Existence and Uniqueness of Positive Solution for 𝑝 -Laplacian Kirchhoff-Schrödinger-Type Equation

We study the existence and uniqueness of positive solution for the following 𝑝 -Laplacian-Kirchhoff-Schr¨odinger-type equation: {−(𝑎 + 𝑏∫ Ω |∇𝑢| 𝑝 )󳵻 𝑝 𝑢 + 𝜆 V (𝑥)|𝑢| 𝑝−2 𝑢 = ℎ𝑓(𝑢) − 𝜇𝑔(𝑢), in Ω, 𝑢 > 0, in Ω, 𝑢 = 0, on 𝜕Ω} , where Ω ⊂ 𝑅 𝑁 (𝑁 ≥ 3) , 𝜆,𝜇 ≥ 0 are parameters, V (𝑥), 𝑓(𝑢),𝑔(𝑢) and ℎ are under some suitable assumptions. For the purpose of overcoming the difficulty caused by the appearance of the Schr¨odinger term and general singularity, we use the variational method and some mathematical skills to obtain the existence and uniqueness of the solution to this problem.

However, up to now, no paper has appeared in the literature which discusses the existence and uniqueness of the positive solution for the -Laplacian-Kirchhoff-Schrödingertype problem.This paper attempts to fill this gap in the literature.Inspired by the above works, in this paper, we try to study the existence and uniqueness of solution to the problem (1) by using the variational method. 0 () < ∞, and there exists ,  ∈ (0, 1) such that lim →0 +  ()   = +∞, lim →∞  ()   = 0. (2) ()  ∈ ( + ,  + ) and there exists a constant  > 0, such that  () ≤  ( −1 +   * −1 ) ,  ∈  + . (3) ) > 0 and the minimum of V() can be achieved in Ω.In other words, there exists a constant   , such that   = inf ∈Ω V().
In this paper, we will make full use of the following definitions.
Remark 3. The result obtained in the paper is an expanding study of the Kirchhoff-Schrödinger-type equation ( = 2); the difficulty is posed by the degenerate quasilinear elliptic operator.We mainly use the variational method to solve the problem.

Preliminary
To prove the main results in this paper, we will employ the following important lemma.
On the basis of the definition of , we can deduce that there exists a sequence {  } ⊂   such that lim →∞ (  ) = .Since  is coercive and  < 0, {  } is bounded in   .Going if necessary to a subsequence, still denoted by {  }, there exists as  → ∞.It follows from (8) and Sobolev embedding theorem that {(  )} is bounded in   * /(1+) .Moreover, from the continuity of , we can get that (  ()) → (  ()), a.e. ∈ Ω.Thus, we obtain Moreover, by Fatou's lemma, we have lim inf According to the weakly lower semicontinuity of the norm, ( 18) and ( 19), we have which yields ( 0 ) = .The proof is completed.
Proof of Theorem 2. Assume that  1 is also a solution of problem (1).Letting ] =  0 −  1 , according to the definition of the weak solution and (26), we can get Advances in Mathematical Physics which implies Next, we will make some estimates for the equation.

Data Availability
The data used to support the findings of this study are included within the article.