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Existence and continuation of solutions for a nonlinear Neumann problem

In this article we study the existence, continuation and bifurcation from infinity of nonconstant solutions for a nonlinear Neumann problem. We apply the Leray-Schauder degree and the degree for SO(2)-equivariant gradient operators defined by the second author.

In our theorems we put assumptions on f ′ (z), where z ∈ Z ∪ {∞}. We emphasize that we also treat problems with resonance at constant solutions and at infinity i.e. it can happen that f ′ (z) ∈ σ(−∆; Ω) for some z ∈ Z ∪ {∞}.
The second goal of this article is to prove the sufficient conditions for continuation of solutions of the following problem where Ω ⊂ R n is an open bounded domain with C 1 − -boundary and f ∈ C 1 (R × R, R).
The third goal of this paper is to study global bifurcations from infinity of solutions of problem (1.3).
It is worth in pointing out that application of classical invariants like the Conley index technique and the Morse theory does not ensure the existence of closed connected sets of critical points of variational problems, see Ambrosetti [2], Böhme [3], Ize [13], Marino [17], Takens [23] for examples and discussion.
In other words one can not apply these invariants in order to prove continuation and global bifurcation of solutions of problem (1.3).
Since the gradient ∇Φ ∈ C 1 (H 1 (Ω), H 1 (Ω)) is of the form compact perturbation of the identity, we apply the Leray-Schauder degree and the degree for SO(2)-equivariant gradient maps to the study of critical points (critical SO(2)-orbits) of the functional Φ.
The choice of the Leray-Schauder degree and the degree for SO(2)-equivariant gradient maps seems to be the best adapted to our theory.
After this introduction our article is organized as follows.
Since the degree for SO(2)-equivariant gradient maps is not widely known, in Section 2 we have summarized without proofs the relevant material on this invariant, thus making our exposition as self-contained as possible.
In Section 3 we have studied problem (1.2). In Lemma 3.1 we have derived a formula for the Leray-Schauder degree of the gradient ∇ u Ψ ∈ C 1 (H 1 (Ω) × R, H 1 (Ω)) of a functional Ψ ∈ C 2 (H 1 (Ω) × R, R) associated with problem (1.2). Suppose now that R n is an orthogonal SO(2)-representation and that Ω ⊂ R n is SO(2)-invariant. Under these assumptions H 1 (Ω) is an orthogonal SO(2)-representation, the functional Ψ is SO(2)-invariant and its gradient ∇ u Ψ is SO(2)-equivariant. In Lemma 3.2 we have proved a formula for the degree for SO(2)-equivariant gradient maps of ∇ u Ψ.
In Section 4 our main results are stated and proved. Subsection 4.1 is devoted to the study of the existence of nonconstant solutions of problem (1.1). In Theorems 4.1.1-4.1.4 we consider non-degenerate case i.e. we assume that f ′ (z) / ∈ σ(−∆; Ω) for every z ∈ Z ∪ {∞}. These theorems ensure the existence of at least one nonconstant solution of problem (1.1). Notice that in Theorems 4.1.2-4.1. 4 we have assumed that domain Ω is SO(2)-invariant.
We emphasize that in the proofs of In Section 5 we illustrate the main results of this paper. Namely, we consider problem (1.1) with Ω = B 2 and Ω = (0, 1) × B 2 .
For a definition of the Euler ring U(G), where G is any compact Lie group, we refer the reader to [8].
Let V be a real, finite-dimensional and orthogonal SO(2)-representation.
Step 2. The first coordinate of the degree for SO(2)-equivariant gradient maps is defined where deg B denotes the Brouwer degree.

Remark 2.3.
Directly from the definition of the degree for SO(2)-equivariant gradient maps (see [21]) it follows that (1) if H ∈ Υ(SO(2)) is a closed subgroup and SO (2) Below we formulate product formula for the degree for SO(2)-equivariant gradient maps.
For k ∈ N define a map ρ k : SO(2) → GL(2, R) as follows For j, k ∈ N we denote by R[j, k] the direct sum of j copies of (R 2 , ρ k ), we also denote by R[j, 0] the trivial j-dimensional SO(2)-representation. We say that two SO(2)representations V and W are equivalent if there exists an SO(2)-equivariant, linear isomorphism T : V → W. The following classic result gives a complete classification (up to equivalence) of finite-dimensional SO(2)-representations (see [1]).
We will denote by m − (L) the Morse index of a symmetric matrix L.
To apply successfully any degree theory we need computational formulas for this invariant. Below we show how to compute degree for SO(2)-equivariant gradient maps of a linear, self-adjoint, SO(2)-equivariant isomorphism.
-equivariant, linear isomorphism and γ > 0, then in particular, if L = −Id, then Let (H, ·, · H ) be an infinite-dimensional, separable Hilbert space which is an orthogonal SO(2)-representation and let C 1 Combining Theorem 4.5 in [21] with Theorem 2.2 we obtain the following theorem.
Below we formulate the continuation theorem for SO(2)-equivariant gradient operators of the form compact perturbation of the identity. In other words we study continuation of critical orbits of SO(2)-invariant C 1 -functionals. The proof of this theorem is standard, but in the proof we have to replace the Leray-Schauder degree with the degree for SO(2)equivariant gradient operators.
and for both C = C ± the following statements are valid (2.5) (2)) defined as follows In the following theorems we have formulated sufficient conditions for the existence of an unbounded closed connected set of critical orbits bifurcating from infinity. Proofs of this theorems can be found in [10].

Linear equation
Throughout this section we assume that Ω ⊂ R n is a bounded, open set with C 1 −boundary. Consider the following eigenvalue problem Solutions of problem (3.1) are in one to one correspondence with critical points of functional Ψ : Computing the gradient ∇ u Ψ : According to the Riesz theorem there exists linear bounded operator K : given by formula Ku, v H 1 (Ω) = Ω uvdx. By definition K is self adjoint and by the imbedding theorems it is compact.
By the spectral theorem for compact, self-adjoint operators Since ∇ u Ψ(·, λ) is a family of operators of the form compact perturbation of the identity, one can apply the Leray-Schauder deg LS degree to ∇ u Ψ(·, λ). The standard proof of the following lemma is omitted.
which completes the proof.

Results
In this section we formulate and prove the main results of this article. In the first subsection we formulate the sufficient conditions for the existence of nonconstant solutions of the following equation In the second subsection we study continuation of solutions of the following family of equations Finally in the third subsection we study global bifurcations from infinity of solutions of above problem.
In the proofs of theorems of this section as the topological invariants we use the Leray-Schauder degree and the degree for SO(2)-equivariant gradient maps. 4.1. Existence of nonconstant solutions. In this section we study weak solutions of the following equation where Ω ⊂ R n is an open, bounded set with C 1 − -boundary and f ∈ C 1 (R, R) satisfy the following assumption , for n ≥ 3 and 1 < p < ∞ for n = 1, 2. are in one to one correspondence with critical points of a functional Φ ∈ C 2 (H 1 (Ω), R) Let us put the following additional assumption We treat ∞ as a critical point of Φ. We say that ∞ is an isolated critical point of Φ if (∇Φ) −1 (0) is bounded. Assume that all the elements of Z ∪ {∞} are isolated critical points of Φ. From now on γ z denotes a positive real number such that: Proof. It is easy to see that 0)). The rest of the prove is a direct consequence of Lemma 3.1.
Since the proof of the next lemma is similar to the proof of Lemma 4.1.1, we will omit it. Put the following assumptions: In the next theorem we ensure the existence of nonconstant solutions of equation (4.1.1).
Then there exists at least one nonconstant solution of equation (4.1.1).
Proof. By the properties of the Leray-Schauder degree we obtain that Suppose, contrary to our claim, that Let assumption (1) or (2) We thus get #Z e + = 0, a contradiction.    (2)).
Proof. Since ∇Φ is an SO(2)-equivariant operator of the form compact perturbation of the identity and ∇ 2 Φ(∞) = Id − (1 + f ′ (∞))K is an isomorphism, The rest of the proof is a direct consequence of Lemma 3.2.
The following corollary is an immediate consequence of Lemmas 2.  .
In the next three theorems we prove the existence of nonconstant solutions of equation (4.1.1).
Since Ω ⊂ V is SO(2)-invariant, ∇Φ is SO(2)-equivariant. Therefore we use in the proofs the degree for SO(2)-equivariant gradient maps. It is worth to point out that we obtain the existence of nonconstant solutions of equation  Proof. In view of Theorem 4.1.1, to complete the proof, it is enough to assume that ν(f ′ (z)) is odd for all z ∈ Z + . By the properties of the degree for SO(2)-equivariant gradient maps we obtain Therefore, to complete the proof, it remains to prove that Suppose, contrary to our claim, that By assumption ν(f ′ (z)) is odd. Hence from Corollary 4.1.1 we obtain Using the above and (4.1.9) we get ∇ SO(2) −deg Z k ′ (∇Φ, B γ (H 1 (Ω), z)) = 0 for all z ∈ Z + . By the assumption there exists   If z ∈ Z − and k ∈ N, then, by Lemma 4.1.3, we get ∇ SO(2) −deg Z k (∇Φ, B γz (H 1 (Ω), z)) = 0 and (H 1 (Ω), z)). (4.1.10) From Theorem 4.1.1 it follows that, to complete the proof, it suffices to consider the case ν(f ′ (z)) is odd for all z ∈ Z + ∪ {∞}. Therefore, by Corollary 4.1.1, we obtain that Taking together the above inequalities and (4.1.10) we obtain ) is a trivial SO(2)-representation, applying Lemmas 2.1, 4.1.3, we obtain Let j 0 , j ∞ ∈ N be the largest integers such that Since f ′ (z 0 ) < λ i 0 < f ′ (∞), we obtain j 0 < j ∞ . Finally, by Lemmas 2.1, 4.1.3, we obtain ) for every z ∈ Z + and Lemmas 2.1, 4.1.3, for every z ∈ Z. Thus, by the above and (4.1.10), we obtain  )) is even. Additionally assume that there exists z 0 ∈ Z + such that ν(f ′ (z 0 )) is even and one of the following conditions is fulfilled Proof. The proof is similar to that of Theorem 4.1.2. By the properties of the degree for SO(2)-equivariant gradient maps we obtain (H 1 (Ω), z)). (H 1 (Ω), z)).

It remains to prove that
Suppose, contrary to our claim, that (H 1 (Ω), z)).
Thus, by the above and (4.1.11), we obtain In the rest of this section we consider a degenerate case i.e. we allow f ′ (z 0 ) ∈ σ(−∆; Ω) for some z 0 ∈ Z ∪ {∞}. To compute a local index of a degenerate isolated critical point of Φ we combine the splitting lemmas (Lemmas 3.2, 3.3 of [9]) and the product formula for the degree for SO(2)-equivariant gradient maps, Theorem 2.2. The following lemma is a consequence of splitting lemmas of [9].
Proof. Fix z 0 ∈ Z and remind that K : Then ∇η 0 is a compact, SO(2)-equivariant operator and |∇η 0 (u)| = o(|u|) as |u| → 0. Now applying Lemma 3.2 of [9] we obtain α 0 > 0 and ϕ ∈ C 2 SO(2) (V −∆ (f ′ (z 0 )), R) with isolated critical point at the origin and such that Finally, combining Theorem 2.2 with a slightly modified version of Lemma 3.2 (instead of the operator ∇ u Ψ(·, λ) it is enough to consider the operator ∇ 2 Φ(z 0 ) |W 0 ), we obtain which completes the proof. The same proof remains valid for z 0 = ∞ but instead of Lemma 3.2 of [9] we must use Lemma 3.3 of [9]. The details are left to the reader.
Proof. Take α 0 > 0 and ∇ϕ as in Lemma 4.1.5. Then The rest of the proof is a direct consequence of formula (2.2).
We can now proof the analog of Hence, from Corollary 4.1.2, we obtain ).
If f ′ (z 0 ) ∈ σ(−∆; Ω), then, by assumption (1) and Corollary 4.1.3, we get Finally, since λ i 0 < f ′ (z 0 ) and R[1, Remark 4.1.4. Let us notice that in the above theorem the same proof works for assumption (4) replaced by assumptions Therefore, by the properties of the Leray-Schauder degree, we obtain If moreover, assumption (A.5) is fulfilled, then Φ ∈ C 2 SO(2) (H 1 (Ω)×R, R) and U is SO(2)invariant. Therefore, by the properties of the degree for SO(2)-equivariant gradient maps, we obtain and assume that f (·, 0) satisfies assumptions of Theorem 4.1.1. Then there exist closed connected sets C ± such that Proof. Repeating the reasoning from the proof of Theorem 4.1.1 we obtain Applying Theorem 2.5 we obtain the existence of closed connected sets C ± such that    1) with λ = 0 converging to some z ∈ Z 0 or there exist closed connected sets C ± such that Proof. Suppose that doesn't exist a sequence of nonconstant solutions of equation (4.2.1) with λ = 0 converging to some point in Z 0 . Then all the points z ∈ Z 0 are isolated critical points of Φ(·, 0). Repeating the reasoning from the proof of Theorem 4.1.5 we obtain We set U = B γ∞ (H 1 (Ω)) \ where f ∈ C 1 (R × R, R), f (·, λ) satisfies condition (A.1) for every λ ∈ R. Ω ⊂ R n is an open, bounded set with C 1 − -boundary. Moreover, we assume that assumption (A.5) is fulfilled.
Notice that under such an assumption . Therefore operator ∇ u Φ(·, λ ± ) is asymptotically linear at infinity and its derivative at infinity is a linear isomorphism. Thus there exists γ > 0 such that and we can define Bif(∞, [λ − , λ + ]) ∈ U(SO(2)). The following theorem is a direct consequence of Theorem 2.6.
In the next lemma show how to verify that the bifurcation index Bif(∞, [λ − , λ + ]) ∈ U(SO(2)) is nontrivial. In this lemma, for simplicity, we assume that (2)) iff at least one of the following conditions is satisfied: Corollary 4.3.1. Fix λ ± ∈ R and η > 0 as in Theorem 4.3.1. Additionally, assume that f ′ (∞, λ + ) > f ′ (∞, λ − ) and that one of the following conditions is satisfied: Then the statement of Theorem 4.3.1 holds true. It is easy to see that the analogous corollary as above holds true also if We will need the following assumption.
It is clear that under this condition we have (2)). The following lemma is a direct consequence of Lemma 4.3.1.  (2)) iff one of the following conditions holds: The next theorem is the consequence of the above lemma and Theorem 2.7.
Theorem 4.3.2. Let assumption (B.2) holds and let λ 0 , λ ± ∈ R be such that condition 4.3.4 is satisfied. Moreover, assume that Then the statement of Theorem 4.3.1 holds true and moreover C meets (∞, λ 0 ).

Examples
In this section we illustrate the abstract results proved in the previous section.
In the next lemma we show some properties of zeros of derivatives of Bessel functions.