A critical point theorem for perturbed functionals and localization of critical point in bounded convex

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+.