω is a bounded set of RNwith smooth boundary Γ, 1 < q < p with p > N and
g(·, ·) is continuous on Ω¯ × [0, ∞).
The last case, we have established the localization of a critical point of minimum type
of a smooth functional is obtained in a bounded convex conical set defined by a norm and
a concave upper semicontinuous functional. Our abstract result is applied to the following
Periodic Problem:
( −u′′(ut) + (0) −a2uu((Tt) = ) =fu(′u(0) (t)) − uon ′(T(0 ) = 0 , T),.
Where a ̸= 0 and f : R → R is a continuous function with f(R+) ⊂ R+.
Description
Keywords
Critical point, Perturbed Functional, Weak solution, Ekeland’s Principle, Periodic Problem, Positive Solution.