problem, Weighted Sobolev spaces, Discontinuous coefficients, Solvability and weak uniqueness weight function.

The qualitative behavior of solutions to linear and fully nonlinear second-order elliptic equations with discontinuous or singular coefficients has been a central topic in the modern theory of Partial Differential Equations (PDEs). A notable model in this class is the Gilbarg–Serrin equation, which features strongly singular coefficients at the origin. When the coefficient function µ(r) takes large values (i.e.,

lim µ(r) = +∞.

, the qualitative properties of the equation’s solutions deviate significantly from those of standard harmonic functions. In particular, uniqueness and coercivity fail in unweighted Sobolev spaces. To overcome the analytical challenges posed by these singular coefficients, the incorporation of appropriate weight functions becomes essential. Weighted Sobolev spaces provide a natural variational framework to capture the precise boundary behavior and energy estimates of weak solutions. While first boundary value problems (Dirichlet problems) for Gilbarg–Serrin type operators have been extensively studied, the second boundary value problem (Neumann problem) presents additional analytical complications due to the boundary flux conditions and spectral properties of the weighted bilinear form. The main objective of this paper is to establish the weak solvability, uniqueness, and energy estimates for the second boundary value problem associated with the fully nonlinear Gilbarg–Serrin equation in the weighted Sobolev space Wα 1,2(D). By establishing fundamental weighted integral inequalities and employing the Lax–Milgram theorem, we demonstrate that the problem admits a unique weak solution under suitable structural conditions on the coefficient function µ(r).

Let D ⊂ Rn (n ≥ 3) be a bounded domain with a smooth boundary ∂D, containing the origin 0 ∈ D. Consider the fully nonlinear Gilbarg–Serrin operator in D:

∑︂ n xixj ∂2u Lu ≡ ∆u + µ(r) 2 , (1) i,j=1 r ∂xi∂xj (︁∑︁n 2)︁1/2 where r = |x| = i=1 xi , and µ(r) is a continuous function for r > 0.

We consider the second boundary value problem for this operator in D:

⎧ ∑︁ ⎨Lu = g(x) + n i=1 ∂x ∂fi , x ∈ D, ∂u (2) ⎩ = 0, x ∈ ∂D,

where n denotes the outward unit normal vector to the boundary ∂D.

It is clear that under appropriate restrictions on µ(r), the operator L is elliptic in D \ {0}. However, as r → 0, if µ(r) → +∞, the qualitative characteristics of the solutions differ fundamentally from those of harmonic functions. In particular, the uniqueness of the solution to the second boundary value problem fails in standard unweighted Sobolev spaces W 1,2(D). Nevertheless, by introducing a weight function ρ(x) = rα, the existence, weak solvability, and uniqueness of the solution can be guaranteed in the weighted Sobolev space.

First, we introduce the key function spaces and definitions. Let L2 α(D) and Wα 1,2(D) denote the Banach spaces of functions defined on D with the finite norms:

(︃∫︂ )︃1/2 ∥u∥L2 α(D) = r |u(x)| dx , (3)

(︃∫︂ )︃1/2 α (︁ 2 2)︁ ∥u∥Wα 1,2(D) = r |u(x)| + |∇u(x)| dx , (4) (︂ )︂ where ∇u = ∂x ∂u 1 , . . . , ∂x ∂u n is the gradient vector of u(x). Let C0 ∞ ,α(D) be a dense subspace of Wα 1,2(D), consisting of infinitely differentiable functions with compact support in D. Throughout this paper, Ci denotes positive constants depending only on the indicated parameters.

Suppose that the function µ(r) satisfies the following structural conditions:

µ(r) ≥ 0, lim rµ′(r) = 0, (5)

α > 2 − n, lim µ(r) = +∞, r ∈ (0, R]. (6)

Lemma 1. If the coefficients of the operator L are defined in the bounded domain D ⊂ Rn and conditions (5)–(6) hold, then for any smooth function u(x) ∈ C0 ∞ ,α(D), the following inequality holds:

∫︂ ∑︂ n 2 ∫︂ α ⎝ xixj ∂ u ⎠ α 2 r ∆u + µ(r) 2 u dx ≥ C3 r |∇u| dx. (7) D i,j=1 r ∂xi∂xj D

Proof : *Proof.* It suffices to prove inequality (7) for smooth functions u ∈ C0 ∞ ,α(D). For simplicity, let ∑︁n xixj ∂2u Lu = ∆u + µ(r) i,j=1 r2 ∂xi∂xj .

∫︁ α Consider the variational bilinear form A = − D r uLu dx. Integrating by parts, we obtain:

∫︂ ∫︂ ∑︂ n α 2 α xixj ∂u ∂u A = r |∇u| dx + r µ(r) 2 dx + . . . (8) D D i,j=1 r ∂xi ∂xj

Calculating the individual integral terms and substituting them into (8), we get:

∫︂ ∫︂ ⃓ ⃓ ∫︂ α 2 α ⃓∂u⃓ 2 α ′ ∂u A = r |∇u| dx + r µ(r) ⃓ ⃓ dx − r (n + rµ (r)) u dx. (9) ⃓ ∂r ⃓ ∂r

D D D

Taking into account conditions (5) and (6) in (9), we arrive at:

∫︂ ∫︂ ⃓ ⃓ ∫︂ α 2 α ⃓∂u⃓ 2 α 2 A ≥ r |∇u| dx + r µ(r) ⃓ ⃓ dx − ε r |∇u| dx. (10) ⃓ ∂r ⃓

D D D

Therefore,

∫︂ ∫︂ ⃓ ⃓ α 2 α ⃓∂u⃓ 2 A ≥ (1 − ε) r |∇u| dx + r µ(r) ⃓ ⃓ dx. (11) ⃓ ∂r ⃓

By condition (5), as r → 0, there exists a sufficiently small δ > 0 such that 1 − ε ≥ C3 > 0. Thus, we obtain:

A ≥ C3 rα|∇u|2dx. (12)

This completes the proof of Lemma 1. □

Now, we define the bilinear form B(u, v) associated with problem (2) on Wα 1,2(D) × Wα 1,2(D).

Lemma 2. The bilinear form B(u, v) is bounded and coercive in Wα 1,2(D). That is, there exist positive constants C4 and C5 such that:*

|B(u, v)| ≤ C4∥u∥Wα 1,2(D)∥v∥Wα 1,2(D), (13)

B(u, u) ≥ C5∥u∥2 W 1,2(D). (14)

Proof : Applying Lemma 1 and Hölder’s inequality yields C5 = C3 > 0, establishing coercivity. □

Definition 1. A function u(x) ∈ Wα 1,2(D) is called a weak solution to the second boundary value problem (2) if for every test function v(x) ∈ Wα 1,2(D), the following integral identity holds:

∫︂ ∑︂ n ∫︂ α α ∂v B(u, v) = r g(x)v(x) dx − r fi(x) dx. (15) D i=1 D ∂xi

Theorem 1. Let g ∈ L2 α(D) and fi ∈ L2 α(D) (i = 1, . . . , n). Then the second boundary value problem (2) for the fully nonlinear Gilbarg–Serrin equation is weakly solvable in Wα 1,2(D), and there exists a weak solution u(x) ∈ Wα 1,2(D).

Proof : The proof follows directly from Lemma 1, Lemma 2, and the Lax–Milgram Theorem, by demonstrating that the right-hand side of (15) defines a continuous linear functional on Wα 1,2(D). □

Theorem 2. The unique weak solution u(x) ∈ Wα 1,2(D) of the second boundary value problem for the fully nonlinear Gilbarg–Serrin equation satisfies the energy estimate:

∥u∥Wα 1,2(D) ≤ C6 ∥g∥L2 α(D) + ∥fi∥L2 α(D) . (16)

Proof : Setting v = u in integral identity (15), we obtain: □

∫︂ ∑︂ n ∫︂ α α ∂u B(u, u) = r g(x)u(x) dx − r fi(x) dx. (17) D i=1 D ∂xi

By Lemma 2 and Young’s inequality with ε > 0:

1 ∑︂ ε C5∥u∥2 W 1,2(D) ≤ B(u, u) ≤ ∥g∥2 L2 α(D) + ∥fi∥2 L2 α(D) + ∥u∥2 W 1,2(D). (18) α 2ε i=1 2 α

Choosing ε = C5, we arrive at estimate (16) with C6 = C 15 . This completes the proof of Theorem 2.

In this paper, we investigated the second boundary value problem for the fully nonlinear Gilbarg–Serrin equation in weighted Sobolev spaces. Under the imposed assumptions on the coefficients of the operator L, we established a fundamental weighted inequality that provides a lower bound for the corresponding quadratic form in terms of the weighted L2-norm of the gradient.

Using this inequality, we proved the boundedness and coercivity of the associated bilinear form in Wα 1,2(D). Consequently, by applying the Lax–Milgram theorem, we established the existence and uniqueness of a weak solution to the second boundary value problem for data satisfying g, fi ∈ L2 α(D), i = 1, . . . , n.

Furthermore, we derived an a priori energy estimate of the form

∥u∥Wα 1,2(D) ≤ C6 ∥g∥L2 α(D) + ∥fi∥L2 α(D) ,

where C6 > 0 is independent of the solution u and depends on the parameters and constants appearing in the assumptions.

These results establish the weak solvability and stability of the second boundary value problem in the weighted Sobolev space Wα 1,2(D). They also provide a basis for further investigations of boundary value problems for elliptic equations with discontinuous coefficients in weighted function spaces. “‘

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

J. Guliyev Pour: methodology, software, and editing. S.Aliyev: conceptualization. J.Aliyev: writing and editing. All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.

180 Guliyev and Aliyev | K. Multidiscip. Int.Sci.J. 3(2) (2026)175-180