EXISTENCE OF SOLUTIONS OF SEMILINEAR SYSTEMS IN

Let Q : 2 → 2 be a symmetric and positive semi-definite linear operator and fj : R → R (j = 1, 2, ...) be real functions so that, fj(0) = 0 and, for every x = (x1, x2, ....) ∈ , it holds that f(x) := (f1(x1), f2(x2), ...) ∈ . Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f(x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of 2 which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions.


A semilinear problem in 2
Let us consider a Hilbert space H, a positive semi-definite linear operator Q : H → H and a function f : H → H so that f (0) = 0. Associated to the above is the semilinear problem Qx = f (x).(1.1)In many problems of different areas, such as biology, computer science, economy, ecology and difference equations, one searches for the existence of non-trivial solutions of a problem of type (1.1).
In [6,10] sufficient conditions for the existence of non-trivial solutions of (1.1) are provided if H = R n and f (x 1 , ..., x n ) = (f 1 (x 1 ), f 2 (x 2 ), ..., f n (x n )), where f j : R → R are continuous functions so that f j (0) = 0.In particular, the results in there were used to study the existence of solutions of partial difference equations on finite graphs.
The aim of this note is to see that such conditions can be naturally extended to the infinite countable case x j y j , and f (x) := (f 1 (x 1 ), f 2 (x 2 ), ...) where f j : R → R satisfy f j (0) = 0 and, for every x = (x 1 , x 2 , ....) ∈ 2 , it holds that f (x) := (f 1 (x 1 ), f 2 (x 2 ), ...) ∈ 2 .We apply this existence results to study the existence of solutions of partial difference equations on graphs with an infinite countable set of vertices.
A function f : R → R, with f (0) = 0, is said locally increasing at 0 (respectively, locally decreasing at 0) if there is some > 0 so that f restricted to (− , ) is increasing (respectively, decreasing).
Theorem 1.Let Q : 2 → 2 be a symmetric and positive semi-definite linear operator and {0} 6 = W < 2 be an invariant subspace under Q.Assume there are continuous maps f j : R → R, for j = 1, 2, ..., so that they all satisfy the following conditions (1) f j (0) = 0; (2) either, (2.1) for all j, f j is locally decreasing at 0, or (2.2) for all j, f j is locally increasing at 0 and, for small |t|, it holds R t 0 f j (s) ds < t 2 ; (3) lim t→+∞ f j (t) t = +∞, and (4) then the semilinear problem (1.1) has non-trivial solutions in W .Moreover, if G is a group of orthogonal linear automorphisms of 2 so that each transformation in G commutes with Q, then the semilinear problem (1.1) has non-trivial solutions invariant under the action of G.
Remark 2. Condition (2.2) in Theorem 1 trivially holds if, for every j, f j is locally increasing at 0 and, for small |t| 6 = 0, it holds that f (t)/t < 2.
In the finite dimensional situation this is not required.As we may identify R n with the subspace of 2 defined by the condition x j = 0, for j > n, then we have the following consequence that extends a result in [6].
Corollary 3. Let Q : R n → R n be a symmetric and positive semi-definite linear operator and {0} 6 = W < R n be an invariant subspace under Q.Assume there are continuous maps f j : R → R, for j = 1, 2, ...n, so that they all satisfy conditions (1), ( 2) and (3) of Theorem 1, then the semilinear problem (1.1) has non-trivial solutions in W .Moreover, if G is a group of orthogonal linear automorphisms of R n so that each transformation in G commutes with Q, then the semilinear problem (1.1) has non-trivial solutions which are invariant under the action of G.
The proof of Theorem 1 is provided in Section 3. In Section 2 we use the above existence results to the case of partial difference equations in metric graphs which are either finite or they have an infinite countable vertex set.In Sections 4 and 5 we use the above to study two examples.

Application to partial difference equations on graphs
In this section we consider graphs G = (V, E) which are simple and of finite degree (that is, each vertex has finite degree).In general one only need to take care on connected graphs, but we do not impose such a restriction at this point.To each vertex v ∈ V there is associated the set N (v) ⊂ V , called the neighborhood of v, so that w ∈ N (v) if {v, w} ∈ E; w is called a neighbor of v. Now on, we only assume the vertex set V is finite or infinite countable.Associated to the graph G are the real vector space C 0 (G) consisting of the real functions defined on V and its subvector space L 2 (G) consisting of those µ ∈ C 0 (G) so that are any derivations, then we have associated a partial difference equation on the graph given by for every vertex v ∈ V .We only consider partial difference equations of the form where Partial difference equations appear naturally by numerical discretization of partial differential equations on grids (see, for instance, [5] for linear elliptic difference equations on the plane).Due to applications in fields as biology, computer science, economy, ecology, etc., nonlinear partial difference equations has recently attracted a lot of attention.
If we set V = {v 1 , v 2 , ....}, then there is a natural isomorphism of real vector spaces φ : In this way, the partial diference equation (2.2) is equivalent to a semilinear problem (1.1).As a direct consequence of Theorem 1 is the following existence result.
Corollary 4. Let G = (V, E) be an finite degree simple graph with vertex set V either finite or countable infinite, say V = {v 1 , ....}.Let D ∈ Ξ R (G) be a symmetric positive semi-definite derivation, F : V × R → R be continuous in the real variable and set f j (t) = F (v j , t), for each v j ∈ V .If, for each j, f j : R → R satisfies the conditions (1), ( 2), (3) and, for V infinite, also condition (4) of Theorem 1, then the partial difference equation (2.2) has non-trivial solutions.Moreover, if G is a group of symmetries of the graph G, then there are non-trivial solutions which are symmetric respect to G.
A discussion of the above result for the particular case of finite graphs and the discrete Laplacian operator can be found in [10] and, for a more general class of derivations, in [6].
Corollary 5 (Dirichlet's problem).Let G = (V, E) be an finite degree simple graph with vertex set V either finite or countable infinite and W < V so that W 6 = ∅ and Consider real values a j ∈ R, where j ≤ 0, and the infinite column vector Assume, with this enumeration of the vertex set V , that the (infinite size) matrix representation of D is where R is the matrix corresponding to W (of size r×r in (1) and of infinite size in ( 2)).Set b = Ua and, f j (t) = F (v j , t).If each f j : R → R satisfies the conditions f j (0) = b j , for all j, and conditions (2), (3) and, if V is infinite, also condition (4) of Theorem 1, then the Dirichlet problem Proof.In this case, we need to solve Sy = g(y), where g(y) = f (y)−Ua, f (y) is the column vector whose j-th coordinate is f j (y j ) and S is symmetric and positive semi-definite.The existence is now consequence of Theorem 1 using g instead of f . 2 Let us consider a finite degree, simple, connected graph G = (V, E), where we assume that the set V of vertices is either finite or countable infinite.Let us fix some metric on the graph, that is, a positive function As we are assuming N (v) to be finite for each v ∈ V , the sum in the above definition is a finite sum.It is well known that ∆ 2 is a symmetric and positive semi-definite operator (see for instance [2,3,4,9]).

The proof of Theorem 1.
The proof of Theorem 1 is a simple application of the Mountain Pass Theorem, due to A. Ambrosetti and P. Rabinowitz, which we recall below as matter of completeness.

Mountain Pass Theorem 6 ([6]
).Let U a Hilbert space and H : U → R be a C 1 (U ; R) functional which is Lipschitz continuous on bounded sets of (ii) there exists a, r > 0 such that H(x) ≥ a, for kxk = r; (iii) there exists y ∈ U , kyk > r with H(y) < 0.
Then "a" is a critical value of H.

Proof of theorem 1.
Let us first consider the case W = 2 .We consider the real map The function It follows that the (non-trivial) critical points of H Q are exactly the solutions we are searching for.Condition (1) ensures H Q (0) = 0.As f j (0) = 0 and Q is positive semi-definite, condition (2) ensures the existence of positive values a, r > 0 so that H Q (x) ≥ a > 0 for kxk = r.Condition (3) asserts the existence of some y ∈ 2 with kyk > r and H Q (y) < 0. Let R > 0 be fixed and let x, y ∈ B(0; R) ⊂ 2 , that is, Then, by condition (4), for some constant K(R) > 0. All the above and Mountain Pass Theorem (see below) ensure the existence of non-trivial critical points of H Q .
Let now consider a general W < 2 which is invariant under Q.Conjugating by a suitable orthogonal linear automorphism of 2 , we may assume W is given by the conditions x r = 0, for some indices r.As Q still being symmetric and positive semi-definite under such a rotation and its restriction to the subspace W still having the same properties, the existence is provided by the above case.
If G is a group of orthogonal transformations of 2 so that each of its elements commutes with Q and set W = F ix(G) = {x ∈ 2 : g(x) = x, ∀g ∈ G}, then one obtains the existence of non-trivial solutions in F ix(G).

Example I: Matukuma type equations
In this section we consider, as a first example, Matukuma type equations and adapt them to metric graphs.Then we use the results obtained previously in order to assert the existence of non-trivial solutions.

Classical Matukuma equations
In order to have a model to describe the dynamics of globular cluster of star, T. Matukuma [8] proposed the following one (Matukuma equation) where p > 1, µ p = |µ| p−1 µ and µ > 0 is the gravitational potential so that represents the total mass.Many results have been obtained for radial solutions µ a (r), where r = |x| and a > 0 is so that µ a (0) = a.Even, in [8] obtained that µ √ 3 (r) = p 3/(1 + r 2 ).A more general kind of equations that generalize Matukuma equation are the Matukuma type equations where f (x) > 0 for every x ∈ R 3 .A discussion about positive solutions of Makutuma type equation can be found in [11].

Matukuma type equations on graphs
Matukuma type equations can be adapted to connected, finite degree metric graphs as follows.Let us consider a finite degree, simple, connected graph G = (V, E), where we assume that the set V of vertices is either finite or countable infinite.Let us fix some metric on the graph, that is, a positive function d : E → (0, +∞).Equation (4.2) is adapted to this metric graph as the partial difference equation where f : V → (0, +∞).More generally, if W ⊂ V is so that V − W 6 = ∅, then we may consider the Dirichlet problem where f : V → (0, +∞).If W 6 = ∅, then Corollary 5 ensures the existence of non-trivial solutions of equation 9.In the case that W = ∅, Corollary 1 ensures the existence of non-trivial solutions of equation (4.3); moreover, it asserts the existence of non-trivial solutions which are invariant under the group of symmetries of the metric graph (G, d).Next theorem asserts that the non-trivial solutions of the equation (4.3) must be sign changing.Theorem 6. Equation (4.3) has non-trivial solutions and all of them must be a sign changing solution, in particular, there is no non-negative nor nonpositive non-trivial solution.
Note that the sums in the definition of b ij are finite sums as we are assuming the graph G to be of finite degree.Also set A = [w ij ], B = [b ij ] and J 2 = B − A. The discrete Laplace operator ∆ 2 is then defined in R N as the linear operator x 7 → J 2 x, for x ∈ R N .As the sum of each column of J 2 (a finite sum) is equal to 0. It follows that for every µ ∈ C 0 (G) it holds that

Example II: Helmholtz equations
Many problems related to acoustic mechanics, electromagnetism, etc., reduce to study the Helmholtz equation [7].We proceed to adapt such equation to finite degree graphs (with either finite or an infinite countable number of vertices) and we find conditions to ensure the existence and uniqueness of solutions.

Classical Helmholtz equation
Let Ω ⊂ R n be some domain and let h : Ω → R be a function (in general with compact support).Helmholtz equation is given by where ∆ is the Laplace operator and µ : Ω → R is some solution function (with the required regularity properties).
If h = 0, then the existence of non-trivial solutions of is equivalent to have k 2 as an eigenvalue of −∆ (on the domain Ω).In this homogeneous case, the equation can be solved by separation of variables.
One may also consider the corresponding Newmann problem where ∂ n µ denotes the normal derivative respect to the boundary ∂Ω ⊂ R n .

Helmholtz equation on graphs
Let us assume that we are given a simple, finite degree and connected graph G = (V, E), with V either finite or countable infinite, and a metric be no necessarily different ones.Newmann's problem (5.2) may be adapted to the above metric graph as In the above two cases, we also set Ind(V ) = {1, 2, ....}.

Existence of solutions
Let us write the matrix J 2 (representing ∆ 2 ) as

#xExample 8 .
where A and D are positive semi-definite symmetric matrices of sizes |W | and |V − W |, respectively.The function σ may be though as a functionσ : Ind(W ) → Ind(V ).Let us set M σ = [m ij ] (where i ∈ Ind(W ) and j ∈ Ind(V )) bym ij = ( −1, σ(i) = j 0, σ(i) 6 = j .Set µ(v j ) = ( x j , j ∈ Ind(W ) y j , j ∈ Ind(V ) .With the above notation, equation (5.3) is equivalent to" = b − M σ y (D − CM σ − k 2 I n )y = a − Cb .(5.5)It nows easily follows the next result.Theorem 7. Newmann's problem (5.3) has a unique solution if and only if k 2 is not an eigenvalue of D − CM σ .If k 2 is an eigenvalue of D − CM, then (5.3) may not have solution and, if it has at least one, then it must have infinitely many solutions.Let us consider the complete graph K 2 .Set V = {v 1 , v 2 }, w = 1/d({v 1 , v 2 }) > 0. Assume W = {v 1 }, b v 1 = b and a v 2 = a.In this case,