Existence and Uniqueness of Weak Solutions to the Two-Dimensional Stationary Navier–Stokes Exterior Problem

This paper is concerned with the stationary Navier–Stokes equation in two-dimensional exterior domains with external forces and inhomogeneous boundary conditions, and shows the existence of weak solutions. This solution enjoys a new energy inequality, provided the total flux is bounded by an absolute constant. It is also shown that, under the symmetry condition, the weak solutions tend to 0 at infinity. This paper also provides two criteria for the uniqueness of weak solutions under the assumption on the existence of one small solution which vanishes at infinity. In these criteria the aforementioned energy inequality plays a crucial role.


Introduction
Let Ω be an exterior domain in the plane R 2 with C 2+γ -boundary Γ with some γ ∈ (0, 1). We are concerned with the following stationary Navier-Stokes equation in Ω: where the vector-valued unknown function w(x), the scalar-valued unknown function π(x) and the vectorvalued given function f (x) stand for the velocity, the pressure and the external force respectively. As is known in Russo and Simader [30], the solution does not satisfy the boundary condition w(x) → 0 as |x| → ∞ (1.4) in general. Throughout this paper we assume that the external force f (x) is given by the formula with a 2 × 2 matrix F jk (x) 2 j,k=1 ∈ L 2 (Ω) 4 .
Ω is invariant under the mapping π 1 : (x 1 , x 2 ) → (−x 1 , x 2 ) ( R 2 I ) and where w ∞ = (0, c) with some c = 0. This result is improved by Russo [27]. Then Galdi and Simader [12] considered the problem for external force f (x) with little regularity, and Galdi and Sohr [13], Sazonov [31] and Russo [26] obtained precise asymptotic behavior of w(x) and π(x). (For more complete references, see Galdi [11].) For the case w ∞ = 0 with external force, Russo [25] obtained the existence of weak solutions under the assumption that |α| is sufficiently small. Here the total flux α is defined by α = Γ a(x) · n(x) ds(x), where n(x) denotes the unit normal vector of Γ at x outward of Ω. Furthermore, Galdi [10, Section 3] and Pileckas and Russo [24], as well as [25], posed the condition The set Ω is invariant under the mappings with a specific coordinate variables (x 1 , x 2 ) on Ω, and the condition on the external force f (x) = f 1 (x), f 2 (x) , and (D4E) on the boundary value a(x), and showed the existence of a weak solution w(x) = w 1 (x), w 2 (x) satisfying (D4E), which is exactly the same as [10, (3.18)]. Furthermore, [25] showed that w(x) tends to 0 in the average, [Precise definition is given in (2.9).] and that w(x) tends to 0 pointwise if the external force has compact support. We here regard this condition above from the viewpoint of transformation groups, as is in Brandolese [6] which obtained sharp decay order of solutions to the nonstationary Navier-Stokes equations in R 2 and R 3 . The condition (D4I) can be described as π j (Ω) = Ω for j = 1, 2, and the condition (D4E) can be described as f π j (x) = π j f (x) . In other words, the domain Ω is invariant, and the vector field f (x) is equivariant, with respect to the action of the group D 4 = {id, π 1 , π 2 , π 1 π 2 }. Later, Russo [28] relaxed the assumptions (D4I) and (D4E) to The set Ω is invariant under the mapping and f j (−x) = −f j (x) for j = 1, 2 and every x ∈ Ω, and showed the same property satisfying w(x) satisfying (C2E). Russo [29] studied the decay order of the solutions at infinity in detail. In other words, Ω is invariant, and f (x) is equivariant, under the group C 2 generated by x → −x. Most of these works treated the solution w(x), π(x) satisfying ∇w ∈ L 2 (Ω) 4 .
For these problems, the author [33] showed the existence, together with the uniqueness in the small, of the solution of the stationary Navier-Stokes equation on the whole plane under the assumption that the small external force f (x) = ∇ · F (x) decays like |x| −2 as |x| → ∞ and satisfies the condition where x ⊥ = (−x 2 , x 1 ), as well as (D4E). The solutions decays like |x| −1 as x → ∞; in other words, decays like the derivatives of the fundamental solution of the Laplacian. This class is invariant under the scaling w ρ (x) = ρw(ρx) keeping (1.1) invariant with suitable scaling for p(x) and f (x). We call these solutions critically decaying. (Its definition in somewhat generalized form, which does not imply pointwise estimate, is given in Remark 2.13.) In the terminology of function spaces, critically decaying solutions belong to the weak-L 2 space. It is also shown that, if F (x) decays more rapidly, then w(x) decays more rapidly (up to |x| −2 ). This result is also obtained by Guillod [17,Section 3] without the assumption that f (x) is of the form f (x) = ∇ · F (x). Very recently, Decaster and Iftimie [8] obtained sharp decay and asymptotic profiles of the solutions. Then [34] showed that the same result as [33] holds for the exterior problem provided the domain Ω satisfies the symmetry condition The set Ω is invariant under the mapping σ : as well as (D4I), and the external force f (x) and the boundary value a(x) satisfies (D4E) and (C4E) with α = 0. Recently, Guillod [17] obtained sharp asymptotic behavior at infinity of the solution above. Notice that (D4I) and (C4I) imply the invariance of Ω for the square dihedral group D 4 , and (D4E) and (C4E) imply the equivariance for D 4 . Then Nakatsuka [22] proved the weak-strong uniqueness; namely, he showed that, if the exterior domain satisfies (D4I) and if there exists a sufficiently small critically decaying solution of (1.1)-(1.4) with a(x) ≡ 0 satisfying the condition (D4E), then every weak solution of (1.1)-(1.4) satisfying the energy inequality and the same symmetry property coincides with the critically decaying solution. He also showed that, if there exists a sufficiently small supercritically decaying solution of (1.1)-(1.4) with a(x) ≡ 0, every weak solution satisfying the energy inequality coincides with the supercritically decaying solution. For the proof he first showed that the critically decaying solutions satisfy the energy identity.
Further, Galdi and Yamazaki [14] showed that the solutions above are stable under initial L 2 -perturbation with the symmetry property (D4E) with no restriction on the size, and the author [35] gave convergence rate in various function spaces. The author [34] also showed that, if f (x) decays more rapidly, then the solution decays faster than the derivatives of the fundamental solution. We call these solutions supercritically decaying. In this case the stability above holds true for initial L 2 -perturbation without symmetry or size restriction. Very recently, Guillod [18] showed that the critically decaying solution is stable under general initial L 2 -perturbation. (The precise definition of supercritically decaying solutions, which does not imply pointwise estimate, is given in Remark 2.10 in somewhat generalized form.) Note that the divergence theorem implies that the outflow condition α = 0 is necessary for the existence of supercritically decaying solutions.
On the other handy, the author [36] proved, assuming only (C4I) on the domain Ω, and assuming only (C4E) on a(x) and f (x) = ∇F (x), where a(x) and F (x) are small in appropriate function spaces, the unique existence of small critically decreasing solutions satisfying the symmetry condition (C4E). In this work we imposed the invariance and equivariance under the cyclic group C 4 .
For the existence of critically decreasing solutions, the condition α = 0 is not necessary. The author [36] also showed that the solutions above become supercritically decreasing if F (x) satisfies a sharper decay condition and if, as well as the outflow condition α = 0, the data a(x) and F (x) are sufficiently small. We observe that the conditions (C4I) and (C4E) are independent of the choice of the coordinate axes, and includes rotationally symmetric functions naturally. However, the results above implies only the uniqueness of small solutions. The main difficulty in the two-dimensional case is that the function with finite Dirichlet integral is not bounded in general, and we must modify Hardy's inequality (see Propositions 3.5 and 7.1) in general. (see Adimurthi et al. [1].) The purpose of this paper is twofold. The first one is a slight generalization of constructing weak solutions with external force f (x) = ∇ · F (x) with neither any sort of symmetry conditions nor the smallness conditions. In order to treat large a(x), we construct a corrector potential G(x) so that there exists a solution v(x) of the Stokes equation with given boundary condition and the external force ∇G(x) is critically decreasing, and consider the equation for u(x) = w(x) − v(x) with homogeneous boundary condition. The correction potential G(x) is a compactly supported function which cancels the angular momentum of the boundary value, and the function v(x) is critically decreasing function whose profile is determined by α, However, these functions do not cancel the momentum, and hence v(x) does not describe the asymptotic profile of w(x). Indeed, for the steady flow w(x) ≡ w 0 = 0 outside a disk with F (x) ≡ 0 and w(x) = w 0 on the boundary, we construct G(x) so that v(x) is supercritically decreasing, and the asymptotic profile w(x) = w 0 at infinity reappears as the homogeneous boundary value problem with external force −∇G(x). This fact also implies that, even if the boundary value vanishes and the external force is compactly supported, the asymptotic behavior of the solution may not be trivial. Hence our existence theorem covers the case where w(x) is not decreasing as |x| → ∞. We then apply the fixed point theorem on disks, whose centers converge to the image of v(x) by the orthogonal projection to H 1 0,σ (Ω), which is denoted by v. In this case the solution does not necessarily satisfy (1.4).
If the domain Ω satisfies the symmetry condition (C2I) and the external force and the boundary conditions satisfies (C2E), then we can construct a solution w(x) satisfying (C2E). Since C 2 is a subgroup of C 4 , these conditions are also independent of the choice of coordinate axes. In this case the solution satisfies (1.4) in the sense that w(x) tends to 0 in the average. Moreover, the solution satisfies an energy inequality provided |α| is bounded by an absolute constant. This inequality is a generalization of the one employed in [22], where the case a(x) = 0. This inequality in this paper seems to be new since it is applicable to solutions which fail to satisfy (1.4), and plays a crucial role in the study of weak-strong uniqueness. In order to treat the case where the angular momentum of the solution is not zero, we use the result of Coifman et al. [7].
The second purpose is to give a condition for the weak-strong uniqueness under non-homogeneous boundary conditions and with external force under the assumption that a(x) is small and that w(x) decays sufficiently fast. Namely, suppose that w(x) is a weak solution such that u(x) = w(x) − v(x) is small and supercritically decaying. (This implies the smallness of a(x) and f (x).) Then every weak solution w (x) such that u (x) = w (x) − v(x) satisfies the energy inequality but not necessarily satisfies (1.4), however large it may be, coincides with w(x). In particular, the weak solution constructed in the previous results coincides with w(x). Under the symmetry condition the assumption on decay property can be relaxed. Namely, suppose that Ω satisfies (C2I), and that ∇F and a(x) satisfies (C2E). Moreover, suppose that w(x) is a weak solution such that u(x) = w(x) − v(x) is a small critically decaying function satisfying (C2E). Then every weak solution w (x) satisfying (C2E) such that u (x) = w (x) − v(x) satisfies (C2E) and the energy inequality, must coincide with w(x). This result implies the uniqueness of the solutions obtained in [36] as well as those in [33,34]. In particular, if w (x) is the weak solution constructed in the previous result such that u (x) = w (x) − v(x) satisfies (C2E), then w (x) coincides with w(x). With (C2I) and (C2E) we have Hardy's inequality. Notice that u(x) need not satisfy an assumption of pointwise estimate in either case.
In the case Ω satisfies (D4I), f (x) satisfies (D4E) and a(x) = 0, our assumption on the smallness of the weighted L p -norm is a slight generalization of the assumption on the smallness of the pointwise estimate in [22]. Moreover, we prove the energy identity under weaker assumption in which no pointwise estimate is necessary. To this end we prove a sharp version of Hardy's inequality.
In addition to the property above on the uniqueness, the results in [14,35] on the stability under initial L 2 -perturbation with no restriction on the size holds, and we can replace the symmetry condition (D4E) by (C2E) by applying the improved Hardy's inequality. In other words, solutions in [33,34,36] have similar property on uniqueness and stability as physically reasonable solutions in the three-dimensional setting. This paper is organized as follows. In Sect. 2 the notation is fixed and main results are stated. In Sect. 3 we list up some facts necessary in the proof. Proof of some lemmata are given in the Appendix. Then corrector potentials and weak solutions are constructed in Sects. 4 and 5 respectively. The uniqueness is proved in Sect. 6. Finally in Sect. 7 we state the improvement in symmetric cases.

Notations and Main Results
We first introduce some function spaces. For a domain U ⊂ R 2 , let C ∞ 0 (U ) denote the set of infinitely differentiable functions on U supported by a compact subset of U , and let C ∞ 0,σ (U ) denote the set of vector-valued functions ϕ( is denoted by · q . For p ∈ (1, ∞) and r ∈ [1, ∞], let L q,r (U ) denote the set of Lorentz space. The norm of L q,r (U ) m for m ∈ N is denoted by · q,r .
Then we have the following properties. (see Bergh and Löfström [4] or Triebel [32] for example.) First, there exist a inclusion relation L q,r (U ) ⊂ L q,s (U ) provided r < s. Second, the space L q,q (U ) coincides with the Lebesgue space L q (U ), and the space L q,∞ (U ) coincides with the weak-L q space on U . Third, for 1 ≤ r < ∞, the space Suppose that U is either a whole plane R 2 , a bounded domain or an exterior domain with C 2+γ boundary, and let Γ denote the boundary of U . Then, for every q ∈ (1, ∞), there exists a direct sum Let P q denote the projection on L q (U ) 2 onto L q σ (U ) associated with the decomposition above. Then we have P q0 = P q1 on L qo (U ) ∩ L q1 (U ) 2 . Hence we can define the projection P q,r on L q,r (Ω) 2 for every q ∈ (1, ∞) and r ∈ [1, ∞] by real interpolation. Let L q,r σ (U ) denote the range of P q,r . For s ∈ R, let H s (R 2 ) andḢ s (R 2 ) denote the Sobolev space and the homogeneous Sobolev space, equipped with the norms , where s 0 = s 1 , 0 < θ < 1 and s = (1 − θ)s 0 + θs 1 . In general homogeneous spaces are defined only modulo polynomials, but if either s < n/p, or s = n/p and q = 1 for the Besov space, the modulo classes inḢ s p andḂ s p,q have a canonical representative which decays as |x| → ∞, and hence these spaces can be considered as function spaces on R n . For a domain U 2024 M. Yamazaki JMFM in R 2 and k ∈ N, let H k (U ) andḢ k (U ) denote the set of the restrictions of the elements of H k (R 2 ) anḋ H k (R 2 ) on U equipped with the norms respectively, and letḢ k 0 (U ) denote the closure of C ∞ 0 (U ) inḢ k (U ). In particular, if U is bounded, the spaceḢ k 0 (U ) is defined as a set of functions even if k ≥ 1. Let H k 0 (U ) denote this space. Furthermore, leṫ satisfying ∇ · u ≡ 0 on Ω belongs toḢ 1 0,σ (Ω), as is shown in Heywood [19,Section 2] for example.
For a domain U and s > 0 such that s / ∈ N, Let C s (U ) denote the Hölder space on U . For a closed curve Γ of C 2+γ class and s > 0, let H s (Γ) denote the Sobolev space on Γ.
Finally, for a scalar-valued function f (x) we write and a vector-valued function u( In the sequel let Ω be a fixed exterior domain with C 2+γ -boundary Γ. We introduce the notion of weak solutions. We construct weak solutions of (1. 4 with a bounded support of satisfying the estimate G 2 ≤ C a H 1/2 (Γ) with some positive constant C. This function is the corrector potential. Fix a positive integer J such that and g(r) = log r for r ≥ 2 J . Furthermore, for q such that 2 < q ≤ ∞, we define a function λ q (t) on [0, ∞) We now state the existence the corrector functions satisfying (2.2)-(2.4) with suitable G(x).

Proposition 2.3. Suppose that the exterior domain Ω satisfies (2.5). Then there exist a positive constant
C and a bounded linear mapping , where c ∈ R 2 \Ω and there exists a positive number Remark 2.4. Note that, as is stated in [30], the solution of the equation (2.2) does not decay as |x| → ∞, but here we construct G(x) so that there exists a solution v(x) which enjoys (2.3)-(2.4) as well. This behavior is independent of the asymptotic profile of w(x) in general.
Then our result concerning the existence and the energy inequality is the following theorem.
Note that α = 0 is necessary in general, at least formally, so that the term (u ⊗ v, ∇u) to be welldefined. Thanks to the estimate (5.11) the term Φ(u, u) is well defined for u ∈ H 1 0,σ (Ω). Although v(x) is critically decreasing, the solution u(x) does not decay as |x| → ∞ in general. Hence v(x) does not describe the asymptotic behavior of w(x) at infinity, but it plays a crucial role in the proof of the existence.
Although the choice of the pair G(x), v(x) is not unique, the validity of the energy inequality (2.7) is independent of the choice of the pair. In fact, we have the following proposition. Proposition 2.6. Suppose that Ω satisfies (2.5) and that w(x) is a weak solution of (1. We next give a condition sufficient for the equality in (2.7).

Theorem 2.7. Suppose that w(x) is a weak solution of the system
with equality, and in this case the left-hand side of (2.7) can be written as Next we state our results on the uniqueness of the solutions for domains which do not necessarily satisfy (C2I) or (C2E).

Theorem 2.9.
Let Ω be an exterior domain satisfying (2.5). Then there exists a positive constant C Ω such that, for q such that 2 < q ≤ ∞, there exists a positive constant δ q,Ω such that the following assertion holds. Suppose that a( Here the smallness of a implies the smallness of |α|. Observe that there exists no restriction on the size of w (x).
We next consider the results in the class of functions satisfying (C2E) under the assumption that Ω satisfies (C2I). In this case we modify the choice of I 1 as follows: If 0 ∈ R 2 \Ω, then we take c = 0.
Otherwise, if c ∈ R 2 \Ω. In this case we take Then v (j) (x) satisfies (C2E). In this case we have the following existence theorem, which states that our solution decays in the average as |x| → ∞. Theorem 2.11. In addition to the assumption of Theorem 2.5, we assume that Ω satisfies Note that the expression (2.8) makes sense in this case. For the uniqueness of solutions satisfying (C2E), we have the following theorem. Note that, as in Theorem 2.9, there is no restriction on the size of w (x).
Vol. 20 (2018) 2D Navier-Stokes Exterior Problem 2027 Theorem 2.12. There exists a positive constant C Ω such that, for every 2 < q ≤ ∞, there exists a positive constant δ q such that the following assertion holds. Suppose that a H 1/2 (Ω) < C Ω and that Ω satisfies (C2I) and (2.5), and that a( Remark 2.13. The condition with some q ∈ (2, ∞] is the precise definition of critically decaying functions in the symmetric setting. If (C q ) holds with some q 0 ∈ (2, ∞], then (C q1 ) holds for every q 1 ∈ (2, q), as we see from the estimate Remark 2.14. We cannot take q = 2 in Theorems 2.9 and 2.12, since the estimate (6.3) fails in this case.

Preparatory Lemmata
We first give the sharp Hölder estimate, which is stated by O'Neil [23] without rigorous proof. We give a simple proof in Appendix A.
We also have the generalized Gagliardo-Nirenberg inequality.

Lemma 3.2.
Let U be a domain in R 2 with C 2 boundary, and suppose that p, q ∈ (1, ∞) and that max{p, q} < r ≤ ∞. We also assume that 2/q − 1 < 2/r. Then there exists a positive constant C such that, for every u ∈ L p,∞ (U ) satisfying ∇u ∈ L q,∞ (U ) 2 , we have u ∈ L r,1 (U ) in the case r < ∞, and Proof. First, choose s such that max{p, q} < s < r. There exists a functionũ on R 2 such thatũ| U = u and that the inequalities ũ p,∞ ≤ C u p,∞ and ∇ũ q,∞ ≤ C ∇u q,∞ hold with a positive constant C independent of u. Hence, by the Sobolev embedding theorem, we haveũ ∈Ḃ In view of the equality we can apply real interpolation to obtain from (3.1) and (3.2). If r = ∞, we obtain the conclusion from the estimate above and the inclusion relationḂ From the choice of δ, we have 2/s − 2/r − δ > 0 and 2/s − 2/r + δ < 2/s. Hence, putting r 0 = 2r/(2 + δr) and r 1 = 2r/(2 − δr), we have and the Sobolev embedding theorem implieṡ It follows from these facts and (3.4) thaṫ

The conclusion follows from this inclusion relation and (3.3).
We next recall the Bogovskii lemma, which is shown by Bogovskii [5]. From this lemma we have the following decomposition. Put U = {x ∈ R 2 | |x| > 2 J+2 /3}. Then we have the following lemma. Lemma 3.4. There exists a positive constant C such that the following assertion holds.
We also have From these estimates we have Since u| Γ = 0, we can apply the Poincaré inequality to u(x) onΩ to conclude that there exists a positive integer C such that the estimate u L 2 (Ω) ≤ C ∇u 2 holds. Substituting this estimate into (3.5) we obtain ∇u 1 2 ≤ C ∇u 2 . This inequality yields ∇u 2 2 ≤ C ∇u 2 since u 2 (x) = u(x) − u 1 (x). We turn to the proof of Assertion (ii). Applying the Poincaré inequality and the Sobolev embedding theorem to u 1 (x), we obtain u 1 q ≤ C ∇u 2 with a positive constant C depending on q. If u(x) ∈ L q σ (Ω) as well, it follows that u 2 q ≤ u q + u 1 q ≤ u q + C ∇u 2 .
Assertion (iii) follows from Assertion (ii) and real interpolation.
At the end of this section we give a refined version of Hardy's inequality, whose proof is given in Appendix B.

Further, we have the expression
It follows that there exists a positive constant C such that |I 3 [a]| ≤ C |x − c| 2 on Ω, which implies Assertion (iii).
We now prove the above proposition. Since u ∈ H 1 0,σ (Ω) with compact support, we obtain by integrating by parts. Hence It follows that We next observe that, for every j ≥ J + 1 and every G ∈ L 2 (Ω j ) 4 , the functional onḢ 1 σ (Ω j ) defined by ϕ → (G, ∇ϕ) is bounded. Hence we can define a bounded linear operator S j from L 2 (Ω j ) 4 to H 1 0,σ (Ω j ) defined by the equality (∇S j [G], ∇ϕ) = (G, ∇ϕ) for every ϕ ∈ H 1 0,σ (Ω j ). We now introduce the mapping T j from the space Then T j is a continuous mapping into the space Y j = X j ∩ H 1 0,σ (Ω j ). Since the inclusion Y j → X j is a compact operator, the operator T j restricted on Y j is a compact mapping into itself.
Then we have the following lemma.

M. Yamazaki JMFM
Proof of Theorem 2.5. We first observe that v j converges strongly to v inḢ 1 0,σ (Ω We also see that the with a constant C independent of k. Substituting this estimate into (5.8), we see that converges weakly in H 1 0 (Ω j+1 ) 2 , and hence strongly in L 4 (Ω j+1 ) 2 , to χ 2 −j |x| u(x). It follows that u j(k) ∞ k=1 converges to u strongly We now show that this u(x) is a weak solution. Suppose that ϕ(x) ∈ C ∞ 0,σ (Ω). Then there exists an converges strongly in , and hence lim k→∞ ∇u j(k) , ∇ϕ = (∇u, ∇ϕ) and From these facts and the equality for every k such that j(k) ≥ j, we conclude (2.6). Since ϕ ∈ C ∞ 0,σ (Ω) is arbitrary, the function u(x) is a weak solution. It remains only to show (2.7). Integrating by parts and substituting ϕ = u j(k) in (5.10), we obtain We easily see that I We next verify that u j(k) ⊗ṽ converges strongly to u ⊗ṽ in L 2 (Ω) 4 . We first observe that Proposition 3.5 implies that the set is bounded in L 4 (Ω) 2 , and the estimate This implies that, for every ε > 0, we can take R > 0 so large that Since ε > 0 is arbitrary, it follows that u j(k) ⊗ṽ converges strongly to u ⊗ṽ in L 2 (Ω) 4 . This implies that we see that I We finally consider −αΦ u j(k) , u j(k) . Applying Fefferman-Stein duality and the result of [7], we obtain the estimate with absolute constants C 1 and C 2 , where ϕ and ψ are identified with their zero extensions to R 2 . In the sequel we assume that |α| < 1/C 2 .
If lim k→∞ ∇u j(k) converges to −Φ(u, u). Hence > ∇u 2 2 + ε. We next have Since the mapping ϕ → Φ(ϕ, u) is a bounded linear functional on H 0, In the same way we have ∇u j(k) , ∇u → ∇u 2 2 as k → ∞. By taking K larger if necessary,, we may assume that k ≥ K implies By taking K larger if necessary, we may assume that k ≥ K implies It follows that This implies (5.12).

Uniqueness of Weak Solutions
In this section we prove Theorems 2.7 and 2.9. We start with the following proposition, which provides a series of test functions approximating weak solutions. This proposition corresponds to the result in [22,Proposition 3.1], in which the existence of a sequence converging with respect to the weak- * topology of L ∞ is proved. Proposition 6.1. Suppose that 2 < q ≤ ∞ and that u ∈ H 1 0,σ (Ω). Then we have the following assertions. (i) Suppose in addition that q = ∞ and that u(x) ∈ L q σ (Ω). Then there exists a sequence Proof. Assertion (i) is proved in Kozono and Sohr [21, Theorem 2]. We next prove Assertion (ii). We first decompose u(x) = u 1 (x) + u 2 (x) as in Lemma 3.4. Then there exists a sequence {ϕ (1) Then the Poincaré inequality yields ϕ 2 as j → ∞ as well.
Then we have Hence we have Hence, for every ε > 0, we can choose L so large that ≥ L implies For every ≤ L , we substitute the estimates (6.8) and (6.9) into (6.7) to obtain g(x), λ q (|x|)u 2 (x) − λ q (|x|)ϕ k ,δ (x) < ε. This implies that the sequence {λ q (|x|)φ (x)} ∞ =1 converges to λ q (|x|)u 2 (x) weakly ( * -weakly if q = ∞) in L q (Ω) 2 as → ∞. This completes the proof of Assertion (ii). We finally prove Assertion (iii). By the symmetry we can easily see that the sequence satisfies all the requirements for −u(−x). Then, for every j ∈ N, 2 We can do this in the same way as Assertion (ii), by replacing λ q (|x|) by |x| 1−2/q and This completes proof of Assertion (iii).
Hence we see that Since Substituting this formula and (6.11) into (6.10), we obtain (2.8). Integration by parts yields − u, u), from which we obtain the conclusion.
We next verify Proposition 2.6. To this end we assume that the pairs v(x), G(x) and v (x), G (x) satisfy the conclusion of Proposition 4.1, and that u( and ∇ϕ j → ∇t in L 2 (Ω) 4 as j → ∞. Then, for every j, we put where Since u + v is a weak solution, the equality (2.6) implies I j,1 = 0. Next we obtain I j,3 = 0 by integrating in parts. We finally consider I j,2 . From the equality Since ϕ j → t in H 1 0,σ (Ω) as j → ∞, we conclude that lim j→∞ I j,2 = lim j→∞ ∇(−t + ϕ j ), ∇(u + ϕ j ) = 0.

Results on Symmetric Solutions
In this section we prove the results on the situation when Ω satisfies (C2I) and a(x) satisfies (C2E). Since the argument is almost the same, we shall give main differences.
We first introduce Hardy's inequality with symmetry, whose proof is given in Appendix C.
Proof. For each , either U is different from −U or U = −U . Moreover, if U = −U , then 0 ∈ U . Indeed, if C is a curve contained in U which connects P ∈ U to −P ∈ −U = U , then C ∪ −C is a closed curve in U surrounding 0. Since U is simply connected, it follows that 0 ∈ U . In this case choose c as follows: If U m = −U holds for some and m such that = m, choose c m = −c , and if 0 ∈ U , choose c = 0. Moreover, if a(x) satisfies (C2E) as well, then the equality α = α m holds for , m such that U m = −U . In this case we have α = α m Hence v (1) (x) satisfies (C2E) in this case. Furthermore, in this case b(x) satisfies (C2E), and so does w(x). It follows that g(x) satisfies (C2AE). Hence we can assume that ϕ(x) satisfies (C2E), by replacing ϕ(x) by ϕ(x) − ϕ(−x) /2 if necessary. In this case ∇ × ϕ(x) and h(x) satisfy (C2AE). We thus conclude that G(x) satisfies (C2AE). The modification of I 3 is given in the introduction.
We can prove Theorem 2.11 in the same way as Theorem 2.5. However, in order to obtain a solution satisfying (C2E), we modify the proof. We replace the space X j by f ∈ L 4 σ (Ω j ) f satisfies (C2E) and making use of the fact that since u(x) satisfies (C2E) implies that so does U [u] in Lemma 5.2.