Positive Evanescent Solutions of Singular Elliptic Problems in Exterior Domains

We investigate the existence of positive solutions for the following class of nonlinear elliptic problems div(a(x)∇u(x)) + f (x, u(x)) − (u(x)) −α ∇u(x) β + g(x)x · ∇u(x) = 0, where x ∈ R n and x > R, with the condition lim x→∞ u(x) = 0. We present the approach based on the subsolution and supersolution method for bounded subdomains and a certain convergence procedure. Our results cover both sublinear and superlinear cases of f. The speed of decaying of solutions will be also characterized more precisely.


Introduction
We consider the solvability of the following singular elliptic equation div(a( x )∇u(x)) + f (x, u(x)) − (u(x)) −α ∇u(x) β + g( x )x · ∇u(x) = 0, (1.1) for x ∈ Ω R , in the case when we look for solutions satisfying the condition where n > 2, R > 1, 0 < 2α ≤ β ≤ 2, for all x, y ∈ R n , x := ∑ n i=1 x 2 i , and x · y := ∑ n i=1 x i y i , Ω R = {x ∈ R n , x > R}. Precisely, we ask about sufficient conditions which guarantee the existence of function u of C 2+α loc class, which satisfies (1.1) at each point x from a certain neighborhood of infinity and we require the solution vanishes when the Euclidian norm of arguments tends to infinity. Our next aim is to describe more precisely how quickly solutions decay.

A. Orpel
Similar problems without any singular part were widely discussed, among others in [4][5][6][10][11][12][13][14][15]. On the other hand, there are many papers devoted to the singular elliptic problems with Laplace operator, similar singularity at zero and subquadratic growth with respect to the gradient. Here we have to mention paper [8] due to D. P. Covei, who looks for positive solutions of C 2+α loc class for the following problem −∆u + c(x)u −1 ∇u 2 = a(x) for x ∈ R N , u > 0, which is a special case of (1.3) and of (1.1), where we consider the singularity (u(x)) −α ∇u(x) β , with 0 < 2α ≤ β ≤ 2. It appears that this assumption plays the special role. First of all, the inequality 2α ≤ β allows us to obtain the subsolution of our problem on a bounded domain with the help of an eigenfunction of a certain linear problem. On the other hand, the condition β ≤ 2 is necessary to apply the technical tools described in [9]. For the reader's convenience we describe the paper [9], where the existence and nonexistence results are discussed for PDE with singular nonlinearities on a bounded domain Ω ⊂ R N with sufficiently smooth boundary. The author applied his general results, among others, to the problem which comes from stochastic process theory and leads (for q = 1, a ≡ 2, b ≡ 1), via substitution u = 1/v, to the problem There are also many results concerning weak solutions. Here it is worth mentioning the paper [20] written by Wen-Shu Zhou who considers the existence and multiplicity of positive weak solutions for the following singular PDE where Ω ⊂ R N is bounded, N ≥ 2, m > 1 and λ = 0 and f is a nonnegative measurable function. The results are also based on the subsolution and supersolution method. We can meet such problems in fluid mechanics (see e.g. [17] and references therein). Further, D. Arcoya, S. Barile, P. J. Martínez-Aparicio (in [2]) investigate the problem of the form where Ω ⊂ R N is a bounded domain, with N ≥ 3, a ∈ L q , with q > N/2 and g is a Carathéodory function in Ω × (0, +∞) which can have a singularity at zero. The authors consider a sequence of approximated problems to (1.5) and show the existence of a sequence (w n ) of their solutions which tends to a positive solution of (1.5) in H 1 loc (Ω). In the end we recall the results presented by D. Arcoya et al. in [3], where we can find the more general problem with f being strictly positive on every compact subset of Ω and a Carathéodory function g : Ω × (0, +∞) → R, which can be singular at 0. The authors also prove that for the following special case of (1.6) with γ > 0, the condition γ < 2 is necessary and sufficient for the existence of distributional solution of (1.7).
We also want to join in this discussion and deal with positive solutions for (1.1)-(1.2) and their asymptotic behavior. We start with the definitions of solution of our problem. We have to emphasize that we use standard definitions based on the ideas from the seventies and the eighties (described e.g. by Amann or Noussair and Swanson in [1] and [18]) which are met also in papers mentioned above. Definition 1.1. As a solution of our problem we understand a function u ∈ C 2+α loc (Ω R ) which satisfies (1.1) at every point x ∈ Ω R and condition (1.2).
Our results are based on the following assumptions (A_g) g : [1, +∞) → R is continuously differentiable and there exists r 0 ≥ 1 such that g(r) ≥ 0 for all r ≥ r 0 .

Supersolution on exterior domain
Our task is now to obtain the existence of function v of the class for x ∈ Ω R , and lim x →∞ v(x) = 0. In the sequel we call such function v a supersolution of (1.1)- (1.2). To this effect we use the ideas presented in the paper [19] and consider the auxiliary linear elliptic problem for function M given in (A_f). We show that there exists a radial positive solution of (2.1) which is a supersolution of (1.1)-(1.2) in a certain neighborhood Ω R of infinity. To prove its existence we employ the standard reasoning applying suitable transformation. Then the problem of the existence of radial solutions for (2.1) leads to the existence of positive solutions of the following singular Dirichlet problem where Taking into account the properties of functions M and a, one can infer that h and a satisfy conditions: (A_h) h : (0, 1) → (0, +∞) is continuous and for all t ∈ (0, 1) and where a min := inf t∈(0,1) a(t).
Applying the approach described in [12] and [19] we prove existence of a positive radial solution v of (2.1) having the special properties, which allow us to show that v is a supersolution of our problem on each bounded domain Ω ⊂ Ω R . We start with the singular ODE. Lemma 2.1. If conditions (A_h) and (A_ a) are satisfied then we state the existence of at least one positive classical solution z of (2.2) such that where φ is any function φ ∈ C 1 (0, 1) such that lim t→1 − φ(t) = 0 and lim t→1 − φ (t) = +∞.
Proof. Firstly, we note that the function z given by the formula is a solution of (2.2), when G is the Green's function Our task is now to show the existence of t 0 such that z (t) ≤ 0 for all t ∈ (t 0 , 1) and the positivity of z in (0, 1). To show the first assertion we state the existence of t 0 ∈ (0, 1) such that Now we prove that z > 0 in (0, 1). By (2.7), we know that z is nonnegative. Suppose that there exists at least one argument t ∈ (0, 1) at which z( t) = 0. Here we can use Rolle's theorem again which leads to the existence of numbers t 1 ∈ (0, t) and t 1 ∈ ( t, 1) such that . Now the iteration process gives us two sequences: (t m ) m∈N ⊂ (0, 1), which is decreasing, and t m m∈N ⊂ (0, 1), which is increasing, and such that z ≡ 0 in [t m , t m ]. The properties of both sequences lead to the existence of their limits. Let t := lim m→∞ t m and t := lim m→∞ t m . Since z is continuous It is easy to show that t = 0 and t = 1, which means that z ≡ 0 in [0, 1]. We get a contradiction to (A_h). Thus z > 0 in (0, 1).

A. Orpel
As a consequence of the above lemma we get the existence of supersolutions for (1.1)-(1.2).
Proof. Applying Lemma 2.1 we state that there exists at least one positive radial solution v(x) = z(1 − x 2−n ) > 0 for x ∈ Ω 1 , of (2.1), where z is a positive solution of (2.2). The first part of Lemma 2.1 guarantees the existence of t 0 ∈ (0, 1) such that z (t) ≤ 0 for all t ∈ (t 0 , 1). Let us put R 0 := (1 − t 0 ) 1 2−n > 1. Then for all x ∈ R n such that x ≥ R 0 , we have the following estimate Moreover for all x ≥ r 0 , g( x ) ≥ 0. Finally, we have for all x ∈ R n such that x ≥ R, namely v is a supersolution of our singular problem in Ω R . Applying assertions (2.5) and (2.6) and the definition of v we obtain (2.9) and (2.10).

Solutions on bounded domain
Let Ω ⊂ R n be a bounded domain with C 2+α boundary such that Ω ⊂ Ω R . Our task is now to prove the existence of a positive solution of the elliptic singular PDE (1.1) in Ω. To this end we use the ideas presented by S. Cui in [9] and formulate the lemma which gives us the solvability of our problem in Ω. For the reader's convenient we recall subsolution and supersolution results from [9]. We start with the following operator where a i,j , b i ∈ C α Ω , for some α ∈ (0, 1) , a i,j (x) = a i,j (x) in Ω, and there exists a constant λ 0 > 0 such that for all x ∈ Ω and ζ ∈ R n , ∑ n i,j=1 a i,j (x)ζ i ζ j ≥ λ 0 |ζ| 2 . Let us consider the function F satisfying the following assumptions (D1) F is locally Hölder continuous in Ω × (0, +∞) × R n and continuously differentiable with respect to the variables u and ξ; (D2) for bounded domain Q ⊂⊂ Ω and any a, b ∈ (0, +∞) , a < b, there exists a corresponding constant C = C(Q, a, b) > 0 such that for all x ∈ Q, u ∈ [a, b] , ξ ∈ R n , |F(x, u, ξ)| ≤ C(1 + |ξ| 2 ).
In spite of the fact that in our case assumptions (D1) and (D2) are satisfied, we have to emphasize that we cannot apply directly the above result. As we see in the next lemma we will construct a subsolution which is equal to zero on the boundary of Ω. On the other hand the supersolution v of our problem will be positive on ∂Ω. Thus condition (3) in Lemma 3.1 does not hold. But it appears that a small modification of the proof of Lemma 3.1 gives us the required assertion.
It is clear that (1.1) is a particular case of (3.
where Ω is a bounded domain. We say that Analogously, we say that v ∈ C 2 (Ω) ∩ C(Ω) is a supersolution of (3.3) in Ω if, at each point of Applying the steps of the reasoning described in the proof of the Lemma 3.1 (see [9, Lemma 3]) we can prove the below result.
Corollary 2.2 gives the existence of the supersolution v of (3.3) on Ω. We have to emphasize that v is independent of the set Ω, namely for each bounded domain Ω ⊂ Ω R , v is the supersolution of (3.3). Our task is now to find a positive subsolution for (3.3) in Ω.

Lemma 3.3.
There exists a positive subsolution w Ω of the problem (3.3) on Ω, such that w Ω ≤ v in Ω.
Proof. To this effect we consider ϕ being the eigenfunction corresponding to the real eigenvalue λ 1 > 0 of the following operator Lu : We know that ϕ ∈ C 2+α (Ω) ∩ C 1 Ω is positive in Ω. We show that function w Ω = sϕ 2 with s satisfying is a subsolution of (3.3). We start with the proof that w Ω (x) < v(x) < d for all x ∈ Ω, which allows us to use properties of f in Ω × [0, d] and, in consequence, we will be able to show that w Ω is the subsolution of (3.3) in Ω. To this effect we note that the following chain of inequalities holds for all x ∈ Ω. By the maximum principle we get v(x) ≥ w Ω (x) on Ω.
Finally we have the positive function w Ω = sϕ 2 , such that w Ω is a subsolution of (3.3) on Ω.
Summarizing, we proved the existence the subsolution w Ω (Lemma 3.3) and the supersolution v (Corollary 2.2) of (3.3) on Ω such that w Ω ≤ v in Ω. Thus as consequence of Lemma 3.2 we get the below result.
Proof. Let us take any bounded domain Ω ⊂⊂ Ω R with C 2+α -smooth boundary and sets Ω 1 , Ω 2 , Ω 3 also with C 2+α -smooth boundary such that Ω ⊂⊂ Ω 1 ⊂⊂ Ω 2 ⊂⊂ Ω 3 ⊂⊂ B m 0 ∩ Ω R , for B m := {x ∈ R n , x < m} and m 0 sufficiently large. For each m ∈ N, Theorem 3.4 implies the existence of solution u m ∈ C 2+α (B m ∩ Ω R ) of (3.3) such that for all m ≥ m 0 , where w m and v are given in Lemma 3.3 and Corollary 2.2, respectively. Let us consider the function we state, by the interior gradient estimate theorem of Ladyzenskaya and Ural'tseva [16], that there exists a positive constant C 1 independent of m such that Therefore (∇u m ) ∞ m=m 0 is uniformly bounded on Ω 2 , and further, (h m ) ∞ m=m 0 is uniformly bounded on Ω 2 which implies the boundedness of (h m ) ∞ m=m 0 in L p (Ω 2 ) for any p > 1. Thus (see e.g. [7,Lemma 2.3]) there exists C 2 > 0 independent of m, such that u m W 2,p (Ω 1 ) ≤ C 2 h m L p (Ω 2 ) + u m L p (Ω 2 ) , for all m ≥ m 0 , and consequently, (u m ) ∞ m=m 0 is bounded in W 2,p (Ω 1 ). Let us choose p > n 1−α . Then Sobolev's imbedding theorem gives the existence of C 3 > 0 such that u m C 1+α (Ω 1 ) < C 3 for all m ≥ m 0 (see e.g. [7, Lemma 2.1]). Moreover, we get h m ∈ C α (Ω 1 ) and there exists C 4 > 0 such that h m C α (Ω 1 ) < C 4 for all m ≥ m 0 . Applying the Schauder estimates for solutions of elliptic equations (see e.g. [7, Lemma 2.2]) we have the existence of C 5 > 0 independent of m and such that for all m ≥ m 0 Thus, using the Ascoli-Arzelà theorem we infer the existence of a subsequence (still denoted by u m ) such that (u m ) ∞ m=m 0 tends to u in C 2 (Ω ). It is clear that on Ω . Applying the Schauder estimates for solutions of elliptic equations we have u ∈ C 2+α (Ω ). Since Ω was arbitrary bounded subset of Ω R , we state that u ∈ C 2+α loc (Ω R ), Since v satisfies (2.8), (2.9), (2.10) and u ≤ v in Ω R , we state that (4.1), (4.2), (4.3) also hold. Now we give an explicit example of (1.1) to illustrate the application of Theorem 4.1.
Proof. We start with the observation that in our case we have functions a(l) = l 4 l 4 + 1 , g(r) = r 6 − 1 and f (x, u) = (x 1 + x 2 ) 2 (u − 5) (u − 6) (u + 1)u 80 x 8 + (x 2 + x 3 ) 2 24 x 6 e u which are sufficiently smooth. Moreover, we get lim l→+∞ a(l) = 1 and ∞ 1 l 1−n a(l) dl = 6 5 . Thus a satisfies (A_a). It is also clear that g(r) = r 6 − 1 is positive for all r > 1, thus (A_g) holds. Our task is now to show that f satisfies (A_f). To this effect we estimate f on the product For the continuous function M(r) := 1 r 6 + 1 4r 4 with r > 1, we have Final remark. The natural question is whether the term (u(x)) −α ∇u(x) β can be replaced by more general singularity. We answer immediately that it is possible to consider the term b(x)(u(x)) −α ∇u(x) β , where b is a sufficiently smooth, bounded and positive function. On the other hand, it is obvious that the approach presented in this paper can be applied only for the singular function satisfying the assumption described by Cui in [9]. His results allow us to obtain the existence of a smooth solution. It seems that more general singularities could imply less regularity of solution, e.g. in [1] we have a Carathéodory function g(x, u) instead of the term u −α , where g may have a singularity at 0. In this case the authors obtain the existence of weak solutions for the similar problem.