By combining the embedding arguments and the variational methods, we obtain infinitely many solutions for a class of superlinear elliptic problems with the Robin boundary value under weaker conditions.

1. Introduction

In this paper, we consider the following equation:

(1.1)

where Ω is a bounded domain in ℝⁿ with smooth boundary ∂Ω and 0 ≤ b ∈ L^∞(∂Ω). Denote

(1.2)

and let λ₁ ≤ λ₂ ≤ ⋯≤λ_j < ⋯ be the eigenvalues of −Δ with the Robin boundary conditions. We assume that the following hold:

f₁ , ∃q ∈ (2, 2^*) such that
(1.3)

where 1 ≤ s < 2N/(N − 2), N ≥ 3. If N = 1,2, let 2^* = ∞;

f₂ f(x, s)s ≥ 0, lim _|s|→+∞(f(x, s)s)/|s|² = +∞ uniformly for x ∈ Ω.
f₃ there exist θ ≥ 1, s ∈ [0,1] s.t.

(1.4)

f₄ f(x, −t) = −f(x, t), (x, t) ∈ Ω × ℝ.

Because of (f₂), (1.1) is usually called a superlinear problem. In [1, 2], the author obtained infinitely many solutions of (1.1) with Dirichlet boundary value condition under (f₁), (f₄) and

(AR) ∃ μ > 2, R > 0 such that

(1.5)

Obviously, (f₂) can be deduced form (AR). Under (AR), the (PS) sequence can be deduced bounded. However, it is easy to see that the example [3]

(1.6)

does not satisfy (AR), while it satisfies the aforementioned conditions (take θ = 1 in (f₃)). (f₃) is from [3, 4].

We need the following condition (C), see [3, 5, 6].

Definition 1.1. Assume that X is a Banach space, we say that J ∈ C¹(X, ℝ) satisfies Cerami condition (C), if for all c ∈ ℝ:

(i)
any bounded sequence {u_n} ⊂ X satisfying J(u_n) → c, J^′(u_n) → 0 possesses a convergent subsequence;
(ii)
there exist σ, R, β > 0 s.t. for any u ∈ J⁻¹([c − σ, c + σ]) with ∥u∥≥R, ∥J^′(u)∥∥u∥≥β.

In the work in [2, 7], the Fountain theorem was obtained under the condition (PS). Though condition (C) is weaker than (PS), the well-known deformation theorem is still true under condition (C) (see [5]). There is the following Fountain theorem under condition (C).

Assume , where X_j are finite dimensional subspace of X. For each k ∈ ℕ, let

(1.7)

Denote S_ρ = {u ∈ X : ∥u∥ = ρ}.

Proposition 1.2. Assume that J ∈ C¹(X, ℝ) satisfies condition (C), and J(−u) = J(u). For each k ∈ ℕ, there exist ρ_k > r_k > 0 such that

(i)
, k → ∞,
(ii)
.

Then J has a sequence of critical points u_n, such that J(u_n)→+∞ as n → ∞.

As a particular linking theorem, Fountain theorem is a version of the symmetric Mountain-Pass theorem. Using the aforementioned theorem, the author in [6] proved multiple solutions for the problem (1.1) with Neumann boundary value condition; the author in [3] proved multiple solutions for the problem (1.1) with Dirichlet boundary value condition. In the present paper, we also use the theorem to give infinitely many solutions for problem (1.1). The main results are follows.

Theorem 1.3. Under assumptions (f₁)–(f₄), problem (1.1) has infinitely many solutions.

Remark 1.4. In the work in [1, 2], they got infinitely many solutions for problem (1.1) with Dirichlet boundary value condition under condition (AR).

Remark 1.5. In the work in [8], they showed the existence of one nontrivial solution for problem (1.1), while we get its infinitely many solutions under weaker conditions than [8].

Remark 1.6. In the work in [9], they also obtained infinitely many solutions for problem (1.1) with Dirichlet boundary value condition under stronger conditions than the aforementioned (f₂) and (f₃) above. Furthermore, function (1.6) does not satisfy all conditions in [9]. Therefore, Theorem 1.3 applied to Dirichlet boundary value problem improves those results in [1, 2, 8, 9].

2. Preliminaries

Let the Sobolev space X = H¹(Ω). Denote

(2.1)

to be the norm of u in X, and |u|_q the norm of u in L^q(Ω). Consider the functional J : X → ℝ:

(2.2)

Then by (f₁), J is C¹ and

(2.3)

The critical point of J is just the weak solution of problem (1.1).

Since we do not assume condition (AR), we have to prove that the functional J satisfies condition (C) instead of condition (PS).

Lemma 2.1. Under (f₁)–(f₃), J satisfies condition (C).

Proof. For all c ∈ ℝ, we assume that {u_n} ⊂ X is bounded and

(2.4)

Going, if necessary, to a subsequence, we can assume that u_n⇀u in X, then

(2.5)

that is,

(2.6)

Since the Sobolev imbedding W^1,2(Ω)↪L^γ(Ω) (1 ≤ γ < 2^*) is compact, we have the right-hand side of (2.6) converges to 0. While ∫_∂Ωb(x)(u_n − u) ²dS ≥ 0, we have ∥u_n − u∥² → 0. It follows that u_n → u in X and J^′(u) = 0, that is, condition (i) of Definition 1.1 holds.

Next, we prove condition (ii) of Definition 1.1, if not, there exist c ∈ ℝ and {u_n} ⊂ X satisfying, as n → ∞

(2.7)

then we have

(2.8)

Denote v_n = u_n/∥u_n∥, then ∥v_n∥ = 1, that is, {v_n} is bounded in X, thus for some v ∈ X, we get

(2.9)

If v = 0, define a sequence {t_n} ⊂ ℝ as in [4]

(2.10)

If for some n ∈ ℕ, there is a number of t_n satisfying (2.10), we choose one of them. For all m > 0, let

, it follows by v_n(x) → v(x) = 0 a.e. x ∈ Ω that

(2.11)

Then for n large enough, by (2.9), (2.11), and

, we have

(2.12)

That is, lim _n→∞J(t_nu_n) = +∞. Since J(0) = 0 and J(u_n) → c, then 0 < t_n < 1. Thus

(2.13)

We see that

(2.14)

From the aforementioned, we infer that

(2.15)

which contradicts (2.8).

If v≢0, by (2.7)

(2.16)

That is,

(2.17)

Since

exists, and by v_n⇀v in X (the weakly convergent sequence is bounded), we get

(2.18)

where C is the constant of Sobolev Trace imbedding from H¹(Ω) → L²(∂Ω), see [10]. We have

(2.19)

For x ∈ Ω^′ : = {x ∈ Ω : v(x) ≠ 0}, we get |u_n(x)|→+∞. Then by (f₂)

(2.20)

By using Fatou lemma, since the Lebesgue measure |Ω^′| > 0,

(2.21)

On the other hand, by (f₂), there exists γ > −∞, such that (f(x, s)s)/|s|² ≥ γ for (x, s) ∈ Ω × ℝ. Moreover,

(2.22)

Now, there is Λ > −∞ s.t.

(2.23)

Together with (2.19) and (2.21), (2.23), it is a contradiction.

This proves that J satisfies condition (C).

3. Proof of Theorem 1.3

We will apply the Fountain theorem of Proposition 1.2 to the functional in (2.2). Let

(3.1)

then

. It shows that J ∈ C¹(X, ℝ) by (f₁) and satisfies condition (C) by Lemma 2.1.

(i)
After integrating, we obtain from (f₁) that there exist c₁ > 0 such that

(3.2)

Let us define . By [2, Lemma 3.8], we get β_k → 0 as k → ∞. Since |u|₂ ≤ C(Ω) | u|_q, let c = c₁ + (1/2)C(Ω), and , then by (3.2), for u ∈ Z_k with ∥u∥ = r_k, we have

(3.3)

Notice that β_k → 0 and q > 2, we infer that

(3.4)

(ii)
While

(3.5)

we can deduce that ∫_Ω | ∇u|²dx + ∫_∂Ωα(x)u²dS is the equivalent norm of ∥u∥² in X. Since dim Y_k < +∞ and all norms are equivalent in the finite-dimensional space, there exists C_k > 0, for all u ∈ Y_k, we get

(3.6)

Next by (f₂), there is R_k > 0 such that F(x, s) ≥ 2C_k | s|² for |s | ≥ R_k. Take M_k := max {0,

, then for all (x, s) ∈ Ω × ℝ, we obtain

(3.7)

It follows from (3.6), (3.7), for all u ∈ Y_k that

(3.8)

Therefore, we get that for ρ_k large enough (ρ_k > r_k),

(3.9)

By Fountain theorem of Proposition 1.2, J has a sequence of critical points u_n ∈ X, such that J(u_n)→+∞ as n → ∞, that is, (1.1) has infinitely many solutions.

Remark 3.1. By Theorem 1.3, the following equation:

(3.10)

has infinitely many solutions, while the results cannot be obtained by [1, 2, 8, 9]

Remark 3.2. In the next paper, we wish to consider the sign-changing solutions for problem (1.1).

Acknowledgments

We thank the referee for useful comments. C. Li is supported by NSFC (10601058, 10471098, 10571096). This work was supported by the Chinese National Science Foundation (10726003), the National Science Foundation of Shandong (Q2008A03), and the Foundation of Qufu Normal University.

References

1 Rabinowitz P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, 1986, 65, American Mathematical Society, Washington, DC, USA, CBMS Regional Conference Series in Mathematics, MR845785.
10.1090/cbms/065
Google Scholar
2 Willem M., Minimax Theorems, 1996, 24, Birkhäuser, Boston, Mass, USA, Progress in Nonlinear Differential Equations and Their Applications, MR1400007.
10.1007/978-1-4612-4146-1
Web of Science® Google Scholar
3 Liu S. B. and Li S. J., Infinitely many solutions for a superlinear elliptic equation, Acta Mathematica Sinica. Chinese Series. (2003) 46, no. 4, 625–630, MR2010750, ZBL1081.35043.
Google Scholar
4 Jeanjean L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ℝ^N, Proceedings of the Royal Society of Edinburgh. Section A. (1999) 129, no. 4, 787–809, MR1718530, ZBL0935.35044.
10.1017/S0308210500013147
Web of Science® Google Scholar
5 Bartolo P., Benci V., and Fortunato D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Analysis: Theory, Methods & Applications. (1983) 7, no. 9, 981–1012, https://doi.org/10.1016/0362-546X(83)90115-3, MR713209, ZBL0522.58012.
10.1016/0362-546X(83)90115-3
Web of Science® Google Scholar
6 Qian A. X., Existence of infinitely many nodal solutions for a Neumann boundary value problem, Boundary Value Problem. (2005) 3, 329–335.
Google Scholar
7 Li S. and Wang Z.-Q., Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Transactions of the American Mathematical Society. (2002) 354, no. 8, 3207–3227, https://doi.org/10.1090/S0002-9947-02-03031-3, MR1897397.
10.1090/S0002-9947-02-03031-3
Web of Science® Google Scholar
8 Shi S. and Li S., Existence of solutions for a class of semilinear elliptic equations with the Robin boundary value condition, Nonlinear Analysis: Theory, Methods & Applications. (2009) 71, no. 7-8, 3292–3298, https://doi.org/10.1016/j.na.2009.01.209, MR2532751.
10.1016/j.na.2009.01.209
Google Scholar
9 Zou W., Variant fountain theorems and their applications, Manuscripta Mathematica. (2001) 104, no. 3, 343–358, https://doi.org/10.1007/s002290170032, MR1828880, ZBL0976.35026.
10.1007/s002290170032
Web of Science® Google Scholar
10 Evans L. C., Partial Differential Equations, 1998, 19, American Mathematical Society, Providence, RI, USA, Graduate Studies in Mathematics, MR1625845.
10.1090/gsm/019
Google Scholar

All articles

Infinitely Many Solutions for a Robin Boundary Value Problem

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.3

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Infinitely Many Solutions for a Robin Boundary Value Problem

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.3

Acknowledgments

References

References

Related

Information