Bifurcation at higher eigenvalues of a class of potential operators and application

Received: 21-Dec-2022, Manuscript No. PULJPAM-22-5960; Editor assigned: 23-Dec-2022, Pre QC No. PULJPAM-22-5960 (PQ); Reviewed: 06-Jan-2023 QC No. PULJPAM-22-5960; Revised: 22-Mar-2023, Manuscript No. PULJPAM-22-5960 (R); Published: 30-Mar-2023, DOI: 10.37532/PULJPAM.23.7(3).140

Citation: Mokhtari A. Bifurcation at higher eigenvalues of a class of potential operators and application. J Pure Appl Math 2023;7(3):140.

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Abstract

We study the existence of a bifurcation branch at the second and higher eigen values of a class of potential operators which possesses the Palais- Smale condition. We will also give an application of our result to a class of semi linear elliptic equations with a critical Sobolev exponent.

Keywords

Hilbert space; Bifurcation point; Potential operator; Higher eigenvalues; Boundary value problem

Introduction

In E. Tones the author consider a bifurcation at the principal eigenvalue of a class of gradient operators which possesses the Palais-Smale condition. The existence of the bifurcation branch is verified by using the Palais Smale condition [1,2].

Also the author have modified and proved the following basic bifurcation result by Krasnoselskii for gradient mappings by removing the requirement of complete continuity of ΦJ and weak continuity of Φ.

Theorem 1.1: Assume that Φ: H→R is weakly continuous and uniformly differentiable in a neighbourhood of 0 and assume that T=ΦJ:H→ H is completely continuous. Then, if T is differentiable at 0, every eigenvalue μ^*≠0 of the derivative T^J(0) is a bifurcation point for:

T (w)=μw.

We need also the following definition.

Definition 1.2: Let Φ ∈ C¹(H,R), and suppose (un) is a sequence in H satisfying Φ(u_n)→c and Φj(u_n)→0 in H^*. Then (u_n) is called a Palais-Smale sequence at level c. If every Palais-Smale sequence at level c contains a strongly convergent subsequence, then Φ is said to satisfy the Palais-Smale condition at level c, and denote by (PS)_c.

The main result proved in E. Tones is:

Theorem 1.3: Let Φ^J (u) be a linear operator which is the gradient of a C¹ functional Φ (u). Let Φ^J have a Fre´chet derivative T at the origin in H, where T is a selfadjoint, completely continuous operator [3,4]. Suppose μ1 is the largest eigenvalue (i.e., λ₁=1/μ₁ is the smallest positive characteristic value) of T. Suppose that for some ξ>0 the family of functionals:

I_λ (u)=1/2 ǁuǁ²−λΦ(u)

Satisfies the (PS)_c condition for λ₁ ξ<λ<λ₁+ξ and for c ∈ R in a neighbourhood of 0.

Then λ₁ is a bifurcation point for Φ^J.

We need also the following result

Theorem 1.4: (Spectral theorem for compact operators).

Let T be a compact, selfadjoint linear operator on a infinite dimension separable Hilbert space H. Then H admits an orthonormal basis (e_n)_n consisting of eigenvectors for T [5,6].

The infinite sequence of basis vectors (e_n)_n can be chosen such that the sequence of corresponding eigenvalues (λ_n) decreases numerically,

|λ₁| ≥ |λ₂| ≥,..., |λ_n| ≥ ... and λ_n→∞, for n→+∞.

Main result

Our main result in this paper is:

Theorem 2.1: Let Φ^J(u) be a linear operator which is the gradient of a C¹ functional Φ(u). Let Φ^J have a Frechet derivative T at the origin in H, where T is a positive selfadjoint, completely continuous operator. Suppose μ₂ is the second largest eigenvalue (i.e. λ₂=1/μ₂ is the second smallest positive characteristic value) of T. suppose that for some ξ>0 the family of functionals:

I_λ (u) = 1/2 ǁuǁ² − λΦ(u)

Satisfies the (PS)_c condition for λ₂ ξ<λ<λ₂+ξ and for c ∈ R in a neighbourhood of 0.

Then λ₂ is a bifurcation point for Φ^J.

Proof. Since T is a positive compact, self-adjoint linear operator, then T admits at least one eigenvalue, namely:

μ₁=max{|(Tu, u)|, u ∈ H, ǁuǁ=1}·

Choose a corresponding normalized eigenvector e₁∈ H.

Let Q₁={e₁}⊥ be the orthogonal complement to the 1-dimensional subspace spanned by the vector e₁. Being an orthogonal complement, Q₁ is a closed subspace of the Hilbert space. For u ∈ Q₁, we have the following computation:

(Tu,e₁)=(u,Te₁)=μ1(u,e₁)=0,

Showing that Q₁ is invariant under T in the sense that T_u ∈ Q₁ if u ∈ Q₁. Hence T can be considered as a compact, self-adjoint linear operator on the Hilbert space Q₁. Working inside Q₁, we then get a second eigenvalue for T,

μ₂=max{|(Tu,u)|, u ∈ Q₁, ǁuǁ=1}

and a normalized eigenvector e₂ ∈ Q₁ for μ₂. Clearly, μ₁ ≥ μ₂ and e₁⊥e₂.

Now, we consider the operator T|Q₁: Q₁→Q₁ and we apply theorem 1.3 for this operator.

We remark that the first eigenvalue of the operator T_|Q1 correspond to the second eigenvalue of the operator T defined on H.

Remark 2.2: One can generalize the above result to eigenvalues of higher orders.

Application

We consider the problem presented by E. Tonkes.

Let Ω⊂R^N a smooth bounded domain and solutions are sought in the Sobolev

Space H¹ 0 (Ω), endowed with norm ǁuǁ=(∫Ω |∇_u| ²)^1\2,

−au=λ(u+|u|^2∗−2u), on Ω;

u(x)=0, x ∈ ∂Ω.

Solutions to problem (3.1) correspond to critical points of the functional

J_λ (u)=1/2ǁuǁ²−λ(1/2 ∫ u²−1/2^* ∫ |u|2^*)

In Tonkes say the functional J_λ satisfies the (PS)_c for some condition on the level c by the same steps as in namely obtained the following lemma.

Lemma 3.1: For λ>0, J_λ satisfies the (PS)_c condition for c<c^*_λ=1/N S ^N/2/ λ^(N−2)/2.

The following result states that the principal characteristic value of the linear problem −au=λu), u ∈ H¹, is a bifurcation point.

Theorem 3.2: For sufficiently small r>0, there exists a solution (λ_r, u_r) to (3.1) with ǁu_rǁ=r. One has λ_r→λ0 a s r→0.

Consider the linear eigenvalue problem

−au(x)=λu(x),x ∈ Ω;

u(x)=0, x ∈ ∂Ω. (3.2)

Theorem 3.3: Equation (3.2) has a sequence of eigenvalues λ_n such that

0<λ₁<λ₂ ≤ λ₃ ≤ ... ≤ λ_n↑+∞.

Here we use the convention that multiple eigenvalues are repeated according to their multiplicity. The first eigenvalues λ₁ is simple and the corresponding Eigen functions do not change sign in Ω. Moreover, λ₁ is the only eigenvalue with this property.

We will denote by ϕ₁ the Eigen function corresponding Eigen function corresponding to λ₁, such that ϕ1(x)>0 and ǁϕ1ǁ=1.

From the above result, we can see that each characteristic value λ_n of the Dirichlet problem associated to (3.2) forms a bifurcation point for the problem (3.1). By theorem 2.1 and remark 2.2, one can generalize theorem 3.2 to the following.

Theorem 3.4: For sufficiently small r>0, there exists a solution (λ^r , u^r ) to (3.1)

with ǁu^rǁ=r. One has λ^r→λn _as r→0.

References

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