Abstract
In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.
Highlights
The original one-dimensional Kirchhoff equation was introduced by Kirchhoff [1] in 1883 as an generalization of the well-known d’Alembert’s wave
His model takes into account the changes in length of the strings produced by transverse vibrations
L is the length of the string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density and P0 is the initial tension
Summary
Definition 1 Let c ∈ , E a Banach space and I ∈C1 ( E, ). Lemma 1 Let X Banach space, and J ∈C1 ( X , ) verifying the Palais-Smale condition. Nehari Manifold ( ) It is well known that the functional J is of class C1 in 01 3 and the solutions of (1.1) are the critical points of J which is not bounded below on ( ) 01 3. Lemma 2 J is coercive and bounded from below on. Lemma 3 Suppose that u0 is a local minimizer for J on. If u0 is a local minimizer for J on , u0 is a solution of the optimization problem min J (u).
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