Multiplicity of Homoclinic Solutions for a Class of Nonperiodic Fourth-Order Differential Equations with General Perturbation

Abstract

In this paper, we investigate a class of nonperiodic fourth-order differential equations with general perturbation. By using the mountain pass theorem and the Ekeland variational principle, we obtain that such equations possess two homoclinic solutions. Recent results in the literature are generalized and significantly improved.

Infinitely Many Homoclinic Solutions for Fourth Order p-Laplacian Differential Equations

“…They are known in phase transitions models as ground states or pulses (see [1]). The existence of homoclinic and heteroclinic solutions of fourth-order equations is studied by various authors (see [2][3][4][5][6][7][8][9][10][11][12] and references therein). Sun and Wu [4] obtained existence of two homoclinic solutions for a class of fourth-order differential equations:…”

“…Yang [8] studies the existence of infinitely many homoclinic solutions for a the fourth-order differential equation:…”

Infinitely many homoclinic solutions for fourth-order differential equations with locally defined potentials

“…This equation arises from some problems associated to mathematical models for the study of pattern formation in physics and mechanics, for example, the well-known extended Fisher-Kolmogorov equation proposed by Coulet, Elphick and Repeaux [4], in the study of phase transitions, the fourth-order elastic beam equation in describing a large class of elastic deflection [13], the Swift-Hohenberg equation which is a general model for patternforming process derived in [5] to describe vandom thermal fluctuations in the Boussinesque equation and the propagation of lazers [7]. With the aid of variational methods and the critical point theory, the existence and the multiplicity of homoclinic solutions for (F ) have been extensively investigated in the literature over the past years, see, e.g., [1,2,3,8,9,10,15,11,12,14,16,17,18,19,20] and the references therein. Many results are on the existence of homoclinic solutions to equation (F ) when a(x) and f (x, u) are independent of x or periodic in x; see [1,2,3,8,14] and the references cited therein.…”

“…After the work of Li [9], there are some results concerning the nonperiodic case; E-mail address: m timoumi@yahoo.com. [9,10,15,11,12,16,17,18,19,20] and the references therein. For this case, function a plays an important role.…”

Multiple Homoclinic Solutions for a Class of Superquadratic Fourth-order Differential Equations

“…In this kind of problems, the function a plays an important role. Compared with the case of a(x) and f (x, u) being periodic in x, there is less literature available for the case where a(x) and f (x, u) are nonperiodic in x, see [9][10][11][12][13][17][18][19][20]. We notice that, for the case that equation (1.1) is not periodic, to obtain the existence of homoclinic solutions, the following coercive condition on a is often needed: (A 0 ) a : R −→ R is a continuous function, and there exists a constant r 0 such that…”