Minimal positive solutions for systems of semilinear elliptic equations

The paper is devoted to a system of semilinear PDEs containing gradient terms. Applying the approach based on Sattinger’s iteration procedure we use sub and supersolutions methods to prove the existence of positive solutions with minimal growth. These results can be applied for both sublinear and superlinear problems.

There exists rich literature devoted to similar problems which arise in many applications e.g. in pseudoplastic fluids [6], reaction-diffusion processes or chemical heterogeneous catalysts [3], heat conduction in electrically conducting materials [7].
Here we have to mention also Constantin's results (see e.g.[8,9]) describing the case of a single equation and further papers (e.g.[14][15][16][17][18]).Recently the research concerning the existence and properties of positive solutions of systems of nonlinear elliptic problems has been very active and enjoying of increasing interest (see e.g.[10][11][12][13]21,22,25,28] and references therein).The sub and supersolutions methods are applied in many of these papers (e.g.[11,12, A. Orpel 21,22] or [25]).Other techniques are also met.In [13] the results are based on approximations.In [10] we can find the variational approach which allowed to show the existence of a few solutions of the following problem −∆U(x) = ∇H(x, U(x)) in Ω U(x) = 0 on ∂Ω, where U = (u 1 , u 2 ) : Ω → R 2 , Ω is a bounded regular domain in R n , the right-hand side is a Carathéodory function and satisfies, among others, some growth conditions.The main result of the paper [10, Theorem 1.1] says that the above problem possesses at least nine nontrivial solutions U = (u 1 , u 2 ), satisfying 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 components is of the one sign while the other is of changing sign, and finally both components change their sing in the ninth solution.
We also have to mention the paper [5], where we can find, among others, the results concerning the existence or nonexistence of radially symmetric solutions for the Emden-Fowler system involving p-Laplace operators and some real parameters.The approach is based on suitable transformations which play a crucial role in the reduction of the main problem to a quadratic system.Moreover, in the case when the main system is variational the behaviour of the ground states was described.Considering parameters satisfying additional conditions and applying a new type of energy function, the authors investigate the existence of ground states also in the case when the system is not variational.
Two further papers [20] and [19] are devoted to more general problems associated with elliptic inequalities.The first one describes the existence and nonexistence of nonnegative and nontrivial entire weak solutions for a single inequality.The approach is based on a generalized version of the Keller-Ossermann condition (see e.g.[4] and [23] ).In the latter paper the system of elliptic inequalities of divergence type is investigated.The author obtains the results employing the method of test functions (see e.g. in [24]).These results can be applied in the case of p-Laplace operators as well as mean curvature operators.
Covei's results are also based on the sub and supersolutions method which was introduced by Keller and Amann.The existence result for solutions of elliptic problems under the assumption concerning the existence of subsolutions and supersolutions was first proved by H. Amann in [1].One year later in [27], Sattinger proved similar theorems for regular elliptic boundary value problems using the L p estimates of Agmon, Douglis and Nirenberg ( [2]).In our paper we want to employ these classical ideas.Motivated by [12], we are interested in another type of nonlinearities.It is worth emphasizing that we do not require our system to be either potential or radial symmetric.Moreover we do not need either growth conditions on f i or the equality f i (•, 0, 0) ≡ 0, i = 1, 2. Precisely, in the sequel we assume the following conditions (A1) for i = 1, 2, g i : [1, +∞) → R belongs to C 1 ([1, +∞)) and there exists l 0 ≥ 1 such that g i (l) ≥ 0 for all l ≥ l 0 ; (A2) , is locally Hölder continuous with exponent α ∈ (0, 1), x → f i (x, 0, 0) is positive and there exist d 1 , d 2 > 0 and continuous functions f i such that sup Remark 1.1.Assumptions (A1)-(A4) are not too restrictive and many elementary functions satisfy them.It is easy to find many examples of f 1 , f 2 satisfying (A2), (A3) and (A4) among functions of the form where q > n, a is positive and sufficiently smooth and f i is a polynomial, exponential or ratio- nal function or their combinations, e.g.
. We can also investigate problems of the Emden-Fowler type when with α, β, M > 0. We will discuss an example of the problem with exponential and rational nonlinearities at the end of this paper.
It is worth emphasizing that we need the monotonicity and differentiability of f i in u and/or v only on some right-hand neighborhood of the origin.Moreover, we can consider both sublinear and superlinear f i which is associated with the fact that we have to control only the value of nonlinearities.Thus we can omit growth conditions concerning second and third variables.In the proof of the existence of a positive solution of (1.1) we do not need (A4).We use this condition only to show that the solution is minimal.Our main tool is Theorem 1.2 which says that the existence of a sub-subsolution (u, v) and a super-supersolution (u, v) of our problem such that 0 ≤ u ≤ u ≤ d 1 and 0 ≤ v ≤ v ≤ d 2 implies the existence of solution (u, v) of (1.1) which is squeezed between them.In the proof of Theorem 1.2 we apply classical ideas based on Sattinger's monotone iteration procedure.The iteration scheme is patterned after that in [22] where the existence results are proved in the case when the problem does not contain the gradient term.Thus we start with standard definitions of a solution, a supersupersolution and a sub-subsolution of (1.1)-(1.2) (see e.g.[22]).
By a solution of our problem we understand a pair (u, ) and vanishing at infinity, that is conditions (1.2).We say that a positive solution of our problem is minimal when the functions: x → x n−2 u(x) and x → x n−2 v(x) are bounded above and below by positive constants in some exterior domain (see, among others, [25]).
By a super-supersolution of (1.1) Analogously, as for a sub-subsolution (u, v) of (1.1)-(1.2) in G R , the sign of the inequality should be reversed. Since , is the trivial subsubsolution.We obtain the super-supersolution of our system as a radial solution of a certain auxiliary linear problem considered in the complement of the unit ball centered at the origin.Now we formulate the theorem which will be our main tool in the proof of the existence result.
Remark 1.3.To prove the above result we will use the ideas described by Kawano in [22] for the case when the elliptic problem contains the gradient terms.We can follow his steps because of the fact that the differential operator is also linear in our problem.Although the proof of Theorem 1.2 is standard we present the sketch of the reasoning in the Appendix for the reader's convenience.

The existence of a super-supersolution
In this section we show the existence of a positive super-supersolution of (1.1) as radial solutions w i , i = 1, 2, of the following auxiliary linear problems We prove that w i is radially decreasing in exterior domain G R for R sufficiently large.We also describe more precisely the asymptotic behaviour of w i .These properties will play the crucial role in the proof of the fact that (w 1 , w 2 ) is a super-supersolution of our problem in a certain exterior domain and our solution of (1.1) has a minimal growth at infinity.
Proof.Fix i ∈ {1, 2}.We start with the well-known fact that, via a suitable transformation, the investigation of the existence of a radial solution for the problem (2.1) leads, to the solvability of the Dirichlet problem with singularity at the end-point 1, namely for all t ∈ (0, 1).Precisely, when we have a radial solution w(x) = z( x ) with z : [1, +∞) → R of (2.1), we get the solution for (2.2) of the form z(t) = z (1 − t) Conversely, having a solution z of (2.2) we derive that w(x) = z(1 − x 2−n ) satisfies (2.1).Assumptions made on f i allow us to state that h i is continuous, h i (•) > 0 and Simple calculation leads to the conclusion that where Taking into account the facts that the solution z i of (2.2) is nontrivial and concave, we can state that z i is positive in (0, 1).As in [17] and [26], we prove the existence of t i ∈ (0, 1) such that z i (t) ≤ 0 for all t ∈ [t, 1).Applying Rolle's theorem, we get that the set S := {t ∈ (0, 1); A. Orpel z i (t) = 0} is nonempty.Since z i in nonincreasing in (0, 1) , we have z i (t) = 0 in [t 1 , t 2 ] for all t 1 , t 2 ∈ S such that t 1 ≤ t 2 .Thus S is an interval.Let t i := sup S. It is easy to show that t i = 1.Indeed, if t i = 1 then, we derive the existence of a sequence {t m } ⊂ S such that lim m→∞ t m = 1.Without loss of generality we can assume that for all m ∈ N, t 1 ≤ t m .Thus for all m ∈ N, z i (t) = 0 in [t 1 , t m ], and consequently z i (t) = 0 in [t 1 , 1), which gives, by the continuity of z i in [0, 1], that z i (t) = 0 in [t 1 , 1].But it is impossible with respect to the fact that z i is positive.In consequence, for all t ∈ [t, 1), z (t) ≤ 0. Now we try to describe more precisely the behaviour of z i in the left-hand neighborhood of 1.First we note that z i (t) = − 1 0 sh i (s)ds + 1 t h i (s)ds and further .
To sum up, we proved the existence of the positive solution z i of (2.2) which satisfies the following conditions z i (t) ≤ 0 for all t ∈ [t i , 1) (2.5) with certain t i ∈ (0, 1) and Coming to the radial solution of (2.1), we use the positive solution z i of (2.2) described above. Then This implies that there exists L i > 0 such that for all x ∈ R n , x > L i , which means that w i is the minimal solution of (2.1).Now, taking into account (2.5), we have for all where the last inequality follows from the fact that for x ≥ R we have t : If we assume additionally (A4) then the solution (u 0 , v 0 ) is minimal.
Proof.Lemma 2.1 gives the existence of a positive super-supersolution (w 1 , w 2 ) such that each function x → x n−2 w i (x) is bonded above and below by positive constants in some exterior domain.On the other hand (1.1) possesses the trivial sub-subsolution in G R .Applying Theorem 1.2, we derive that there exists a solution of (1.1) such that Our last task is to show that the solution is minimal (see also [25]).We start with the proof that u 0 and v 0 are bounded below by functions of the form x → B x 2−n in a certain exterior domain.To this end we apply (A4) which gives the existence of L 1 > 1 and A 1 > 0 such that for all u ∈ [0, We can assume (without loss of generality) that L 1 > max{R, l 0 }.Now we consider w(x) = B 1 x 2−n with B 1 > 0 and B 1 < min{A 1 , L n−2 1 min x =L 1 u 0 (x)}.Let us note that assumption (A4) guarantees for all x ∈ R n with the norm x > L 1 the following chain of assertions Finally, the maximum principle allows us to state that u 0 (x) − w(x) ≥ 0 for all x ∈ R n , x > L 1 , which gives On the other hand, by the asymptotics of the super-supersolution, we can state that for all x ∈ R n , x > L 1 , we have Finally, the function x → x n−2 u 0 (x) is bounded below and above by respectively, B 1 and In the same way we can obtain the similar conclusion for v 0 .To sum up, we have shown that the solution (u 0 , v 0 ) of our problem is minimal.

Example 2.3. Let us consider the following problem
It is obvious that for g i , i = 1, 2, (A1) holds.Moreover, in our case Simple calculations allow us to state that for r ≥ 1, sup

Appendix
As in [22], we start with the existence of solutions on bounded domains.
where k ∈ N := {1, 2, . . .}. Assume that conditions (A1)-(A3) hold and suppose that (u, v) and (u, v) are, respectively, a super-supersolution and a sub-subsolution of (1.1) Proof.Let us consider the following auxiliary system with By assumption (A3) we state the existence of positive constants K 1 , K 2 such that . The starting point of the iteration procedure based on the Sattinger's schema is associated with super-supersolution of our problem.Precisely, we take u 0 = u and v 0 = v and obtain the existence of a classical solution (u 1 , v 1 ) of the linear problem considered in Ω k : Since (u, v) is a super-supersolution of (1.1)-(1.2), it is easy to get Taking into account the fact that u 1 (x) = ϕ 1 (x) on ∂Ω k , we state, by the maximum principle, Let us note that for all m ∈ N, For m = 1, (4.3) was proved.Fix integer m ≥ 1.If we assume that u m (x) ≤ u m−1 (x) and v m (x) ≤ v m−1 (x) in Ω k , the properties of f 1 and f 2 give two assertions: for all x ∈ Ω k , Applying the boundary conditions and again the maximum principle one infers u m+1 (x) ≤ u m (x) and v m+1 (x) ≤ v m (x) in Ω k .Finally, the induction principle allows us to state that for all m ∈ N, (4.3) holds.
Taking the sub-subsolution of (1.1)-(1.2) in G R as a starting point of our procedure we can use the same reasoning and construct another sequence Applying again properties of f 1 and f 2 , the maximum principle and the induction, one also proves the following chains of inequalities To sum up, we have constructed two monotonic and bounded sequences {(u m , v m )} and {(u m , v m )}.They are both pointwisely convergent in Ω k to some vector functions u k , v k and u k , v k respectively.Our task is now to show that u k , v k and u k , v k are solutions of (4.1).The standard reasoning, based on the L p -estimates of Agmon-Douglis-Nirenberg, allows us to prove the existence of C > 0 such that for all m ∈ N, [22] or [25] for details).Therefore the compact embedding , and consequently, u k , v k is a solution of (4.1).Analogously we can show that u k , v k satisfies (4.1).By (4.4) we get u ≤ Proof of Theorem 1.2.Applying Lemma 4.1 with ϕ 1 = u and ϕ 2 = v, we state for each k ∈ N, the existence of a solution u k , v k of the problem Now we investigate the properties of the sequence { u k , v k }.Employing the reasoning similar to that in the proof of Lemma 4.1 one proves that for any integer fixed k 0 > 0 and k > k 0 + 1, this sequence is bounded in C 2+α (Ω k 0 ) × C 2+α (Ω k 0 ), namely there exists a positive constant C 1 such that for all k > k 0 + 1, In the next step we use the compact embedding C 2+α (Ω 1 ) × C 2+α (Ω 1 ) into C 2 (Ω 1 ) × C 2 (Ω 1 ), which allows us to state the existence of a subsequence u k l 1 , v k l 1 of u k , v k which tends to a certain u 1 , v 1 in C 2 (Ω 1 ) × C 2 (Ω 1 ).It is obvious that u 1 , v 1 satisfies (1.1) in Ω 1 .Moreover, u ≤ u 1 ≤ u and v ≤ v 1 ≤ v in Ω 1 and u 1 = u and v 1 = u on {x ∈ R n , x = R}.Applying the same reasoning we obtain the subsequence u k l 2 , v k l 2 of u k l 1 , v k l 1 such that u k l 2 , v k l 2 converges in C 2 (Ω 2 ) × C 2 (Ω 2 ) to u 2 , v 2 being a solution of (1.1) in Ω 2 .We also have u ≤ u 2 ≤ u, v ≤ v 2 ≤ u in Ω 2 and u 2 = u and v 2 = v on {x ∈ R n , x = R}.Iterate this process for each m ∈ N, we construct a sequence { u k lm , v k lm } which is convergent in C 2 (Ω m ) × C 2 (Ω m ) and such that u k lm , v k lm is a subsequence of u k lm −1 , v k l m−1 .Let u m (x) := lim l m →∞ u k lm (x) and v m (x) := lim l m →∞ v k lm (x) in Ω m .To sum up, one can state that ( u m , v m ) is a solution Then (U, V) ∈ C 2+α loc (G R ) × C 2+α loc (G R ) and satisfies (1.1) in G R .Moreover we have u ≤ U ≤ u, v ≤ V ≤ v in Ω m , U = u and V = v on {x ∈ R n , x = R}.The asymptotics of u and v implies that lim x →∞ U(x) = 0, lim x →∞ V(x) = 0.
(1.1) inΩ m , u ≤ u m ≤ u and v ≤ v m ≤ v in Ω m and u m = u and v m = v on {x ∈ R n , x = R}.By the construction we know that u m | Ω m−1 = u m−1 and v m | Ω m−1 = v m−1 .Finally, we define U, V : G R → Rin the following way U(x) := u m and V(x) := v m in Ω m .