The Dirichlet Problem for a Class of Anisotropic Schr¨odinger-Kirchhoff-Type Equations with Critical Exponent

. In this paper, our focus lies in addressing the Dirichlet problem associated with a specific class of critical anisotropic elliptic equations of Schr¨odinger-Kirchhoff type. These equations incorporate variable exponents and a real positive parameter. Our objective is to establish the existence of at least one solution to this problem


Introduction and main results
In this paper, our focus is on a class of critical anisotropic Schrödinger-Kirchhofftype equations with variable growth conditions.This type of nonlinear partial differential equation describes the behavior of waves in an anisotropic (direction-dependent) system by combining the Schrödinger equation [22], which describes the quantum mechanical behavior of particles, and the Kirchhoff equation [18], which describes the behavior of waves in a medium.This equation includes variable exponent terms, which allow for a more flexible and accurate description of the wave behavior in the anisotropic system.This is especially important in systems that have complex and dynamic properties, such as materials with varying levels of anisotropy or systems that are subject to changes in temperature, pressure, or other external factors.And also the critical nonlinearities in these equations can have important implications for the behavior of the wave, such as the formation of singularities, the collapse of the wave, and the generation of shock waves.These behaviors are important for understanding the behavior of waves in complex media and for predicting how these media will interact with light, for more details, we refer the reader to [1,20,21,23].More precisely, we show the existence of nontrivial solutions for the following class of equations, where Ω ⊂ R N (N ≥ 2) is a bounded domain with a Lipschitz boundary ∂Ω, w ∈ L ∞ (Ω) satisfies w 0 := ess inf x∈Ω w(x) > 0, and λ is a positive parameter.− → p : Ω → R N is a vector function defined as − → p (x) = (p 1 (x), . . ., p N (x)) , with each component , for all x∈Ω.
The operator is referred to as the − → p (x)-Laplacian operator, which is a natural extension of the Laplacian operator when all p i (x) = 2.The functions M : R + → R + and f : Ω × R → R are continuous and satisfy the following conditions: (H M1 ) There exists M 0 > 0 such that M (s) ≥ M 0 for all s ≥ 0; (H M2 ) There exists γ ∈ ( (H f1 ) f (x, s) = o(|s| p + M −1 ) as s → 0, uniformly for x ∈ Ω; (H f2 ) There exist a positive continuous function ℓ(x) (H f3 ) There exists α ∈ (p + M /γ, p * − m ) such that 0 < αF (x, s) ≤ sf (x, s) for all x ∈ Ω and s ̸ = 0, where F (x, s) = s 0 f (x, t)dt and γ is given by (H M2 ) below.
Our approach to tackling problem (1.1), inspired by the ideas in [3], is primarily variational in nature, and we employ minimax critical point theorems as our primary tool.The main challenge we face arises from the absence of compactness in the embedding W Consequently, it becomes unfeasible to directly verify the Palais-Smale condition for the associated energy functional.To address this hurdle, we turn to the new version of the Lions concentration-compactness principle [19], specifically designed for anisotropic variable exponent Sobolev spaces.This adaptation was introduced by Chems Eddine et al. in [9], and it plays a crucial role in addressing this challenge effectively.

Proof of the main result
We define the energy functional associated with problem (1.1) as (Ω) represents the anisotropic Sobolev space, and its norm is given by ∥u∥ Proposition 1 [see [9]].The embedding W The following Poincaré-type inequality holds: where C is a positive constant independent of u ∈ W 1, − → p (x) 0 (Ω) (see [12, Theorem 2.6]).
Through standard calculus, it can be observed that E λ is a function in (Ω), R) and its Fréchet derivative is expressed as follows: (Ω).Therefore, the weak solutions of (1.1) coincide with the critical points of E λ .Consequently, our focus is on establishing the existence of these critical points.
To apply variational methods, we present certain results related to the Palais-Smale compactness condition.It's important to note that a sequence Notations: Strong convergence is denoted by →, while weak convergence is denoted by ⇀.Constants are represented by C, C i , and C ′ i , which can vary from one line to another and depend on specific conditions.The symbol δ xj represents the Dirac mass at x j .For any ρ > 0 and x ∈ Ω, B(x, ρ) denotes the ball with radius ρ centered at x.
In the following, we prove that the functional E λ exhibits the mountain pass geometry.This assertion is established in the forthcoming lemmas.
Proof.First, from assumptions (H f1 ) and (H f2 ), for any ε > 0, there exists a positive constant C(ε) such that the following inequality holds for almost every x ∈ Ω and all s ∈ R: Next, by using [13, Theorem 1.3] and Jensen's inequality on the convex function q(t) = t pm,M for pm,M > 1, we obtain that Math.Model.Anal., 29(2):254-267, 2024.
Clearly ϕ * ∈ Γ , then, by using assumption (H f3 ), we have Since the function M is continuous and δ 0 = 0, we get So, by relation (2.6), it follows that lim n→∞ c λn = 0.Moreover, by virtue of assumption (H f3 ), we can deduce that {c λ } λ forms a monotone sequence.Consequently, we can establish that lim λ→∞ c λ = 0. ⊓ ⊔ Let S * denote the optimal positive constant of the Sobolev embedding W , which can be expressed as Proof.[Proof of Theorem 1] From Lemmas 1, 2 and 3, we can establish the existence of a sequence Moreover, by the assumption (H f3 ) and for sufficiently large n, it follows from the assumptions (H M1 ) and (H M2 ) that On the other hand, for each n, let us denote by B n1 and B n2 the indices sets Then, we have Hence, by using Jensen's inequality (2.3) (applied to the convex function h : , for n large enough we have Since α > p + M /γ, {u n } is bounded.Therefore, up to a subsequence, we may assume that µ i (weak*-sense of measures), where µ and ν are nonnegative bounded measures on Ω.Then, according to the new version of Lions's concentration-compactness principle for anisotropic variable exponents [9], there exists an index set J which is at most countable, such that with δ xj is the Dirac measure mass at x j ∈ Ω.
For j ∈ J and ε > 0, let ψ j,ε (x) = ψ( First, we will show that (2.10) By applying the Hölder inequality and considering the boundedness of (Ω), we obtain Therefore, by Lebesgue's Dominated Convergence Theorem, we get Moreover, by Hölder inequality .
Note that with S N is the surface area of an N -dimensional unit sphere.We have with C is a positive constant that doesn't depend on ε.Therefore, , ∥|u| pi(x) ∥ . But, From this, it follows that lim ε→0 lim sup n→∞ Ω u n |∂ xi u n | pi(x)−1 ∂x i ψ j,ε dx = 0 for all i ∈ {1, 2, . . ., N }.
(2.11) Since {u n } n is bounded and the function M is continuous, we can choose a subsequence, and there exists t 0 ≥ 0 such that as n → ∞.Then, by (2.11), we obtain that (2.10) is proved.

A special case
Now, we consider a special case of the problem given by Equation (1.1).The problem is described as follows: