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Transactions on Engineering and Computing Sciences - Vol. 12, No. 4

Publication Date: August 25, 2024

DOI:10.14738/tecs.124.17316.

Jin, Z. (2024). Index Theory and Multiple Solutions for a Fractional Laplacian Equation with an Asymptotically Linear Term.

Transactions on Engineering and Computing Sciences, 12(4). 45-58.

Services for Science and Education – United Kingdom

Index Theory and Multiple Solutions for a Fractional Laplacian

Equation with an Asymptotically Linear Term

Ziqing Jin

ABSTRACT

In this paper, we investigate the existence and multiplicity of solutions for the

asymptotically linear fractional Laplacian equation with homogeneous Dirichlet

boundary conditions. We will construct an index theory for the associated linear

fractional Laplacian equation. Using results from critical point theory, we show how

the behavior of the nonlinearity near zero and at infinity affects the number of

solutions via the index.

Keywords: Fractional Laplacian equation, Homogeneous Dirichlet boundary condition,

Index theory, Variational techniques.

INTRODUCTION AND MAIN RESULTS

The goal of this paper is to establish the existence and multiplicity of solutions to the following

nonlocal elliptic equation

(−∆)

su = f(x, u) in Ω , (1.1)

u = 0 in RN\Ω ,

where s ∈ (0,1) is fixed, N > 2s, Ω ⊂ RN is an open bounded set with Lipschitz boundary. Here,

(−∆)

is the fractional Laplace operator, which (up to normalization factors) is defined as

−(−∆)

su(x) = ∫

u(x+y)+u(x−y)−2u(x)

RN |y|N+2s

dy, x ∈ RN.

The nonlocal equations have been experiencing impressive applications in different subjects,

such as the thin obstacle problem, phase transitions, stratified materials, anomalous diffusion,

crystal dislocation, soft thin films, semipermeable membranes and flame propagation,

conservation laws, ultrarelativistic limits of quantum mechanics, quasigeostrophic flows,

multiple scattering, minimal surfaces, materials science, water waves, elliptic problems with

measure data, optimization, finance, etc. See [24] and the references therein.

In the literature, many papers are devoted to the study of the non-local fractional Laplacian

equation (1.1) with nonlinearities having subcritical or critical growth, see [2] [3], [4], [5],

[6][7] [8] [22][23][24],[26] [27] [28] [29] [30] and references therein. In this paper, we assume

that the nonlinearity f is continuous in Ω̅ × R and verify the following conditions:

(f0) f(x, u) = B0(x)u + f1(x, u) , with B0 a continuous function in Ω̅ and f1(x, u) = o(|u|)

uniformly in x as |u| → 0.

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Transactions on Engineering and Computing Sciences (TECS) Vol 12, Issue 4, August - 2024

Services for Science and Education – United Kingdom

(f∞) f(x, u) = B∞(x)u + f2(x, u) , with B∞ a continuous function in Ω̅ and f2(x, u) = o(|u|)

uniformly in x as |u| → ∞.

Conditions (f0) and (f∞) are referred to in the literature as asymptotically linear for the

nonlinearity at the origin and at infinity. A quantitative way to measure the difference between

B0 and B∞ is given by the index, which was introduced in [9][12][17] [18] [19] [20] [33] [34]

by Amann, Zehnder, Ekeland and Long, and developed in [10] [11] [16] [31] for the study of

Hamiltonian system in relation with Morse theory and critical point theory. We refer to the

books of Ekeland [13] and Long [21] for a more detailed account of the concept. We think that

a natural question is whether or not these index theories may be adapted to the fractional

Laplacian equation Eq (1.1). The aim of this paper is to consider extending the index theory to

fractional Laplacian equation Eq (1.1) (see Section 2.2) and show how the behavior of the

nonlinearity f at the origin and at the infinity affects the number of solutions.

In order to state our main result of this paper, we briefly introduce the index theory for the

linear fractional Schrödinger equation. For any function B(x) continuous in Ω , the index is a

pair of integers, denoted by (i(B), v(B)) where v(B) is the dimension of the solution space of

(−∆)

su − B(x)u = 0 (see Lemma 2.3), and i(B) is the dimension of the eigenvector space

associated with all the negative eigenvalues (see Lemma 2.4). Moreover, i(B) is nondecreasing

with respect to B.

Our results read as follows.

Theorem 1.1. Let (f0) and (f∞) satisfy v(B0) = 0 and v(B∞) = 0 . If f(x, u) is odd in u, then Eq.

(1.1) has at least |i(B∞) − i(B0)| pairs of solutions.

Remark 1.1. (1) It is worth pointing that the condition v(B∞) = 0 means the problem (1.1) is

non-resonant at infinity. For the resonant case, we refer to [14].

(2) Consider the following eigenvalue problem

(−∆)

su = λu in Ω, (1.2)

u = 0 in RN\Ω.

Let the eigenvalues of this problem be denoted by

λ1 < λ2 < λ3 ⋯

counting their multiplicity. Let dim(λi) denote the dimension of the eigenvector space associate

with the eigenvalue λi

. Set dk = ∑ dim(λi)

i=1 and d0 = 0. If λm < B0(x) < λm+1, λn < B0(x) <

λn+1, ∀x ∈ Ω, we have

i(B0) = dm, v(B0) = 0, i(B∞) = dn, v(B∞) = 0.

And Eq. (1.1) has at least |dm − dn

| pairs of nontrivial solutions.

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Jin, Z. (2024). Index Theory and Multiple Solutions for a Fractional Laplacian Equation with an Asymptotically Linear Term. Transactions on

Engineering and Computing Sciences, 12(4). 45-58.

URL: http://dx.doi.org/10.14738/tecs.124.17316

This paper is organized as follows. In Section 2, we first introduce a variational setting for (1.2)

and show the associated Euler-Lagrange functional. Then we construct an index theory for the

fractional Schrödinger equations. Finally, in Section 3, we give the proof of Theorem 1.1.

PRELIMILARIES

Variational Settings

We first recall some definitions on the spaces X and X0. In the sequel we set Q = R2N\Λ where

Λ = (CΩ) × (CΩ) ⊂ R2N,

and CΩ = RN\Ω. The space X is endowed with the norm defined as

‖u‖X = ‖u‖L

(Ω) + (∫

|u(x) − u(y)|

|x − y|N+2s

dxdy)

And

X0 = {u ∈ X: u = 0 a.e. in RN\Ω}. (2.1)

Throughout this paper we use the following norm

‖u‖X0 = (∫

|u(x)−u(y)|

Q |x−y|N+2s

dxdy)

2 (2.2)

which is equivalent to the usual one defined as in (2.1) (see [29]).

From [27] [30], we have

Lemma 2.1. X0 embeds continuously into L

(Ω) for all p ∈ [1,

N−2s

], and compactly into L

(Ω)

for all p ∈ [1,

N−2s

On X0 we define the functional

I(u) =

∫

(u(x)−u(y))

Q |x−y|N+2s dxdy −

∫ F(x, u)

dx. (2.3)

Our hypotheses on f imply that f verifies the subcritical growth condition:

f(x, u) ≤ a0(1 + |u|

p−1

) for a.e. x ∈ Ω, ∀u ∈ R, (p ∈ (1,

N−2s

)).

And a standard argument shows that I ∈ C

(X0, R) and critical points of I are solutions of (1.1).

Index Theory for Linear Fractional Laplacian Equation