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lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

  • Peiluan Li EMAIL logo and Youlin Shang
Published/Copyright: May 8, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

Keywords: Fractional Schrodinger equations; Variational methods; Infinitely many solutions
MSC 2010: 26A33; 35A15; 35B20

1 Introduction

In this paper, we are concerned with the existence of solutions of the following fractional Schrödinger equations

(−Δ)αu+V(x)u=f(x,u)+λh(x)|u|p−2u,x∈RN, (1)

where 0<α<1,2α<N,1≤p<2,λ>0,f∈C(RN×R,R),h∈L22−p(RN). (− Δ)α is the so-called fractional Laplacian operator of order α ∈ (0,1) and can be defined pointwise for x ∈ RN by

(−Δ)αu(x)=−12∫RNu(x+y)+u(x−y)−2u(x)|y|N+2αdy

along any rapidly decaying function u of class C∞(RN), see Lemma 3.5 of [1]. It can also be characterized as by (− Δ)αu = ℱ− 1(|ξ|2αℱu), ℱ denotes the usual Fourier transform in RN. The potential V satisfies the following conditions

(V0)V∈C(RN,R),infx∈RNV(x)=V0>0 and lim|x|→∞V(x)=∞.

In [1], the authors proved that (− Δ)α reduces to the standard Laplacian − Δ as α → 1. When α = 1, the problem (1) reduces to the generalized integer order Schrödinger equation. Over the past decades, with the aid of different methods, for various conditions of the potential V and the nonlinear term f, the existence and multiplicity of nontrivial solutions for the classical Schrödinger equation have been extensively investigated.

Recently, a great attention has been focused on the study of problems involving the fractional Laplacian. Fractional calculus provide a powerful tool for the description of hereditary properties of various materials and memory processes (see [2, 3]). Fractional differential equations play important roles in the modeling of medical, physical, economical and technical sciences. For more details on fractional calculus theory, we refer the readers to the monographs of Kilbas et al. [4], Lakshmikantham et al. [5], Podlubny [6] and Tarasov [7]. Fractional differential equations involving the Rieman-Liouville fractional derivative or Caputo fractional derivative have received more and more attention.

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Laskin [8, 9] as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths, where the Feynman path integral leads to the classical Schrödinger equation, and the path integral over Lévy trajectories leads to the fractional Schrödinger equation. Recently, some researchers have investigated the following fractional Schrödinger equations (2) for the different cases of the potential V (for example, V = 1 or V = u) and the nonlinear term f (for example, f(x, u) = |u|p − 1u) under the suitable assumptions. We refer the interested readers to [10–20] and the references therein.

(−Δ)αu+V(x)u=f(x,u),x∈RN, (2)

In [12], Felmer et al. studied the existence and regularity of solutions for the fractional Schrödinger equations (2) under the famous Ambrosetti-Rabinowitz (A-R) condition, i.e., there exists θ > 2 such that 0 < θ F(x, t) < t f(x, t), where F(x,u)=∫0uf(x,τ)dτ. In [17], by virtue of some nonlinear analysis techniques, the existence of weak solutions for (2) is obtained. In [18], for the case that V = 1 and f(x, u) = f(u), the authors proved that (2) has at least two nontrivial radial solutions without assuming the A-R condition by variational methods and concentration compactness principle. In [19], using variant Fountain theorems, the author proved the existence of infinitely many nontrivial high or small energy solutions of (2). In [20], the authors obtained the existence of a sequence of radial solutions for N ≥ 2, a sequence of non-radial solutions for N = 4 or N ≥ 6, and a non-radial solution for N = 5 by the variational methods.

Some researchers also investigated the related problems, see [21–26].

In [27], by using the mountain pass theorem and Ekeland’s variational principle, the author showed that the generalized fractional Schrödinger equations (1) possess two solutions.

Motivated by the work above, in the present paper, we consider the generalized fractional Schrödinger equations (1). By the variational methods, we obtain the existence of infinitely many solutions without the A-R condition. The form of problem (1) is more general than (2). In (1), the nonlinearity involves a combination of superlinear or asymptotically linear terms and a sublinear perturbation. To our best knowledge, the problem (1) has received considerably less attention. We get the existence of infinitely many solutions for (1), which generalize and improve the recent results in the literature.

To state our results, we make the following assumptions.

(H1) Let f ∈ C(RN × R, R) . There exist constants a1,a2≥0,q∈[2,2N+4αN)⊂[2,2α∗) with a12S22+a2qSqq<12, such that

|f(x,u)|≤a1|u|+a2|u|q−1,∀(x,u)∈RN×R,

where 2α∗=2NN−2α with 2α < N, Sr is the best constant for the embedding of X ⊂ Lq(RN), the details of Sr and the definition of X will be given in the section 2.

(H2) lim|u|→∞F(x,u)|u|2=∞ uniformly in x ∈ RN and there exists r0 > 0 such that F(x, u) ≥ 0, for any x ∈ RN, u ∈ R and |u| ≥ r0, where F(x,u)=∫0uf(x,τ)dτ.

(H3)2F(x, u) < uf(x, s), ∀(x, u) ∈ RN × R.

The main results are as follows.

Theorem 1.1

Assume the hypotheses (V0), (H1)-(H3) hold. Suppose that F(x, − u) = F(x, u) for all (x, u) ∈ RN×Randh∈L22−p(RN), then there exists a constant λ0 > 0, for λ ∈ (0, λ0), such that (1) possesses infinitely many solutions.

Remark 1.2

A-R condition plays an important role in variational methods, which could guarantee boundedness of the sequence. The problem in bounded domain with A-R condition or weaker conditions was studied in many works.

However, in the unbounded domain RN, there are few papers that considered the problem without A-R condition. Furthermore, there are very few papers which replaced A-R condition by (H3) in RN. Moreover, the form of (1) is more general. Hence, our results can be viewed as extension to the related results of the fractional Schrödinger equation.

Remark 1.3

We can identify (− Δ)α with − Δ when α = 1. Hence, Theorem 1.1 is also valid for α = 1.

The rest of this paper is organized as follows. In Section 2, some definitions and lemmas which are essential to prove our main results are stated. In Section 3, we give the main results. Finally, one example is offered to demonstrate the application of our main results.

2 Preliminaries

At first, we present the necessary definitions for the fractional calculus theory and several lemmas which will be used further in this paper.

In order to establish a variational structure which enables us to reduce the existence of solutions of problem (1) to one of finding critical points of corresponding functional, it is necessary to construct appropriate function spaces.

Let us recall that for any fixed t ∈ RN and 1< q ≤ ∞,

∥u∥∞=maxt∈RN|u(t)|,∥u∥Lq(RN)=(∫RN∥u(s)∥qds)1q.

Throughout this paper, we denote by ∥u∥q the Lq-norm, for 1 < q ≤ ∞.

Let

H=Hα(RN):={u∈L2(RN):∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz<+∞}

with the inner product and the norm

〈u,v〉H=∫RN∫RN[u(x)−u(z)][v(x)−v(z)]|x−z|N+2αdxdz+∫RNu(x)v(x)dx,∥u∥H=〈u,u〉H12,

while [u]Hα=(∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz)12 is the Gagliardo(semi) norm. The space Hα(RN) can also be described by means of the Fourier transform, which can be denoted by

Hα(RN):={u∈L2(RN):∫RN(1+|ζ|2)α(|Fu(ζ)|2)dζ<+∞},

and the norm is defined as

∥u∥H=(∫RN(1+|ξ|2)α(|Fu(ξ)|2)dξ)12.

In the following, we introduce the definition of Schwartz function δ(is dense in Hα(RN)), that is, the rapidly decreasing C∞ function on RN. If u ∈ Δ, the fractional Laplacian (− δ)α acts on u as

(−Δ)αu(x)=C(N,α)P.V.∫RNu(x)−u(y)|x−y|N+2αdy=C(N,α)limε→0+∫RN∖B(0,ε)u(x)−u(y)|x−y|N+2αdy

where the symbol P.V. represents the principal value integrals, the constant C(N, α) depends only on the space dimension N and the order α, and it is explicitly given by the formula

1C(N,α)=∫RN1−cos⁡ζ|ζ|N+2αdζ.

In [1], the authors show that for all u ∈ δ,

(−Δ)α=F−1(|ζ|2αFu)

and

[u]Hα=(2C(N,α)∫RN|ζ|2α|Fu|2dζ)12.

Furthermore, from the Plancherel formula in Fourier analysis, we can easily see that

[u]Hα2=2C(N,α)∥(−Δ)α2∥22.

Hence, the norms on Hα(RN) defined below are all equivalent:

∥u∥H=(∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz+∫RN|u(x)|2dx)12,

∥u∥H=(∫RN(1+|ξ|2)α(|Fu(ξ)|2)dξ)12,∥u∥H=(∫RN|u(x)|2dx+∥(−Δ)α2∥22)12,

∥u∥H=(∫RN|u(x)|2dx+∫RN|ξ|2α|Fu(ξ)|2dξ)12.

In order to investigate the problem (1), we define the following space

X={u∈L2(RN):∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz+∫RNV(x)u(x)2dx<+∞}

with the inner product and the norm

〈u,v〉=∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz+∫RNV(x)u(x)v(x)dx,∥u∥2=〈u,u〉.

It is easy to see that X is a Hilbert space with the inner product 〈u, v〉, X ⊂ H and X ⊂ Lq(RN) for all q∈[2,2α∗) with the embeddings being continuous.

Lemma 2.1

([27]). Assume that the condition (V0) holds. Then there exists a constant c0 > 0 such that

∫RN∫RN|u(x)−u(z)|2|x−z|N+2αdxdz+∫RNV(x)u(x)2dx≥c0∥u∥H2,∀u∈H. (3)

Lemma 2.2

([27]). Assume that the condition (V0) holds. Then X is compactly embedded in X ⊂ Lq(RN) for all q∈[2,2α∗) .

Remark 2.3

Lemma 2.2 implies that Sr ∥u∥r ≤ ∥u∥, where Sr > 0 is the best constant for the embedding of X ⊂ Lq(RN).

We consider the functional φ: X → R, defined by

φ(u)=12∥u∥2−∫RNF(x,u)dx−λp∫RNh(x)|u|pdx. (4)

Then φ is continuously differentiable under the assumption (H1), and

〈φ′(u),v〉=∫RN∫RN[u(x)−u(z)][v(x)−v(z)]|x−z|N+2αdxdz+∫RNV(x)u(x)v(x)dx−∫RNf(x,u(x))v(x)dx−λ∫RNh(x)|u(x)|p−2u(x)v(x)dx, (5)

for all u, v ∈ X Hence, the critical point of φ is the solution of problem (1). Next, we only need to consider the critical point of φ.

Definition 2.4

Let (E, ∥•∥) be a Banach space and φ ∈ C1(E, R). We say φ satisfies (Cerami)C condition, if any sequence {um} ⊂ E for which φ(um) → c and ∥φ′(um)∥(1+∥um∥) → 0 as m → 0 posses a convergent subsequence, and {um} is called a(Cerami)c sequence.

From now on, we will denote (Cerami)C condition by the abbreviation (C)C condition.

Lemma 2.5

([28] Theorem 9.12]). Let (E, ∥•∥) be an infinite dimensional Banach space, X = Y ⊕ Z with dim Y < ∞. Let φ ∈ C1(X, R) be an even functional, which satisfies the (C)C condition and φ(0) = 0. In addition, if φ satisfies:

  1. There exist a, ρ > 0 such that φ|∂ Bρ ∩ Z ≥ a where Bρ = {u ∈ X :∥u∥ < ρ},

  2. For any finite dimensional subspace W ⊂ X, there exists R = R(W) such that φ(u) ≤ 0 on W ∖ BR(W).

Then, φ possesses an unbounded sequence of critical values.

3 Main results

Without loss of generality, if we take a subsequence {um}, we also use the same notation {um}.

Lemma 3.1

Assume that (V0), (H1)-(H3) hold. Then there exists a constant λ0 > 0, for any λ ∈ (0, λ0), φ satisfies the (C)c condition.

Proof

Let {um} ⊂ X be a (C)c sequence, that is

φ(un)→c,∥φ′(un)∥(1+∥un∥)→0, as n→∞, (6)

which also implies

〈φ′(un),un〉→0, as n→∞. (7)

First, we prove that {um} is bounded.

We argue it by contradiction. If {um} is unbounded in X, then there exists a subsequence {um} with ∥un∥ → ∞, as n → ∞. From (4)-(5), (H3) and the Hölder inequality, one has

φ(un)−〈φ′(un),un〉2=−∫RNF(x,un)dx−(1p−12)λ∫RNh(x)|un|pdx+12∫RNf(x,un)undx≥−(1p−12)λ∫RNh(x)|un|pdx≥−(1p−12)λ0∥h∥22−p∥un∥2p.

By (6)-(7), we can easily get that ∥un∥2 is bounded.

Also from (4)-(5), we have

〈φ′(un),un〉=∥un∥2−∫RNf(x,un)undx−λ∫RNh(x)|un|pdx.

Then, it follows from (H1) that

∥un∥2≤〈φ′(un),un〉+∫RNf(x,un)undx+λ0∥h∥22−p∥un∥2p≤〈φ′(un),un〉+a1∥un∥22+a2∥un∥qq+λ0∥h∥22−p∥un∥2p. (8)

By the Fractional Gagliardo-Nirenberg inequality ([29], corollary 2.3]) and the definition of the norm in X, we know that

∥u∥Lq(RN)≤ζsq∥u∥sq∥u∥L2(RN)1−sq, (9)

where ζ=2−απ−α2(Γ(N)Γ(N/2))αNΓ(N−α2)/Γ(N+α2) and s(12−αN)+q−s2=1.

For 0<s=(q−2)N2α<q∈[2,2N+4αN)⊂[2,2α∗), it is easy to see that

0<s<2. (10)

Then it follows (8)-(10) that

1=∥un||2||un||2≤〈φ′(un),un〉∥un∥2+a1∥un∥22∥un∥2+a2∥un∥qq∥un∥2+λ0∥h||22−p∥un∥2p||un∥2≤〈φ′(un),un〉∥un∥2+a1∥un∥22∥un∥2+a2ζs∥un∥s||un∥2q−s||un||2+λ0∥h||22−p∥un∥2p||un∥2→0, as n→∞.

This is a contradiction. Hence, we know that {un} is bounded in X.

Next we prove φ satisfies the (C)c condition.

For any {un} ⊂ X being a (C)c sequence, from the boundedness of {un}, we know there exists a weakly convergent subsequence {un} such that un ⇀ u weakly in X. From Lemma 2.2, we can obtain that un → u strongly in Lq(RN) for q∈[2,2N+4αN).

Then we prove that un → u in X.

From (4)-(5), we have

∥un−u∥2=〈φ′(un)−φ′(u),un−u〉+∫RN[f(x,un(x))−f(x,u(x))](un−u)dx+λ∫RNh(x)|un−u|pdx. (11)

It is easy to see that

〈φ′(un)−φ′(u),un−u〉→0, as n→∞. (12)

Based on the fact that h∈L22−p(RN),un→u in Lq(RN) and Hölder inequality, one has

λ∫RNh(x)|un−u|pdx≤λ0∥h∥22−p∥un∥2p→0, as n→∞. (13)

From (H1) and the Hölder inequality, we can obtain

∫RN[f(x,un(x))−f(x,u(x))](un−u)dx≤∫RN|f(x,un(x))−f(x,u(x))||un−u|dx≤∫RNa1(u+un)+a2(|u|q−1+|un|q−1)|un−u|dx≤a1(∥un∥2+∥u∥2)∥un−u∥2+a2(∥un∥qq−1+∥u∥qq−1)∥un−u∥q→0, as n→∞. (14)

It follows from (11)-(14) that ∥un − u ∥2 → 0, which shows that un → u in X. Hence, φ satisfies the (C)c condition.

We complete the proof of Lemma 3.1.□

Let {ej} be a total orthonormal basis of X. We define

Xj:=span{ej},Yk:=⊕j=1kXj and Zk=⊕j=k+1∞Xj,¯k∈N.

Clearly, X = Yk ⊕ Zk with dim Yk < ∞.

Lemma 3.2

Assume that (V0), (H1)-(H3) hold. Then there exist constants a, ρ > 0 such that φ|∂ Bρ ∩ Zk ≥ a, where Bρ = {u ∈ X :∥u∥ < ρ}.

Proof

From (H1), (4) and Hölder inequality, for any u∈Zk,q∈[2,2N+4αN), we have

φ(u)=12∥u∥2−∫RNF(x,u)dx−λp∫RNh(x)|u|pdx≥12∥u∥2−a12∥un∥22−a2q∥un∥qq−λ0∥h∥22−p∥un∥2p12∥u∥2−a12S22∥un∥2−a2qSqq∥un∥q−λ0∥h∥22−p∥un∥2p.

We denote c=12−(a12S22+a2qSqq), then (H1) implies c > 0. Let

αk=supu∈Zk,||u||=1∥u∥r, for r∈[1,2α∗).

By a similar proof to the Lemma 3.2 of [19], we can deduce that αk → 0 as k → ∞. Assume 0 < ρ≤ 1, then we have

φ|∂Bρ∩Zk≥c∥u∥2−λ0∥h∥22−pαkp∥u∥p=∥u∥p[c∥u∥2−p−λ0∥h∥22−pαkp].

Then it is easy to see that we can choose constant ρ ∈ (0,1] such that φ|∂ Bρ ∩ Zk > 0, as k →∞, which implies the conclusion of Lemma 3.2.□

Lemma 3.3

Assume that (V0), (H1)-(H3) hold. Then for any finite dimensional subspace W ⊂ X, there exists R = R(W) such that φ(u) ≤ 0 on W ∖ BR(W).

Proof

First, we prove that for any finite dimensional subspace W ⊂ X and ∥u∥ →∞, u ∈ W, there holds φ(u) → − ∞.

On the contrary, we assume that for some sequence {un} ⊂ W with ∥un∥ →∞, there exists a positive constant M > 0 such that

φ(un)≥−M. (15)

Let vn=un||un||, then ∥vn∥ = 1. From the boundedness of vn, there exits a weakly convergent subsequence vn such that vn ⇀ v weakly in X. Then for the finite dimensional subspace W ⊂ X, we know that vn → v strongly in W. By the equivalence of finite dimensional spaces, we can get that vn → v a.e. in RN, which also implies ∥v∥ = 1. Let V : = {x ∈ RN:v(x) ≠ 0}, then we know the measure of the set V is positive, i.e., meas(V) > 0. Hence, for x ∈ V, from un = ∥un∥vn we can deduce

|un|→∞, as n→∞. (16)

For any 0 < x1 < x2, we denote Ωn(x1, x2) = {x ∈ RN:x1 ≤ |un| < x2}, which implies Ωn(x1, x2) ⊂ V. Then for n large enough, it easy to see that χΩn(r0,∞)(x) = 1, where χΩn(r0,∞) is the characteristic function on Ωn(r0,∞), r0 > 0 is given in (H2). Hence, we have

χ Ω n ( r 0 , ∞ ) ( x ) v n → v ,  as  n → ∞ ,  for  a . e . x ∈ Ω n ( r 0 , ∞ ) . (17)

On one hand, from (4), (15) and Remark 2.3, for any λ ∈ (0, λ0), we have

limn→+∞∫RNF(x,un)dx∥un∥2=limn→+∞12∥un∥2−λp∫RNh(x)|un|pdx−φ(un)∥un∥2≤limn→+∞12∥un∥2+λ0S2p∥h||22−p∥u∥p+M∥un||2=12. (18)

On the other hand, (H2) shows there exists r0 > 0 such that F(x, u) ≥ 0, for |u| ≥ r0. From (16), we know that

F(x,un)≥0,fornlargeenough. (19)

Then it follows from (17), (19), (H2) and Fatou’s Lemma that

limn→∞∫RNF(x,un)dx∥un∥2≥limn→∞∫VF(x,un)dx∥un∥2=limn→∞∫VF(x,un)vn2dxun2≥liminfn→∞⁡∫VF(x,un)vn2un2dx≥∫Vliminfn→∞⁡F(x,un)vn2un2dx≥∫Ωn(r0,∞)liminfn→∞⁡F(x,un)vn2un2dx=∫Ωn(r0,∞)liminfn→∞⁡F(x,un)vn2|un|2χΩn(r0,∞)2(x)vn2dx=∞,

which is a contradiction with (18). Therefore φ(u) → −∞ for ‖u‖ → ∞, u ∈ W. Hence, we can easily choose R = R(W) such that φ(u) ≤ 0 on W \ BR(W). Then we complete the proof.□

Proof of Theorem 1.1

Let Y = Yk, Z = Zk, then X = Y ⊕ Z with dim Y < ∞. From the condition that F(x,−u)=F(x,u)andh∈L22−p(RN), we know φ is even and φ(0) = 0. Lemma 3.1-3.3 imply that φ satisfies other conditions of Lemma 2.5. Consequently, we can deduce that φ possesses an unbounded sequence of critical values, which are the solutions of the fractional Schrödinger equation (1).□

Finally, we give one example to illustrate the usefulness of our main result. Consider the following fractional Schrödinger equations.

Example 3.4

(−Δ)12u+(1+x2)u=8S8/38/3(sin2⁡x)u5/39+ln⁡(1+|sin⁡x2|)e|x|(1+x2)|u|−1u,x∈R2. (20)

Obviously, α=1/2,N=2,p=1,λ=1,f(x,u)=8S8/38/3(sin2⁡x)u5/39 is continuous, V(x) = 1 + x2 and h(x)=ln⁡(1+|sin⁡x2|)e|x|(1+x2) is a L2 integrable function.

First, we can see that 2α=1<N=2,F(x,u)=S8/38/3(sin2⁡x)u8/33, then we have

|f(x,u)|=|8S8/38/3(sin2⁡x)u5/39|≤S8/38/3|u|83−1,

with 83=q∈[2,3)⊂[2,2α∗=4),a1=0,a2=S8/38/3anda12S22+a2qSqq=38<12, which shows that (H1) of Theorem 1.1 holds.

From the fact that 2F(x,u)=2S8/38/3(sin2⁡x)u8/33<8S8/38/3(sin2⁡x)u8/39=uf(x,u), we can verify (H3) of Theorem 1.1 is also satisfied.

It is also easy to check that the hypotheses (V0), (H2) and other conditions of Theorem 1.1 hold. Then all the conditions in Theorem 1.1 are satisfied. In virtue of Theorem 1.1, we conclude that (20) possesses infinitely many solutions.

Acknowledgement

The authors thank the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper.We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contribute to the perfection of the paper.

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Received: 2017-2-13
Accepted: 2017-3-22
Published Online: 2017-5-8

© 2017 Li and Shang

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0053
Keywords for this article
Fractional Schrodinger equations; Variational methods; Infinitely many solutions
Creative Commons
BY-NC-ND 3.0

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  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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