Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 52, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.52 EXISTENCE, UNIQUENESS AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS MEIQIANG FENG, YICHEN LU Abstract. We study the existence of nontrivial weak solutions for biharmonic equations with Navier and with Dirichlet boundary conditions. This is done by using critical point theory for even functionals, and the theory of strongly monotone operators. Also we analyze the existence of infinitely many weak solutions. This is probably the first time that the theory of strongly monotone operator is used to study biharmonic equations. 1. Introduction Let Ω denote a smooth bounded domain in RN (N > 4). We study the biharmonic problems ∆2u = f(x, u) in Ω, u = ∆u = 0 on ∂Ω, (1.1) and ∆2u = f(x, u) in Ω, u = ∂u ∂ν = 0 on ∂Ω, (1.2) where ∆2u = ∆(∆u) denotes the biharmonic operator, and the nonlinearity f satisfies the Carathéodory conditions. More assumptions of f will be specified later on in the explicit state- ments of the theorems. For problem (1.1), we say a function u ∈ H2(Ω) ∩H1 0 (Ω) is a weak solution of (1.1) if∫ Ω ∆u ·∆v dx = ∫ Ω f(x, u)v dx for all v ∈ H2(Ω) ∩H1 0 (Ω), (1.3) where H2(Ω) ∩H1 0 (Ω) denotes the Hilbert space, endowed with the scalar product (u, v)2 = ∫ Ω ∆u∆vdx. This product induces the norm ∥u∥ = ∥∆u∥L2(Ω), which is equivalent to the standard norm of H2(Ω), see[42, Remarks 2.1 and 2.2]. For problem (1.2), we say a function u ∈ H2 0 (Ω) is a weak solution of (1.2) if∫ Ω ∆u ·∆v dx = ∫ Ω f(x, u)v dx for all v ∈ H2 0 (Ω), (1.4) where H2 0 (Ω) denotes the standard Sobolev space with the norm ∥∆u∥L2(Ω), see Corollary 9.10 of Gilbarg and Trudinger [24]. 2020 Mathematics Subject Classification. 35J35, 35J40. Key words and phrases. Biharmonic equation; Carathéodory conditions; monotone mapping; mountain pass lemma; existence, uniqueness and multiplicity. ©2025. This work is licensed under a CC BY 4.0 license. Submitted February 4, 2025. Published May 24, 2025. 1 2 M. FENG, Y. LU EJDE-2025/52 Such problems can describe static deflection of a bending beam [32], traveling waves in sus- pension bridges [12] and other physical applications. For instance, when we consider the clamped plate problem ∆2u = f in Ω, u = ∂u ∂ν = 0 on ∂Ω, we want to find out whether the positivity of the datum implies the positivity of the solution. Or, in a physical sense, does upwards pushing of a clamped plate generate upwards bending? Because of their profusion of applications and beautiful theory, biharmonic problems have drawn the attention of many mathematicians and have become a subject of current interest, see [27, 14, 38, 15, 19, 21, 20, 46, 49, 35, 22, 45, 11, 18, 39, 40, 48, 52, 54, 51, 53, 16, 33, 50, 36, 28, 29, 30, 31] and the references cited therein. Some special situations of (1.1) and (1.2) have been explored. For example, Abid and Baraketin [1] demonstrated the existence of singular solution for problem (1.1) when f(x, u) = up. Assuming that Σ is a compact submanifold of Ω without boundary of dimension (N −m) and 4 < m < N . They verified that problem (1.1) admits at least one solution which is singular on Σ when p > m m−4 and close enough to this value. In [23], Gazzola, Grunau, and Squassina studied problem (1.1) and (1.2) when f(x, u) = λu + |u|2∗−2u, where λ ≥ 0 and 2∗ = 2N N−4 denotes the critical Sobolev exponent for the embeddingH2(RN ) ↪→ L2∗(RN ). They analyzed, by a decomposition method and a careful application of concentration compactness lemmas, the existence of nontrivial solutions and nonexistence of positive solutions for these problems. In [6], when f(x, u) = λ|u|q−2u+|u|2∗−2u (where λ > 0 is a parameter and 1 < q < 2), Bernis, Garcia-Azorero and Peral proved that there exists λ0 > 0 such that the above problems admit infinitely many solutions for 0 < λ < λ0. When f(x, u) = λu + |u|2∗−2u + g(x) (where λ ∈ R is a given constant and g(x) is a given function), Deng and Wang [17] considered the existence of multiple solutions for problem (1.2) via Mountain Pass Lemma and the Ekeland’s variational principle. Liu and Wang [37] applied a variant version of Mountain Pass Lemma to verify the existence and nonexistence of positive solution for problem (1.1) and problem (1.2) when f(x, u) satisfies the fundamental condition f(x, u) ∈ C(Ω̄,R) and other appropriate conditions. Recently, when f(x, u) = f(u), Feng [19] respectively studied the existence of positive solution for problem (1.1) by using a fixed point theorem on cone, the uniqueness and approximation of positive radial solution to problem (1.1) via iterations of the solution and the multiplicity of positive radial solution to problem (1.1) by employing index theory of fixed points for completely continuous operators. However, the theory of strongly monotone operators has been barely touched on in the literature of biharmonic equations. As one would expect, the main difficulty in using such technique lies in the constructing of strongly monotone operators. In this article, we will overcome the difficulty by using the Riesz theorem of bounded linear functionals in Hilbert space. It should be pointed out here that the use of the theory of strongly monotone operators in the present article will be the ‘starting point’ for such techniques. In this article, we study the existence and multiplicity of nontrivial weak solutions for problems (1.1) and (1.2) by applying the Mountain Pass Lemma and the critical point theory for even functionals. Here, we extend the study of Gazzola, Grunau, and Squassina [23], Bernis, Garcia- Azorero and Peral [6], and Deng and Wang [17] from the special cases to a more general case of f . In addition, comparing with Gazzola, Grunau, and Squassina [23], Bernis, Garcia-Azorero and Peral [6], Deng and Wang [17] and Liu and Wang [37], the uniqueness of weak solution is also considered. This is probably the first time that the theory of strongly monotone operator is to be used to deal with the uniqueness of weak solution for biharmonic equations. This article is organized as follows. In Section 2, we review some definitions and lemmas of Nemytskii operators and monotone mappings, which will be used in the subsequent sections. Section 3 is devoted to analyzing the uniqueness of weak solution to problems (1.1) and (1.2). The results of nontrivial weak solution will be stated and proved in Section 4. In section 5, we will prove that problems (1.1) and (1.2) admit infinitely many nontrivial weak solutions. EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 3 2. Preliminaries In this section, we review some definitions and lemmas of Nemytskii operators and monotone mappings. Let Ω be a measurable subset of RN and 0 < meas Ω ≤ ∞. For x ∈ Ω, −∞ < u <∞, define a Nemytskii operator (Fu)(x) = f(x, u(x)), where f(x, u) meets the Carathéodory conditions, that is, f(x, u) is measurable in u for all fixed x, and is continuous in x for almost all u. We refer to Brezis-Browder [7] and Krasnosel’skii [34] for an exhaustive treatment of the proper- ties for Nemytskii operators. Guo [25] discussed some properties for Nemytskii operator in Orlicz spaces. We refer the reader to [4, 13, 5] for recent developments and applications of Nemytskii type operators. The following properties of Nemytskii operators in Lp space can be found in Krasnosel’skii [34]. Lemma 2.1. Suppose that F maps Lp1(Ω) (p1 ≥ 1) into Lp2(Ω) (p2 ≥ 1), that is Fϕ(x) ∈ Lp2(Ω), for all ϕ ∈ Lp1(Ω). Then F must be continuous. Lemma 2.2. Suppose that F maps Lp1(Ω) (p1 ≥ 1) into Lp2(Ω) (p2 ≥ 1). Then F must be bounded. Lemma 2.3. The operator F maps Lp1(Ω) (p1 ≥ 1) into Lp2(Ω) (p2 ≥ 1) when and only when there are d > 0 and c(x) ≥ 0 with c(x) ∈ Lp2(Ω) such that |f(t, u)| ≤ c(x) + d|u| p1 p2 x ∈ Ω, −∞ < u < +∞. (2.1) In the following, we review some known results of monotone mappings, which can be found in Browder [8], Chang [10], and Minty [41]. Suppose that E is a real Banach space, E∗ is its conjugate space. For every u ∈ E, f ∈ E∗, write (f, u) = f(u). We also assume that 2E ∗ stands for the set of all subsets of space E∗. Definition 2.4 ([10, Def. 2.5.1]). Let D̂ ⊂ E. A set-valued mapping T of D̂ into 2E ∗ is called to be monotone if for all u, v in D̂ we have (Tu− Tv, u− v) ≥ 0. (2.2) A single-valued monotone mapping is said a monotone operator. It is widely known that the requirement of the continuity for monotone operators is very weak. Definition 2.5 ([10, Def. 2.5.2]). A map T : D̂ → E∗ is said hemi-continuous at u0 ∈ D̂, if fo rall v ∈ E and all tn ↓ 0 with (u0 + tnv) ∈ D̂, it holds that T (u0 + tnv)⇁ A(u0) The map T is demi-continuous at u0 ∈ D̂, if for all {un} ⊂ D̂, un → u0 indicates that T (un)⇀ u0. Lemma 2.6 ([9, 41]). Suppose that T is a set-valued monotone mapping of E into 2E ∗ . Then, T is locally bounded in u0 for all u0 ∈ intD(T ), the interior of set D(T ). Lemma 2.7 ([9, 41]). Suppose that E is a reflexive Banach space and T : E → E∗ is hemi- continuous and monotone. If additionally T is coercive: lim ∥u∥→+∞ (Tu, u) ∥u∥ = +∞, then T is surjective. 4 M. FENG, Y. LU EJDE-2025/52 Lemma 2.8 ([9, 41]). Suppose that E is a reflexive Banach space and T : E → E∗ is hemi- continuous, and satisfies (Tu− Tv, u− v) ≥ α(∥u− v∥)∥u− v∥, ∀u, v ∈ E, (2.3) where α(0) = 0, α(t) > 0 for all t > 0, limt→+∞ α(t) = +∞. Then T is surjective, and T maps E into E∗ injectively, that is, for every f ∈ E∗, there exists exactly one solution u in E of the equation Tu = f . Definition 2.9 ([41]). Let H be a real Hilbert space, and that T be a continuous strongly monotone operator, i.e., there is c > 0 so that (Tu− Tv, u− v) ≥ c∥u− v∥2|, ∀u, v ∈ H. Lemma 2.10 ([9],FEB,GJM). Suppose that all the conditions of Lemma 2.8 hold and α(t) is continuous on (0,+∞). If T is the gradient of a functional f , then the following conclusions hold: (i) Tx = θ admits a unique solution x∗; (ii) f admits a lower bound in E, letting d = infx∈E f(x), then f(x∗) = d, f(x) > f(x∗), ∀x ̸= x∗; (iii) If xn ∈ E such that limn→∞ f(xn) = d (xn is a minimizing sequence), then ∥xn − x∗| → 0 (n→ ∞). 3. Uniqueness of nontrivial solutions In this section, we use the following assumptions on f : (A1) f(x, u) Carathéodory conditions for x ∈ Ω and −∞ < u < +∞), and for fixed x ∈ Ω, f(x, u) is a decreasing function, that is f(x, u1) ≥ f(x, u2) when u1 < u2; (A2) There exists 0 < σ ≤ N+4 N−4 such that |f(x, u)| ≤ a(x) + b|u|σ, a(x) ∈ L 2N N+4 , b > 0. (3.1) Next, we discuss the uniqueness of nontrivial weak solutions for problem (1.1) and problem (1.2) by using Lemma 2.10. Theorem 3.1. Suppose that (A1) and (A2) hold. Then problem (1.1) admits a unique nontrivial weak solution u∗ ∈ H2(Ω) ∩H1 0 (Ω), and the following functional on H2(Ω) ∩H1 0 (Ω), ψ(u(x)) = ∫ Ω [1 2 ∆u(x) ·∆u(x)− F (x, u(x)) ] dx (3.2) admits a lower bound, where F (x, u) = ∫ u 0 f(x, v)dv. Moreover, if there exists un(x) ∈ H2(Ω) ∩H1 0 (Ω) such that lim n→∞ ψ(un(x)) = d = inf un∈H2(Ω)∩H1 0 (Ω) ψ(un(x)), then ∥un − u∗∥ → 0 (n→ ∞). Proof. Let α(u, v) = ∫ Ω [∆u ·∆v − f(x, u)v]dx, ∀u, v ∈ H2(Ω) ∩H1 0 (Ω). (3.3) We first prove that α(u, v) iswell defined. In H2(Ω) ∩H1 0 (Ω), the scalar product and norm can be taken as ∥u∥ = ∥u∥L2(Ω) = (∫ Ω |∆u|2dx )1/2 , (u, v)2 = ∫ Ω ∆u ·∆vdx. EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 5 Since N > 4, it follows from embedding theorem that W 2,2(Ω) ↪→ L 2N N−4 (Ω). Hence, for u, v ∈ W 2,2(Ω), we have u, v ∈ L 2N N−4 (Ω), and there exists a constant c > 0 such that ∥u∥ 2N N−4 ≤ c∥u∥2,2, where ∥ · ∥2,2 denotes the norm of W 2,2(Ω). Since H2(Ω) ∩H1 0 (Ω) ⊂W 2,2(Ω), we have ∥u∥ 2N N−4 ≤ c∥u∥. (3.4) On the other hand, by (A2), the Nemytskii operator Fu(x) = f(x, u(x)) maps L 2N N−4 into L 2N N+4 , which is continuous and bounded. This indicates that∫ Ω f(x, u)vdx exits for u, v ∈ H2(Ω) ∩H1 0 (Ω)). So α(u, v) makes sense for u, v ∈ H2(Ω) ∩H1 0 (Ω), and |α(u, v)| ≤ ∥u∥ ∥v∥+ ∥Fu∥ 2N N+4 ∥v∥ 2N N−4 ≤ (∥u∥+ c∥Fu∥ 2N N+4 )∥v∥, (3.5) which shows α(u, ·) is a bounded linear functional in H2(Ω)∩H1 0 (Ω) for fixed u ∈ H2(Ω)∩H1 0 (Ω). Therefore, by the Riesz theorem of bounded linear functionals in Hilbert space, there exists a unique w ∈ H2(Ω) ∩H1 0 (Ω) such that α(u, v) = (w, v)2, ∀v ∈ H2(Ω) ∩H1 0 (Ω). (3.6) Let Tu = w. Then T : H2(Ω) ∩H1 0 (Ω) → H2(Ω) ∩H1 0 (Ω), and α(u, v) = (Tu, v)2, ∀u, v ∈ H2(Ω) ∩H1 0 (Ω). (3.7) Because of the density of C∞ 0 (Ω) in H2(Ω) ∩ H1 0 (Ω), u ∈ H2(Ω) ∩ H1 0 (Ω) is a weak solution of problem (1.1) when and only when Tu = θ. Therefore, we need to demonstrate that Tu = θ admits a unique solution in H2(Ω) ∩H1 0 (Ω). In the following we verify that all conditions of Lemma 2.10 are satisfied. We first verify the continuity of T . Suppose that ∥un − u∥ → 0 (un, u ∈ H2(Ω)∩H1 0 (Ω)). Then it follows from (3.4) that ∥un − u∥ 2N N−4 → 0. Hence, by (3.4), (3.7) and the continuity of F, we obtain ∥Tun − Tu∥ = sup ∥v∥=1 (Tun − Tu, v)2 = sup ∥v∥=1 [α(un, v)− α(u, v)] ≤ sup ∥v∥=1 [∣∣ ∫ Ω [∆(un − u) ·∆vdx ∣∣+ ∣∣ ∫ Ω [f(x, un)− f(x, u)]vdx ∣∣] ≤ sup ∥v∥=1 [ ∥un − u∥ · ∥v∥+ ∥Fun − Fu∥ 2N N+4 · ∥v∥ 2N N−4 ] ≤ ∥un − u∥+ c∥Fun − Fu∥ 2N N+4 → 0 as n→ ∞. This indicates that T is continuous. Next, we declare T is strongly monotonic. Indeed, by using that f(x, u) is a decreasing function for fixed x ∈ Ω, we obtain (Tu− Tv, u− v)2 = α(u, u− v)− α(v, u− v) = ∫ Ω [ ∆(u− v) ·∆(u− v)− [f(x, u)− f(x, v)](u− v)dx ] ≥ ∫ Ω ∆(u− v) ·∆(u− v)dx = ∥u− v∥2L2(Ω) = ∥u− v∥2. This indicates that (2.3) is correct, where α(t) = t is a continuous function. So, Lemma 2.8 yields that Tu = θ admits a unique solution u∗. 6 M. FENG, Y. LU EJDE-2025/52 Finally, we demonstrate that T is the gradient of functional ψ defined in (3.2): T = gradψ. On the one hand, for u ∈ H2(Ω) ∩H1 0 (Ω), it follows from (3.2) that |ψ(u)| = ∣∣ ∫ Ω [1 2 ∆u ·∆u− F (x, u) ] dx ∣∣ = ∣∣ ∫ Ω [1 2 ∆u ·∆u− f(x, τu)u ] dx ∣∣ ≤ 1 2 ∥u∥2L2(Ω) + ∥f(x, τu)∥ 2N N+4 · ∥u∥ 2N N−4 = 1 2 ∥u∥2 + ∥f(x, τu)∥ 2N N+4 · ∥u∥ 2N N−4 < +∞, for 0 ≤ τ(x) ≤ 1, which indicates that ψ(u) is well defined for u ∈ H2(Ω) ∩H1 0 (Ω). On the other hand, for u, h ∈ H2(Ω) ∩H1 0 (Ω), we have ψ(u+ h)− ψ(u)− (Tu, h)2 = ψ(u+ h)− ψ(u)− α(u, h) = ∫ Ω [1 2 ∆(u+ h) ·∆(u+ h)− F (x, (u+ h)) ] dx − ∫ Ω [1 2 ∆u ·∆u− F (x, u) ] dx− ∫ Ω [ ∆u ·∆u− f(x, u)h ] dx = 1 2 ∫ Ω ∆h ·∆h dx− ∫ Ω [f(x, u+ τ∗h)− f(x, u)]h dx, (3.8) where 0 ≤ τ∗(x) ≤ 1. So, (3.4), (3.8) and the continuity of F yield 1 ∥h∥ [ψ(u+ h)− ψ(u)− (Tu, h)2] = 1 ∥h∥ [ψ(u+ h)− ψ(u)− α(u, h)] ≤ 1 ∥h∥ [1 2 ∥h∥2L2(Ω) + ∥F(u+ τ∗h)− F(u)∥ 2N N+4 ∥h∥ 2N N−4 ] = 1 ∥h∥ [1 2 ∥h∥2 + ∥F(u+ τ∗h)− F(u)∥ 2N N+4 ∥h∥ 2N N−4 ] ≤ 1 2 ∥h∥+ ∥F(u+ τ∗h)− F(u)∥ 2N N+2m → 0 as ∥h∥ → 0. This shows that ψ is Fréchet differentiable at u and ψ′(u) = Tu, that is T = gradψ. Therefore, all conditions of Lemma 2.10 are satisfied. This completes the proof. □ Theorem 3.2. Suppose that (A1) and (A2) hold. Then problem (1.2) admits a unique nontrivial weak solution u∗ ∈ H2 0 (Ω), and the functional ψ(u) defined as in (3.2) admits a lower bound. Moreover, if there exists un(x) ∈ H2 0 (Ω) such that lim n→∞ ψ(un(x)) = d = inf un∈H2 0 (Ω) ψ(un(x)), then ∥un − u∗∥ → 0 (n→ ∞). The proof of the above theorem is similar to that of Theorem 3.1. Hence we omit it here. To the authors’ knowledge, this is probably the first time that the theory of strongly monotone operator combining the properties of Nemytskii operator is used to study the uniqueness of weak solution for biharmonic equations. 4. Existence of nontrivial weak solution of problem (1.1) In this section, we analyze the existence of nontrivial weak solution of problem (1.1) by using the Mountain Pass Lemma proposed by Ambrosetti and Rabinowitz in [2], which is different from that used in Section 3. EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 7 Definition 4.1 ([55, Def. 1.39]). Let I denote a C1 functional on a Banach space. Every sequence un satisfying sup n |I(un)| < +∞, I ′(un) → 0 (4.1) is said a Palais-Smale sequence ((PS)-sequence, for short). If any (PS)-sequence of I possesses a convergent subsequence, we say that I satisfies the (PS) condition. Zou [55] pointed out that the (PS) condition was first proposed by Palais [43], Smale [47] and Palais-Smale [44]. Lemma 4.2 (Mountain Pass Lemma). Let E be a real Banach space, f : E → R be a C1 functional and satisfy (PS) condition, x0, x1 ∈ E, Ω be an open neighborhood of x0 and x1 ̸∈ Ω̄. Suppose that max{f(x0), f(x1)} < inf x∈∂Ω f(x). (4.2) Let c = inf h∈Φ max t∈[0,1] f(h(t)), (4.3) where Φ = {h ∈ C([0, 1], E) : h(0) = x0, h(1) = x1 denotes the set of continuous paths joining x0 and x1. Then c must be the critical value of f , that is, there exists x∗ ∈ E such that f ′(x∗) = θ and f(x∗) = c. In this section, we suppose that f satisfies the following assumptions: (A3) f(x, u) satisfies Carathéodory conditions for x ∈ Ω and −∞ < u < +∞), and there exists 0 < σ ≤ N+4 N−4 such that |f(x, u)| ≤ a+ b|u|σ, a > 0, b > 0; (4.4) (A4) There exist 0 ≤ ξ < 1 2 and L > 0 such that F (x, u) = ∫ u 0 f(x, v)dv ≤ ξuf(x, u), ∀|u| ≥ L, x ∈ Ω; (4.5) (A5) The following two limits hold lim u→0 f(x, u) u = 0 uniformly for x ∈ Ω; (4.6) lim u→+∞ f(x, u) u = +∞ uniformly for x ∈ Ω. (4.7) Theorem 4.3. Suppose that (A3–(A5) hold. Then problem (1.1) admits a nontrivial weak solution u ∈ H2(Ω) ∩H1 0 (Ω). Proof. From the proof of Theorem 3.1, it is not difficult to see that we only need to prove that Tu = θ admits a nontrivial solution onH2(Ω)∩H1 0 (Ω). Here, T : H2(Ω)∩H1 0 (Ω) → H2(Ω)∩H1 0 (Ω) is continuous, and (Tu, v)2 = ∫ Ω [∆u ·∆v − f(x, u)v]dx, ∀u, v ∈ H2(Ω) ∩H1 0 (Ω). (4.8) Moreover, Tu = ψ′(u), ∀u ∈ H2(Ω) ∩H1 0 (Ω), (4.9) where ψ : H2(Ω) ∩H1 0 (Ω) → R is a C1 functional, ψ(u(x)) = ∫ Ω [1 2 ∆u(x) ·∆u(x)− F (x, u(x)) ] dx. (4.10) In the following, we show that ψ satisfies the conditions of the Mountain Pass Lemma. We first prove ψ satisfies (PS) condition. Suppose that {un} ⊂ H2(Ω) ∩H1 0 (Ω), |ψ(un)| ≤ γ (n ∈ {1, 2, . . . , }), ψ′(un) → θ (n→ +∞). 8 M. FENG, Y. LU EJDE-2025/52 Let Ωn = {x ∈ Ω : un(x) ≥ L}. Then it follows from (4.4) and (4.5) that there exist constants L1 and L2 such that γ ≥ ψ(un) = 1 2 ∥un∥2L2(Ω) − ∫ Ω F (x, un)dx ≥ 1 2 ∥un∥2L2(Ω) − ∫ Ωn F (x, un)dx− L1 ≥ 1 2 ∥un∥2L2(Ω) − ξ ∫ Ωn unf(x, un)dx− L1 ≥ 1 2 ∥un∥2L2(Ω) − ξ ∫ Ω unf(x, un)dx− L2 = ( 1 2 − ξ)∥un∥2L2(Ω) + ξ ∫ Ω [∆un ·∆un − f(x, un)un]dx− L2 = ( 1 2 − ξ)∥un∥2L2(Ω) + ξ(ψ′(un), un)2 − L2 ≥ ( 1 2 − ξ)∥un∥2L2(Ω) − ξ∥ψ′(un)∥L2(Ω)∥un∥L2(Ω) − L2 = ( 1 2 − ξ)∥un∥2 − ξ∥ψ′(un)∥ ∥un∥ − L2 (n ∈ {1, 2, . . . }). (4.11) Since 0 ≤ ξ < 1/2 and ∥ψ′(un)∥ → 0, it follows from (4.11) that {∥un∥} is bounded. Let s = 2Nσ N+4 . Then 1 < s ≤ 2N N−4 . Hence it follows from embedding theorem that W 2,2 ↪→ Ls(Ω), and {un} admits a convergent subsequence {uni} on Ls(Ω), that is, there exists u∗ ∈ Ls(Ω) such that ∥uni − u∗∥s → 0 as i→ +∞. (4.12) On the one hand, it follows from (4.4) that Nemytskii operator Fu(x) = f(x, u) maps Ls(Ω) into L 2N N+4 is bounded and continuous. So we derive from (4.12) that ∥Funi −Fu∗∥ 2N N+4 → 0 (i→ +∞), which shows ∥Funi − Funj ∥ 2N N+4 → 0 as i, j → +∞. (4.13) On the other hand, it follows from (4.8)-(4.10) that (ψ′(u), v)2 = (u, v)2 − ∫ Ω f(x, u)vdx, ∀u, v ∈ H2(Ω) ∩H1 0 (Ω). (4.14) So (uni − unj , v)2 = (ψ′(uni )− ψ′(unj ), v)2 + ∫ Ω [f(x, uni )− f(x, unj )]v dx. Thus, it follows from (3.4) that |(uni − unj , v)2| ≤ ∥ψ′(uni)− ψ′(unj )∥L2(Ω)∥v∥L2(Ω) + ∥Funi − Funj∥ 2N N+4 ∥v∥ 2N N−4 = ∥ψ′(uni)− ψ′(unj )∥ ∥v∥+ ∥Funi − Funj∥ 2N N+4 ∥v∥ 2N N−4 ≤ ∥ψ′(uni )− ψ′(unj )∥ · ∥v∥ + c∥Funi − Funj ∥ 2N N+4 · ∥v∥. Therefore, ∥uni − unj ∥ = sup ∥v∥=1 |(uni − unj , v)2| ≤ ∥ψ′(uni)− ψ′(unj )∥+ c∥Funi − Funj∥ 2N N+4 . (4.15) Since ∥ψ′(un))∥ → 0 as n → +∞, it follows from (4.13) and (4.15) that ∥uni − unj ∥ → 0 as i, j → +∞, which indicates that {uni } is convergent on H2(Ω) ∩H1 0 (Ω). Hence ψ satisfies (PS) condition. In the following, we look for u0, u1 ∈ H2(Ω) ∩H1 0 (Ω) and the open neighborhood (denoted by Br, where r > 0) of u0 ⊂ H2(Ω) ∩H1 0 (Ω) satisfying u1 ̸∈ Br. By the Friedrichs inequality, there exists τ∗ > 0 such that ∫ Ω |∇u|2dx = (−∆u, u) ≥ τ∗ ∫ Ω u2dx EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 9 for every u ∈ C∞ 0 (Ω). Since∫ Ω |∇u|2dx ≤ ∫ Ω [|∆u|2 + |∇u|2 + |u|2]dx, we obtain ∥u∥22,2 ≥ τ∗∥u∥2L2(Ω). Because the norm ∥u∥ = ∥∆u∥L2(Ω) is equivalent to the standard norm of H2(Ω), there exists τ > 0 such that ∥u∥2 ≥ τ∥u∥2L2(Ω). (4.16) Because of the density of C∞ 0 (Ω) in H2(Ω)∩H1 0 (Ω), (4.16) also holds for every u ∈ H2(Ω)∩H1 0 (Ω) by taking the limit. On the one hand, it yields from (4.6) that there exists δ > 0 such that |f(x, u)| ≤ τ 2 |u|, ∀0 < |u| < δ, x ∈ Ω. So F (x, u) ≤ τ 4 u2, ∀|u| < δ, x ∈ Ω. (4.17) On the other hand, it follows from (4.7) that F (x, u) ≤ a|u|+ b σ + 1 |u|σ+1, ∀ −∞ < u < +∞, x ∈ Ω. (4.18) Since σ + 1 < 2N N−4 , it follows from (4.16) and (4.17) that there exists b1 > 0 such that F (x, u) ≤ τ 4 u2 + b1|u| 2N N−4 , ∀ −∞ < u < +∞, x ∈ Ω. (4.19) Hence, when u ∈ H2(Ω) ∩H1 0 (Ω), it follows from (3.4), (4.16) and (4.19) that∫ Ω F (x, u)dx ≤ τ 4 ∥u∥2L2(Ω) + b1∥u∥α2N N−4 ≤ 1 4 ∥u∥2 + b1c α∥u∥α, (4.20) where α = 2N N−4 . Thus, according to (4.10) again, we derive ψ(u) ≥ 1 4∥u∥ 2 − b1c α∥u∥α, ∀u ∈ H2(Ω) ∩H1 0 (Ω). (4.21) Because α = 2N N−4 > σ + 1 ≥ 2, it follows from (4.21) that there exists r > 0 small enough such that inf x∈∂Br ψ(u) = cr > 0, (4.22) where Br = {u ∈ H2(Ω) ∩H1 0 (Ω) : ∥u∥ < r}. On the one hand, it is clear that the zero element θ ⊂ H2(Ω) ∩H1 0 (Ω) satisfies ψ(θ) = 0. (4.23) On the other hand, take v0 ∈ H2(Ω) ∩H1 0 (Ω), which satisfies that ∥v0∥ = 1 and v0(x) > 0, fo rall x ∈ Ω. Let ∥v0∥L2(Ω) = a0. Then a0 > 0. By (4.7), there exists τ0 > 0 such that f(x, u) ≥ 4 a20 u, ∀u ≥ τ0, x ∈ Ω. (4.24) We define a function ϕ(t) = ψ(tv0). It follows from (4.10) that ϕ(t) = ψ(tv0) = t2 2 ∥∆v0∥2L2(Ω) − ∫ Ω F (x, tv0)dx = t2 2 − ∫ Ω F (x, tv0)dx. (4.25) Take 0 < t1 < t2 < t3 < . . . , tn → +∞, and define Dn = {x ∈ Ω : tnv0(x) ≥ τ0}. Then Ω \Dn = {x ∈ Ω : tnv0(x) < τ0} (n ∈ {1, 2, . . . }). 10 M. FENG, Y. LU EJDE-2025/52 So, by (4.24), we have∫ Ω F (x, tv0)dx = ∫ Dn dx ( ∫ τ0 0 + ∫ t0v0(x) τ0 f(x, v)dv ) + ∫ Ω\Dn dx ( ∫ t0v0(x) 0 f(x, v)dv ) ≥ ∫ Dn dx ∫ t0v0(x) τ0 f(x, v)dv − ∫ Dn dx ∫ τ0 0 |f(x, v)|dv − ∫ Ω\Dn dx ( ∫ τ0 0 |f(x, v)|dv ) ≥ ∫ Dn dx ∫ t0v0(x) τ0 4 a20 v dv − ∫ Ω dx ∫ τ0 0 |f(x, v)|dv ≥ 2 a20 ∫ Dn (t2nv 2 0(x)− τ20 )dx− L3 ≥ 2 a20 t2n ∫ Dn v20(x)dx− L4 (4.26) where L3 = (aτ0 + b σ + 1 τσ+1 0 )measΩ, L4 = L3 + 2τ20 a20 measΩ are positive constants. Obviously, D1 ⊂ D2 ⊂ D3 ⊂ . . . , and Ω = ∪∞ n=1Dn. Hence measDn → measΩ (n→ +∞). So, by the absolute continuity of Lebesgue integral, there exists N0 > 0 satisfying n ≥ N0 such that∫ Dn v20(x)dx > ∫ Ω v20(x)dx− a20 2 = ∥v0∥2L2(Ω) − a20 2 = a20 2 . (4.27) Hence it follows from (4.26) and (4.27) that∫ Ω F (x, tv0)dx > t2n − L4, ∀n > N0. (4.28) Therefore, (4.25) and (4.28) yield ϕ(t) < −1 2 t2n + L4, ∀n > N0. This indicates that ϕ(tn) → −∞ as n→ +∞. (4.29) It is clear that ∥tnv0∥ = tn → +∞ as n→ +∞. (4.30) So it follows from (4.29) and (4.30) that there exists n big enough such that u0 = tnv0 ∈ H2(Ω)∩ H1 0 (Ω) satisfying u0 ̸∈ B̄r, ψ(u0) < 0. (4.31) Therefore, (4.22), (4.23), (4.31) and Lemma 4.2 imply that ψ admits critical value c∗ ≥ cr > 0; that is, there exists u∗ ∈ H2(Ω) ∩H1 0 (Ω) such that ψ(u∗) = c∗ and ψ′(u∗) = Tu∗ = θ. Since ψ(θ) = 0, we obtain u∗ ̸= θ. This completes the proof. □ Theorem 4.4. Suppose that (A3), (A4) (4.6) hold, lim u→−∞ f(x, u) u = +∞ uniformly for x ∈ Ω. (4.32) Then problem (1.1) admits a nontrivial weak solution u ∈ H2(Ω) ∩H1 0 (Ω). The proof of the above theorem is similar to that of Theorem 4.3. Hence we omit it here. Theorem 4.5. Suppose that (A3)–(A5) hold. Then problem (1.2) admits a nontrivial weak solu- tion u ∈ H2 0 (Ω). The proof of the above theorem is similar to that of Theorem 4.3. So we omit it here. EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 11 Remark 4.6. It is not hard to find elementary functions that satisfy (A3)–(A5); For instance f(x, u) = puq, where q > 1 and 0 < p < 1 2 (q + 1) are constants. 5. Existence of infinitely many nontrivial weak solutions In this section, we analyze the existence of infinitely many weak solutions for problem (1.1), using the critical point theory of even functionals proposed by Ambrosetti and Rabinowitz in [2]. This used in different from that in Sections 3 and 4. Definition 5.1 ([2, Def. 1.1]). Let E be a real Banach space. Define Σ(E) = {A : A is a symmetric closed set with respect to θ in E and A ⊂ E \ {θ}. Then A ∈ Σ(E) has genus N (denoted by γ(A) = N) if N is the smallest integer for which there exists T ∈ C(A,RN \ {θ}). γ(A) = ∞ if there exists no finite such n and γ(∅) = 0. Lemma 5.2 ([2, Lemma 1.2]). Let A,B ∈ Σ(E). (1) If there exists an odd T ∈ C(A,B), then γ(A) ≤ γ(B); (2) If A ⊂ B, then γ(A) ≤ γ(B); (3) If there exists an odd homeomorphism h ∈ C(A,B), then γ(A) = γ(B)γ(h(A)); (4) If γ(B) <∞, Then γA \B ≥ γ(A)− γ(B); (5) If A is compact, γ(A) < ∞, and there exists a uniform neighborhood Nδ(A) (al1points within δ of A) of A such that γ(Nδ(A)) = γ(A); (6) If A is homeomorphic by an odd homeomorphism to the boundary of a symmetric bounded open neighborhood of 0 in Rm, then γ(A) = m; (7) A ∈ Σ(E), V be a k dimensional subspace of E, and V ⊥ an algebraically and topologically complementary subspace. If γ(A) > k, then A ∩ V ⊥ = ∅. Let E be a real Banach space. Define Br = {u ∈ E : ∥u∥ < r}, Sr = ∂Br (r > 0), B1 and S1 will be denoted by B and S, respectively. Let f : E → R. We define f (0) = {x ∈ E : f(x) ≥ 0}. Suppose that f(θ) = 0 and satisfies (A6) there exist ρ > 0, a > 0 such that B̄ρ ⊂ f (0), and f(x) ≥ a for all x ∈ Sρ; (A7) if E0 is an infinite dimensional subspace of E, then E0 ∩ f (0) is bounded. We define Γ = {h|h : E → E is an odd homeomorphism, h(B̄1) ⊂ f (0))}. Let h0(x) = ρx for all x ∈ E. Then h0 ∈ Γ. So we derive that Γ ̸= ∅ when f satisfies (A6). We also we define Γm = {K ⊂ E : K is compact, symmetric with respect to θ and γ(K ∩ h(S1)) ≥ m, ∀h ∈ Γ}. where m is a positive integer. Because h(S) ⊂ E \ {θ} is closed and symmetric, γ(K ∩ h(S1)) is defined. Lemma 5.3 ([2, Lemma 2.7]). Suppose that dimE ≥ m, f : E → R satisfies (A6) and (A7). Then (1) Γm ̸= ∅; (2) Γm+1 ⊂ Γm; (3) K ⊂ Γm and A ∈ Σ(E) with γ(A) ≤ r < m implies K −A ∈ Γm−r; (4) If φ : E → E is an odd homeomorphism and φ−1(f (0)) ⊂ f (0), then φ(K) ⊂ Γm, ∀K ∈ Γm. Lemma 5.4 ([2, Theorem 2.8, Corollary 2.9]). Suppose that E is an infinite dimensional real Banach space, f : E → R is a C1 functional and satisfies (A6), (A7) and the (PS) condition. For each m ∈ {1, 2, . . . }, let bm = inf K∈Γm max x∈K f(x). (5.1) 12 M. FENG, Y. LU EJDE-2025/52 Then (1) 0 < a ≤ bm < +∞, bm is a critical value of f m ∈ {1, 2, . . . }; (2) bm = bm+1 = · · · = bm+r−1 = b (r ≥ 1) mplies γ(Kb) ≥ r, where Kb = {x ∈ E : f(x) = b, f ′(x) = θ}; (3) bm ≤ bm+1 (m ∈ {1, 2, . . . }), and bm → +∞ (m→ +∞); (4) f admits infinitely many critical points and admits infinitely many critical values. Theorem 5.5. Under conditions (A3)–(A5), if f satisfies (A8) f(x, u) is an odd function of u, that is f(x,−u) = −f(x, u), ∀x ∈ Ω, −∞ < u < +∞, (5.2) then problem (1.1) admits infinitely many weak solutions in H2(Ω) ∩H1 0 (Ω). Proof. From the proof of Theorem 4.3, it is not difficult to see that we need only to prove the C1 function ψ(u) defined by (4.10) satisfies all the conditions of Lemma 5.4. In fact, it follows from the definition F (x, u) and (5.2) that F (x,−u) = ∫ −u 0 f(x, v)dv = ∫ u 0 f(x,−t)d(−t) (let v = −t) = − ∫ u 0 f(x,−t)dt = ∫ u 0 f(x, t)dt = ∫ u 0 f(x, v)dv = F (x, u), which indicates that F (x, u) is an even functional of u. In addition, in Theorem 4.3, we have verified that ψ satisfies (PS) condition. Thus it follows from (4.21) that (A6) holds. In the following, we show that (A7) also holds. First, by (4.7) and (5.2), we have lim u→+∞ f(x,−u) −u = lim u→+∞ f(x, u) u = +∞ uniformly for x ∈ Ω, which implies lim u→−∞ f(x, u) u = +∞. (5.3) Next, we prove that (A7) also holds by means of reduction to absurdity. We suppose that (A7) does not hold, that is, there exists finite dimensional subspaces X ⊂ H2(Ω) ∩H1 0 (Ω) (let dimX = s) such that X ∩ ψ(0) is unbounded, where ψ(0) = {u ∈ H2(Ω) ∩H1 0 (Ω) : ψ(u) ≥ 0}. So there exists un ∈ X with ∥un∥ → +∞ such that ψ(un) ≥ 0 (n ∈ {1, 2, . . . }). (5.4) Let tn = ∥un∥, vn = 1 tn un ∈ X. Then ψ(un) = ψ(tnvn), ∥vn∥ = 1 (n ∈ {1, 2, . . . }). (5.5) Since X is finite dimensional, the unit sphere in X is compact. Hence {vn} admits convergent subsequence. To simplify the symbol, we assume that vn itself converges, that is, vn → v0 (v0 ∈ X, EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 13 ∥v0∥ = 1), which indicates ∥vn−v0∥ → 0. Moreover, it follows from (4.16) that ∥vn−v0∥L2(Ω) → 0. So, there exists subsequence of vn , which almost everywhere converge to v0 on Ω. To simplify the symbol, we assume that vn itself almost everywhere converge to v0 on Ω. Let Ω0 = {x ∈ Ω : v0(x) ̸= 0 and vn → v0 (n→ +∞)}. Then measΩ0 > 0. We define a0 = (∫ Ω0 v20(x)dx )1/2 . Then a0 > 0. So it follows from (4.7) and (5.3) that there exists τ0 such that f(x, u) ≥ 8 a20 u, ∀u ≥ τ0, x ∈ Ω; (5.6) f(x, u) ≤ 8 a20 u, ∀u ≤ −τ0, x ∈ Ω. (5.7) We define Dn = {x ∈ Ω : tn|vn(x)| ≥ τ0}. Then Ω \Dn = {x ∈ Ω : tn|vn(x)| < τ0}, Dn = D1 n ∪D2 n, D1 n ∩D2 n = ∅, where D1 n = {x ∈ Ω : tnvn(x) ≥ τ0}, D2 n = {x ∈ Ω : tnvn(x) ≤ −τ0}. So, by (4.4), (5.6) and (5.7), we obtain∫ Ω F (x, tnvn)dx = ∫ D1 n dx ( ∫ τ0 0 + ∫ tnvn(x) τ0 f(x, v)dv ) + ∫ D2 n dx ( ∫ −τ0 0 + ∫ tnvn(x) −τ0 f(x, v)dv ) + ∫ Ω\Dn dx ( ∫ tnvn(x) 0 f(x, v)dv ) ≥ ∫ D1 n dx ∫ tnvn(x) τ0 8 a20 vdv − ∫ D1 n dx ∫ τ0 0 |f(x, v)|dv − ∫ D2 n dx ∫ −τ0 tnvn(x) 8 a20 vdv − ∫ D2 n dx ∫ 0 −τ0 |f(x, v)|dx− ∫ Ω\Dn dx ∫ τ0 −τ0 |f(x, v)|dv ≥ ∫ D1 n dx ∫ tnvn(x) τ0 4 a20 (t2n[vn(x)] 2 − τ20 )dv − ∫ Dn dx ∫ τ0 0 |f(x, v)|dv − ∫ D2 n dx ∫ tnvn(x) τ0 4 a20 (t2n[vn(x)] 2 − ∫ Dn dx ∫ 0 −τ0 |f(x, v)|dx− ∫ Ω\Dn dx ∫ τ0 −τ0 |f(x, v)|dv = 4 a20 ∫ Dn dx ∫ tnvn(x) τ0 (t2n[vn(x)] 2 − τ20 )dv − ∫ Ω dx ∫ τ0 −τ0 |f(x, v)|dv ≥ 4 a20 t2n ∫ Dn [vn(x)] 2dx− L5, (5.8) where L5 = 2 (τ20 a20 + aτ0 + bτσ+1 0 σ + 1 ) measΩ is a constant. On the one hand, when n→ +∞, we have∣∣( ∫ Dn v2n(x)dx )1/2 − ( ∫ Dn v20(x)dx )1/2∣∣ ≤ ( ∫ Dn [v2n(x)− v20(x)]dx )1/2 ≤ ( ∫ Ω [v2n(x)− v20(x)]dx )1/2 = ∥vn − v0∥L2(Ω) → 0. 14 M. FENG, Y. LU EJDE-2025/52 Hence, there exists N1 > 0 such that(∫ Dn v2n(x)dx )1/2 > (∫ Dn v20(x)dx )1/2 − a0 4 , ∀n > N1. (5.9) We define D∗ n = ∩∞ k=nDk. Then D ∗ 1 ⊂ D∗ 2 ⊂ D∗ 3 ⊂ . . . . We also define D∗ = ∪∞ n=1Dn. Then Dn ⊃ D∗ n, D ∗ n ⊂ D∗, measD∗ n → measD∗ (n→ ∞). (5.10) So there exists N2 > 0 such that(∫ Dn v20(x)dx )1/2 ≥ (∫ D∗ n v20(x)dx )1/2 > (∫ D∗ v20(x)dx )1/2 − a0 4 , ∀n > N2. (5.11) On the other hand, it follows from the definition of Ω0 that(∫ D∗ v20(x)dx )1/2 ≥ (∫ Ω0 v20(x)dx )1/2 = a0. (5.12) Therefore, (5.9), (5.10) and (5.11) yield(∫ Dn v2n(x)dx )1/2 > a20 2 , ∀n > N = max{N1, N2}. (5.13) From (5.8) and (5.13) it follows that∫ Ω F (x, tnvn)dx > t2n − L5, ∀n > N. (5.14) Therefore, (4.10), (5.5) and (5.14) yield ψ(un) = ψ(tnvn) = t2n 2 ∥vn∥2 − ∫ Ω F (x, tnvn)dx = t2n 2 − ∫ Ω F (x, tnvn)dx = t2n 2 − ∫ Ω F (x, tnvn)dx < − t 2 n 2 + L5, ∀n > N. Since tn → +∞, we obtain lim n→+∞ ψ(un) = −∞. (5.15) which contradicts (5.4). Hence condition (A7) holds. This completes the proof of Theorem 5.5. □ Theorem 5.6. If (A3)–(A5), (A8) hold, then problem (1.2) admits infinitely many weak solutions in H2 0 (Ω). The proof of the above theorem is similar to that of Theorem 5.5. So we omit it here. Remark 5.7. It is easy to find elementary functions that satisfy (A3)–(A5) and (A8). For instance, f(x, u) = puq, where q ≥ 3 is an odd number and 0 < p < 1 2 (q + 1) is a constant. Consider the following biharmonic systems: ∆2u1 = f1(x, u1, u2) in Ω, ∆2u2 = f2(x, u1, u2) in Ω, u1 = u2 = ∆u1 = ∆u2 = 0 on ∂Ω, (5.16) and ∆2u1 = f1(x, u1, u2) in Ω, ∆2u2 = f2(x, u1, u2) in Ω, u1 = u2 = ∂u1 ∂ν = ∂u2 ∂ν = 0 on ∂Ω, (5.17) EJDE-2025/52 NONTRIVIAL SOLUTIONS FOR BIHARMONIC EQUATIONS 15 We believe the conclusions of Theorems 3.1, 3.2, 4.3, 4.5, 5.5, and 5.6 also hold for systems (5.16) and (5.17), but we cannot prove that these results yet. Acknowledgments. This work is sponsored by the Beijing Natural Science Foundation under Grant No. 1212003. The authors want to express their gratitude to the reviewers for careful reading and valuable suggestions which led to an improvement of the original manuscript. References [1] I. Abid, S. Baraket; Construction of singular solutions for elliptic problem of fourth order derivative with a subcritical nonlinearity, Differ. Integral Equ., 21 (2008), 653-664. [2] A. Ambrosetti, P. H. 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Meiqiang Feng School of Applied Science, Beijing Information Science and Technology University, Beijing, 102206, China Email address: meiqiangfeng@sina.com Yichen Lu School of Applied Science, Beijing Information Science and Technology University, Beijing, 102206, China Email address: lycmath@163.com 1. Introduction 2. Preliminaries 3. Uniqueness of nontrivial solutions 4. Existence of nontrivial weak solution of problem (1.1) 5. Existence of infinitely many nontrivial weak solutions Acknowledgments References