Connected sets of positive solutions of elliptic systems in exterior domains

The existence of infinitely many connected sets of positive solutions for a certain elliptic system is investigated in this paper. We consider semilinear equations with perturbed Laplace operators described in an exterior domain. We show that each of these solutions u=(u1,u2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {u}=( u_{1},u_{2})$$\end{document} has the minimal asymptotic decay, namely ui(x)=O(||x||2-n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{i}(x)=O(||x||^{2-n})$$\end{document} as ||x||→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$||x||\rightarrow \infty ,$$\end{document}i=1,2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2,$$\end{document} and finite energy in a neighborhood of infinity. Our main tool is the sub and super-solutions theorem which is based on the Sattinger’s iteration procedure. We do not need any growth assumptions concerning nonlinearities.


Introduction
The purpose of this paper is to formulate conditions which guarantee the existence of continua of positive solutions of the following system involving perturbed Laplace operators div(a 1 (||x||)∇u 1  for x ∈ G R = {x ∈ R n , ||x|| > R} with n > 2, R > 0 and ||x|| := n i=1 x 2 i . We are interested in global solutions vanishing at infinity, i.e. lim||x|| → ∞u i (x) = 0 for i = 1, 2, (2) which are often called evanescent solutions. Many problems modeled by similar systems arise in various areas of applied mathematics, in biological, chemical or physical phenomena, for example in pseudoplastic fluids [9], reaction-diffusion processes, chemical heterogeneous catalysts [4] or heat conduction in electrically conducting materials [18]. Recently the existence and multiplicity of solutions for such elliptic systems considered also in unbounded domains has been widely discussed in the literature (see e.g. [5,[11][12][13][14][23][24][25]27,34,36] and the references therein).
The existence or nonexistence of radially symmetric solutions for the Emden-Fowler system involving p-Laplace operators and some real parameters was discussed e.g. in [8]. There the approach is based on suitable transformations which play a crucial role in the reduction of the main problem to a quadratic system. In [20] and [21] results concerning more general problems associated with the existence of weak solutions of elliptic inequalities can be found. In Covei's paper [13] the following Lane-Emden-Fowler system is investigated for bounded domains ⊂ R n or = R n , in the case when nonlinearities F 1 , F 2 are positive and satisfy some growth condition with respect to the second and third variables. The multiplicity of solutions for system of similar form and their additional properties were discussed, among others, in [11]. In that paper the variational approach allowed the authors to show the existence of at least nine solutions in the case when is a bounded regular domain in R n , the right-hand side is a Carathé odory function and satisfies, among others, some growth conditions (see Th.1.1). Precisely, these solutions U = (u 1 , u 2 ), satisfy the following sign conditions: both u 1 and u 2 are strictly positive or negative in the first four solutions; four others are such that one of the two coordinates is of the one sign while the other is of changing sign, and finally both coordinates change their sing in the ninth solution. There are many results concerning the existence of infinitely many solutions for the case when the system does not depend on ∇u i . For example, vanishing solutions for Schrödinger-Poisson system (also called Schrödinger-Maxwell equations), was discussed in [24], where an approximation methods were employed. Such systems are motivated by many problems arising in quantum mechanic, in plasma physics, semiconductor theory or nonlinear optics (see e.g. [6,7,[15][16][17]30] and the references therein). Based on variational methods Ambrosetti and Ruiz proved the existence of infinitely many radial solutions for the problem when p ∈ (2, 5) and λ > 0 is sufficiently large (see [2] and [3]). In the case when p ∈ (1, 2] and λ > 0 is small enough, the existence of finite number of such solutions was proved there. In 2011 d'Avenia, Pomponio and Vaira constructed infinitely many nonradial solutions of the above system with the first equation with the function K (x) instead of constant parameter λ for p ∈ (1, 5) and K decaying at infinity. In [10,26] and [33] the following general problem was considered for the positive potential V . Applying the variant of the fountain theorem the authors obtain infinitely many high-energy solutions for superlinear f . In the sublinear case and K ≡ 1, Sun constructed infinitely many small-energy solutions for such problem [32]. Similar multiplicity results for sublinear and odd f and positive nonconstant K was proved in [34]. The authors obtained the existence of finite number of nontrivial solution for superlinear case. The existence of a sequence of solutions decaying at infinity of the above system was obtained in [35] in the case when K ≡ 1, λ = 1, f is odd and satisfies particular estimates. We want to join in the discussion concerning the multiplicity of global solutions for (1) under quite mild conditions concerning nonlinearities. We focus on positive solutions with minimal growth and finite energy in a neighborhood of infinity. The aim of this paper is twofold. On the one hand we want to formulate sufficient conditions for the existence of infinitely many unbounded connected sets of global solutions for our problem. Here we have to emphasize that this results will be obtained without assumptions concerning the oddness or growth of nonlinearity f i with respect to the second and/or third variables. These condition are often met in the literature. On the other hand we describe precisely the asymptotics of solutions and their gradients. By a solution of our problem we understand a vector function u = (u 1 , (1) and (2). Moreover, we say that u = (u 1 , u 2 ) is a finite energy solution (or a solution with finite energy) in a neighborhood of infinity when there exits a nonnegative radial function ψ : in the norm ||ϕ|| := ||∇ϕ|| L 2 ( R ) (see, among others, e.g. [27]).
To show the existence of continua of positive solutions with finite energy we assume the following hypotheses and there exist K 1 , K 2 > 1 such that the following inequality holds It is worth emphasizing that we need the monotonicity and differentiability of f i only on some right-hand side neighborhood of the origin. Moreover, we can consider both sublinear and superlinear f i . It is associated with the fact that we have to control only the value of nonlinearities f i (x, ·, ·) in the rectangle [0, d 1 ] × [0, d 2 ]. Thus we can omit growth conditions concerning variable u = (u 1 , u 2 ). Our main tool is Theorem 1 (given below) which says that the existence of a sub-solution u = (u 1 , u 2 ) and a super-solution u = (u 1 , u 2 ) of our problem such that 0 ≤ u i ≤ u i ≤ d i for i = 1, 2, implies the existence of a solution u = (u 1 , u 2 ) of (1) which is squeezed between them. The proof of Theorem 1 is based on classical ideas associated with the Sattinger's monotone iteration procedure (see [31]) and the approach described by Kawano [25] for the system which does not contain the gradient term. The similar theorem was also proved in the [29] (Th. 1.2) but only for the special case when a i ≡ 1, i = 1, 2. Therefore we can not use it directly. For the reader's convenience we present the proof of this result in the "Appendix". Now we formulate precisely the theorem which will be our main tool in the proof of the existence result. The proof of Theorem 1 is given at the end of the paper (in "Appendix").
Theorem 1 Assume that (H1), (H2), (H3) hold and u = (u 1 , u 2 ) and u = (u 1 , u 2 ) are, respectively, a positive super-solution and a nonnegative sub-solution of (1)- (2) such that Then there exists a vector function u = (u 1 , and such that for i = 1, 2, This approach allows us to prove two main results: the first one is associated with the existence of sequences of uncountable sets of solutions and the other one gives additional information concerning the asymptotics of solutions and their gradients.  2 ), where q ∈ R and l ∈ N, described in Theorem 2, satisfy the following conditions:

there exists at least one sequence of continua {S
2 ) has the minimal growth) and for all φ

Existence of continua of super-solutions
Taking into account the fact that f (x, 0, 0) ≥ 0 in R we obtain the trivial sub-solution u ≡ (0, 0) of (1)- (2). Therefore it suffices to construct an increasing sequences of 766 A. Orpel positive super-solutions for (1)- (2). To this effect we consider a sequence of auxiliary linear problems. It appears that their radial positive solutions give us a sequence of super-solutions for our main problem. Let q ∈ R and l ∈ N be fixed. We consider the following two independent linear problems i } l∈N is an increasing sequence of positive numbers, 2. for all l ∈ N, the following inequality holds 3. there exists K i > 1 such that for all l ∈ N, the following inequality holds Proof Taking into account the definition of p q,l i we get that { p q,l i } l∈N ⊂ (0, +∞) and it is an increasing sequence. Moreover, (H3) part (c) implies for all i ∈ {1, 2}, the following chain of inequalities Coming to the last part let us note that Thus, by (H3) part (c), 1 2 which gives (8).
Applying the transformation given by diffeomorphism ψ , are solutions of the following two independent Dirichlet problems with singularity at the end-point 1 with h q,l 2. there exists 0 < ε i < 1 such that for all l ∈ N and q ∈ R where t := 1 − R 2−n ∈ (0, 1) . and for all l ∈ N and q 1 ,
Proof We start with the solutions of the sequence of Dirichlet problems (9). To this end we fix q ∈ R and l ∈ N. It is obvious that taking where Let us note that for all l ∈ N, i = 1, 2, h q,l The above assertion and the boundary condition z q,l Applying the same reasoning we obtain for all l ∈ N and q 1 , Taking into account the facts that the unique solution z q,l i of (9) is nontrivial and concave, we can state that z q,l i is positive in (0, 1). Moreover we get Indeed, owing to (11) we obtain the following chain of assertions Moreover, we can describe the behavior of z and further z q,l Now we consider the vector function u q,l which implies (14). To prove (15) we take an arbitrary φ ∈ C 1 (1, +∞) such that lim r →+∞ φ(r ) = 0 and lim r →+∞ φ (r )r n−1 = +∞. Then function ψ(t) = φ((1 − t) = 0 for i = 1, 2, which gives (15).

Proof of the main results
Now we can prove our main result concerning the existence of sequences of connected sets of positive evanescent solutions of (1). The proof of this theorem is based on the sub-solution and super-solutions method described in Theorem 1 and the iteration process applied in [28]. 2 ) for our problem such that for i = 1, 2, 2 ) of (1)-(2) such that for i = 1, 2,

Proof
The last equality and the fact that u The compactness of the injection 2 ) allowed us to prove that ( u q 1 , u q 2 ) is a positive evanescent solution of (1). Our task is now to prove the other result concerning the asymptotics of solutions and their gradients. Here we employee the ideas described in [19,27,28].
Proof (of Theorem 3) Suppose that (H1), (H2), (H3) hold. Let us fix q ∈ R, l ∈ N, and consider the solution (u q,l 1 , u q,l 2 ) given in Theorem 2. We prove that each (u q,l 1 , u q,l 2 ) has finite energy applying the standard approach (see e.g. [27]). To this end we consider x ∈ R such that ||x|| ≥ 2R, and a ball B(x; r /2) of center x and radius r /2, where r = ||x||. Then, applying (14 ) and the estimates for solutions of elliptic problems ( [22], Th. 6.2), we obtain the existence of L q,l where m := max{m 1 , m 2 }. Finally, taking into account (14 ), we obtain the existence of M q,l According to the definition given in the first section, this assertion implies that (u q,l 1 , u q,l 2 ) has finite energy in a neighborhood of infinity. Assertions (4) and (5) are a simple consequence of (14) and (15).

Final remarks and examples
Employing the Kawano's approach (see [25]) we can obtain the same conclusion as in Theorem 2 also in the case when condition ( b) in (H3) is replaced by the following condition Then in the proof of Theorem 1 we have to choose different starting point of the monotone procedure and consider a super-subsolution (u 1 , u 2 ) of our problem instead of super-solution. In this case we can also prove the existence of sequences of connected sets of solutions. Now two natural questions appear. The first one concerns the existence of functions for which (H3) holds. It turns out that it is easy to find many examples of f 1 , f 2 satisfying (H3) among functions of the form f i (x, u 1 , where q > 2n − 2, k is positive and sufficiently smooth and f i is a polynomial, exponential or rational function or their combinations, e.g. f i (u 1 , u 2 ) = c(u 5 1 + u 4 1 + (u 1 + u 2 ) 2 + 1) or f i (u 1 , u 2 ) = c(e u 1 +u 2 + u 3 1 +u 3 2 (4−u 1 )(5−u 2 ) ). With help of our approach we can also investigate problems of the Emden-Fowler type when At the end of this paper an example of the problem with superlinear f i (x, ·, ·) will be discussed more precisely.
The other question is associated with R. Because of condition (3), the radius R has to be sufficiently large. For n = 3 we can consider R being the complement of large balls because in this case R has to be greater than or equal to 32. But for n = 4, we can consider R with all R ≥ 3. 674 2. It is easy to note that for high dimension n, radius R can be close to 1. This fact is associated with the behavior of each function (1, +∞), where n ∈ N. We can note that for each n ∈ N the minimal value of t n (·) in (1, +∞) is attained at k n = n−2 √ (n − 1) and then Condition (H3) part (c) can be rewritten as follows R > t n (k n  Example 1 Let us consider the following system i +||x|| 4 . We show that Theorems 2 and 3 can be applied for the system given above. To this end we note that for g i and a i , i = 1, 2, (H1) and (H2) hold with c 1 = 1, c 2 = 2. Since, in our case Therefore, it suffices to take m 1 = 2 and m 2 := 1. It is clear that Then we can state that ( H3) also holds. Finally, Theorem 2 leads to the existence of sequences of connected sets of positive solutions (22) with asymptotics described in Theorem 3.
Acknowledgements The author is grateful to anonymous referee for his/her careful reading of the first version of this manuscript and his/her constructive comments.
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Appendix
Now we prove the main tool which allowed us to obtain Theorem 2.
Proof (of Theorem 1) As in [25], we start with the existence of solutions on bounded domains and next we construct a solution on R .
To sum up, we have constructed monotonic (with respect to each coordinate) and bounded sequence {(u m,1 , u m,2 )} m∈N , which is pointwisely convergent in k to some vector functions u k 1 , u k 2 . We will show that u k 1 , u k 2 is a solutions of (23). To this effect we apply the standard reasoning, based on the L p -estimates of Agmon-Douglis-Nirenberg ( [1]), which leads to the existence of C > 0 such that for all m ∈ N, and u 2 gives the required conditions at infinity: lim ||x||→∞ U 1 (x) = 0, lim ||x||→∞ U 2 (x) = 0.