Nonhomogeneous Elliptic Kirchhoff Equations of the P -Laplacian Type
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NONHOMOGENEOUS ELLIPTIC KIRCHHOFF EQUATIONS OF THE P -LAPLACIAN TYPE A. Benaissa1,2 and A. Matallah3
UDC 517.9
We use variational methods to study the existence and multiplicity of solutions for a nonhomogeneous p-Kirchhoff equation with the critical Sobolev exponent.
1. Introduction The present paper deals with the existence and multiplicity of solutions to the following Kirchhoff problem with the critical Sobolev exponent −(akukp + b) ∆p u = up u2W
⇤ −1
1,p
+ λg(x)
in RN , (Pλ )
N
(R ),
where N ≥ 3, 1 < p < N, ∆p is the p-Laplacian operator, k . k is the ordinary norm in W 1,p (RN ) given by p
kuk =
Z
RN
|ru|p dx,
p⇤ = pN/(N − p) is the critical Sobolev exponent of the embedding
kukqq =
Z
⇣ ⌘ � � 1,p N W (R ), k.k ,! Lq (RN ), k.kq
with
q 2 [p, p⇤ ],
|u|q dx is the norm in Lq (RN ), a and b are two positive constants, λ is a positive parameter, and g �⇤ � is a function from W 1,p (RN ) such that Z gu⇤ dx 6= 0, RN
RN
where u⇤ is a function defined below in (1), 1
⇣� ⌘ �⇤ W 1,p (RN ) is the dual of W 1,p (RN ) .
Laboratory of Analysis and Control of PDEs, Djillali Liabes University, Sidi Bel Abbes, Algeria; e-mail: benaissa [email protected]. Corresponding author. 3 ´ Ecole Pr´eparatoire en Sciences Economiques, Commerciales et Sciences de Gestion, Tlemcen, Algeria; e-mail: atikaa [email protected]. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 184–190, February, 2020. Original article submitted February 13, 2017. 0041-5995/20/7202–0203
© 2020
Springer Science+Business Media, LLC
203
A. B ENAISSA
204
AND
A. M ATALLAH
In recent years, various Kirchhoff-type problems in bounded or unbounded domains have been studied in numerous papers by using variational methods. Some interesting studies can be found in [1, 4–6, 8]. Since the Sobolev embedding ⇣ ⌘ � � 1,p N W (R ), k.k ,! Lq (RN ), k.kq
is not compact for all q 2 [p, p⇤ ], numerous authors studied the following Kirchhoff-type problem without the critical Sobolev exponent: − (akukp + b) ∆p u + V (x)u = h(x, u)
RN ,
in
(PV )
� � � � where V 2 C RN , R and h 2 C RN ⇥ R, R is subcritical and satisfies sufficiency conditions required to show the boundedness of any Palais-Smale or Cerami sequence. They imposed certain conditions on the weight function V (x) for recovering the compactness of the Sobolev embedding (see, e.g., [11]). It is worth noting that, to the best of our knowledge, there are no results concerning Kirchhoff equations of p-Laplacian type with nonlinear terms of critical growth but without potential terms of higher dimension. The main result of the present paper is the following theorem: Theorem 1.1. Assume that a > 0, b > 0, N = 3k, p = 2k, and k 2 N⇤ . Then there exists ⇤⇤ > 0 such that the problem (Pλ ) has at least two nontrivial solutions for any λ 2 (0, ⇤⇤ ). The paper is organized as follows: In Section 2, we give some technical results, which allow us to formulate a variational approach to our main result. The main result is proved in Section 3. 2. Auxiliary Results � � In the present p
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