Open Access Research

Sign-changing solutions for some nonlinear problems with strong resonance

Aixia Qian

Author affiliations

School of Mathematic Sciences, Qufu Normal University, Qufu Shandong, 273165, P. R. of China

Citation and License

Boundary Value Problems 2011, 2011:18  doi:10.1186/1687-2770-2011-18


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/18


Received:7 January 2011
Accepted:30 August 2011
Published:30 August 2011

© 2011 Qian; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By means of critical point and index theories, we obtain the existence and multiplicity of sign-changing solutions for some elliptic problems with strong resonance at infinity, under weaker conditions.

2000 Mathematics Subject Classification: 35J65; 58E05.

Keywords:
critical point theory; strong resonance; index theory; Cerami condition

1 Introduction

In this article, we consider the following equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M1">View MathML</a>

(1.1)

where Ω is a bounded domain in ℝn with smooth boundary ∂Ω. In order to explain what we mean, a brief description is necessary. We suppose that f is asymptotically linear, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M2">View MathML</a> exists. If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M3">View MathML</a>

(1.2)

then we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M4">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M5">View MathML</a>

We denote λ1 < λ2 < ⋯ < λj < ⋯ to be the distinct eigenvalues sequence of -Δ with the Dirichlet boundary conditions. We state that problem (1.1) is resonant at infinity if α in (1.2) is an eigenvalue λk. The situation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M6">View MathML</a>

is what we call a strong resonance.

Now we present some of the results of this article. We write (1.1) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M7">View MathML</a>

(1.3)

We assume that g is a smooth function satisfying the following conditions.

(g1) g(t) · t → 0 as |t| → ∞.

(g2) the real function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M8">View MathML</a> is well defined and G(t) → 0 as t → +∞.

(g3) G(t) ≥ 0, ∀t ∈ ℝ.

Theorem 1.1 If (g1) - (g3) hold, then problem (1.1) has at least one solution.

Remark 1.1 Since 0 is a particular point, we cannot make sure those solutions are nontrivial without more conditions.

Theorem 1.2 Let g(0) = 0, and suppose that (g1) - (g3) hold, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M9">View MathML</a>

(1.4)

then problem (1.3) has at least one sign-changing solution.

Theorem 1.3 Assume that (g1)(g3) hold, g is odd, and G(0) ≥ 0. Moreover, suppose that there exists an eigenvalue λh < λk s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M10">View MathML</a>

Then, problem (1.3) possess at least m = dim(Mh ⊕ ⋯ ⊕ Mk) - 1 distinct pairs of sign-changing solutions (Mj denotes the eigenspace corresponding to λj).

Remark 1.2 In the article [1], they only show the existence of solutions to problem (1.3), while we obtain its sign-changing solutions under the same conditions.

The resonance problem has been widely studied by many authors using various methods--see [1-6] and the references therein. We will use critical point and pseudo-index theories to obtain the sign-changing solutions for strong resonant problem (1.3). We also allow the case in which resonance also occurs at zero.

In Section 2, we will give some preliminaries, which are fundamental for this article. In Section 3, we will give some abstract critical point theorems, which are used to prove above theorems in this article. In Section 3, we prove our main theorems, which result in the existence and multiplicity of sign-changing solutions.

2 Preliminaries

We denote by X a real Banach space. BR denotes the closed ball in X centered at the origin and with radius R > 0. J is a continuously Frèchet differentiable map from X to ℝ, i.e., J C1(X, ℝ).

In the literature, deformation theorems have been proved under the assumption that J C1(X, ℝ) satisfies the well-known Palais-Smale condition. In problems which do not have resonance at infinity, the (PS) condition is easy to verify. On the other hand, a weaker condition than the condition (PS) is needed to study problems with strong resonance at infinity.

Definition 2.1 We state that J C1(X, ℝ) satisfies the condition (C) in ]c1, c2[ (-∞ ≤ c1 < c2 ≤ +∞) if

(i) every bounded sequence {uk} ⊂ J-1 (]c1, c2[), for which {J(uk)} is bounded and J'(uk) → 0, possesses a convergent subsequence, and

(ii) ∀c ∈] c1, c2[, ∃σ, R, α > 0 s.t. [c - σ, c + σ] ⊂] c1, c2[ and ∀u J-1([c - σ, c + σ]), ||u|| ≥ R : ||J'(u)|| ||u|| ≥ α.

In the article [1], they propose a deformation theorem under the condition (C). For c ∈ ℝ, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M11">View MathML</a>

Proposition 2.2 [1] Let X be a real Banach space, and let J C1(X, ℝ) satisfy the condition (C) in ]c1, c2[. If c ∈]c1, c2[ and N is any neighborhood of Kc, then there exists a bounded homeomorphism η of X onto X and constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M12">View MathML</a>, s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M13">View MathML</a> satisfying the following properties:

(i) η(Ac+ε\N) ⊂ Ac-ε.

(ii) η(Ac+ε) ⊂ Ac-ε, if Kc = ∅.

(iii) η(x) = x, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M14">View MathML</a>.

Moreover, Let G be a compact group of (linear) unitary transformation on a real Hilbert space H. Then,

(vi) η can be chosen to be G-equivariant, if the functional J is G-invariant. Particularly, η is odd if the functional J is even.

3 Abstract critical point theorems

In this article, we shall obtain solutions of problem (1.3) using the linking-type theorem. Its different definitions can be seen in [1,7,8] and the references therein.

Definition 3.1 Let H be a real Hilbert space and A a closed set in H. Let B be an Hilbert manifold with boundary ∂B, we state that A and ∂B link if

(i) A ∩ ∂B = ∅;

(ii) If ϕ is a continuous map of H into itself s.t. ϕ(u) = u, ∀u ∈ ∂B, then ϕ(B) ∩ A ≠ ∅.

There are some typical examples as following, cf. [1,7,9].

Example 3.1 Let H1 and H2 be two closed subspaces of H such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M15">View MathML</a>

Hence, if A = H1, B = BR H2, then, A and ∂B link.

Example 3.2 Let H1 and H2 be two closed subspaces of H such that H = H1 H2, dim H2 < ∞, and consider e H1, ||e|| = 1, 0 < ρ < R1, R2, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M16">View MathML</a>

Then, A and ∂B link.

Let X H be a Banach space densely embedded in H. Assume that H has a closed convex cone PH and that P := PH X has interior points in X. Let J C1(H, ℝ). In the article [10], those authors construct the pseudo-gradient flow σ for J, and have the same definition as [11].

Definition 3.1 Let W X be an invariant set under σ. W is said to be an admissible invariant set for J if (a) W is the closure of an open set in X; (b) if un = σ(tn, v) → u in H as tn → ∞ for some v W and u K, then un u in X; (c) If un K W is such that un u in H, then un u in X; (d) For any u ∂W\K, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M17">View MathML</a> for t > 0.

Now let S = X\W, W = P ∪ (-P). Similar to the proof described in the article [10], the W is an admissible invariant set for J in the following section 4. We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M18">View MathML</a>

In the article [7], a new linking theorem is given under the condition (PS). Since the deformation still holds under the condition (C) (see [1]), the following theorem also holds.

Theorem 3.1 Suppose that W is an admissible invariant set of J and J C1(H, ℝ) such that

(J1)J satisfies condition (C) in ]0, +∞[;

(J2) There exists a closed subset A H and a Hilbert manifold B H with boundary ∂B satisfying

(a) there exist two constants β > α ≥ 0 s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M19">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M20">View MathML</a>.

(b) A and ∂B link;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M21">View MathML</a>.

Then, a* defines below is a critical value of J

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M22">View MathML</a>

Furthermore, assume 0 ∉ Ka*, then Ka* S ≠ ∅, if a* > b0 and Ka* A ≠ ∅, if a* = b0.

In this article, we shall consider the symmetry given by a ℤ2 action, more precisely even functionals.

Theorem 3.2 Suppose J C1(H, ℝ) and the positive cone P is an admissible invariant for J, Kc ∂P = ∅, for c > 0, such that

(J1) J satisfies condition (C) in ]0, +∞[, and J(0) ≥ 0;

(J2) There exist two closed subspace H+, H- of H, with codim H+ < +∞ and two constants c> c0 > J(0) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M23">View MathML</a>

(J3) J is even.

Hence, if dim H-> codim H++1, then J possesses at least m := dim H- -codim H+ - 1 (m := dim H- -1 resp.) distinct pairs of critical points in X\P ∪ (-P) with critical values belong to [c0, c].

Remark 3.1 The above theorem locates the critical points more precisely than Theorem 3.3 in [10].

We shall use pseudo-index theory to prove Theorem 3.2. First, we need the notation of genus and its properties, see [10,12]. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M24">View MathML</a>

with more preciseness, we denote iX(A) to be the genus of A in X.

Proposition 3.2 Assume that A, B ∈ ∑X, h C(X, X) is an odd homeomorphism, then

(i) iX(A) = 0 if and only if A = ∅;

(ii) A B iX(A) ≤ iX(B) (monotonicity);

(iii) iX(A B) ≤ iX(A) + iX(B) (subadditivity);

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M25">View MathML</a> (supervariancy);

(v) if A is a compact set, then iX(A) < +∞ and there exists δ > 0 s.t. iX(Nδ(A)) = iX(A), where Nδ(A) denotes the closed δ - neighborhood of A (continuity);

(vi) if iX(A) > k, V is a k-dimensional subspace of X, then A V≠ ∅;

(vii) if W is a finite dimensional subspace of X, then iX(h(Sρ) ∩ W ) = dim W.

(viii) Let V, W be two closed subspaces of X with codim V < +∞, dim W < +∞. Hence, if h is bounded odd homeomorphism on X, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M26">View MathML</a>

The proposition is still true when we replace ∑X by ∑H with obvious modification.

Proposition 3.3 [10,11] If A ∈ ∑X with 2 ≤ iX(A) < ∞, then A S ≠ ∅.

Proposition 3.4 Let A ∈ ∑H, then A X ∈ ∑X and iH(A) ≥ iX(A X).

Now, we shall discuss about the notion of pseudo-index.

Definition 3.2 [1] Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M27">View MathML</a> be an index theory on H related to a group G, and B ∈ ∑. We call a pseudo-index theory (related to B and I) a triplet

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M28">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M29">View MathML</a> is a group of homeomorphism on H, and i* : ∑ → ℕ ∪ {+∞} is the map defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M30">View MathML</a>

Proof of Theorem 3.2 Consider the genus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M31">View MathML</a> and the pseudo-index theory relate to I and B = Sρ H+, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M32">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M33">View MathML</a>

Obviously, conditions (a1)(a2) of Theorem 2.9 [1] are satisfied with a = 0, b = +∞ and b = Sρ H+. Now, we prove the condition that (a3) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M34">View MathML</a>. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M35">View MathML</a>, and by property (iv) of genus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M36">View MathML</a>

Now, by (viii) of Proposition 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M37">View MathML</a>

Therefore we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M38">View MathML</a>

Then, by Theorem 2.9 in [11] and Proposition 3.3 above, the numbers

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M39">View MathML</a>

are critical values of J and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M40">View MathML</a>

(3.1)

If for every k, ck ck+1, then we get the conclusion of Theorem 3.2. Assume now that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M41">View MathML</a>

Then, similar to the proof of Theorem 2.9 [11], where Kc is replaced by KcS and A by A S, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M42">View MathML</a>

(3.2)

Now, from Proposition 3.3 and (3.1), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M43">View MathML</a>

(3.3)

Since a finite set (not containing 0) has genus 1, we deduce from (3.2) and (3.3) that Kc above contains infinitely many sign-changing critical points. Therefore, J has at least m := dim H- -codim H+ -1 distinct pairs of sign-changing critical points in X\P ∪ (-P) with critical values belonging to [c0, c].

If codim H+ = 0, then we consider cj for j ≥ 2. As per the above arguments, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M44">View MathML</a> and if c := cj = ⋯ = cj+l for 2 ≤ j j + l ≤ dim H- with l ≥ 1, then i(Kc S) ≥ l + 1 ≥ 2.

Therefore, J has at least dim H- -1 pairs of sign-changing critical points with values belong to [c0, c].   ■

Remark 3.2 Theorem 3.1 above can also be proved by the pseudo-index theory in the same way as Theorem 3.2.

4 Proof of Theorems 1.1-1.3

We shall apply the abstract results of Section 3 to problem (1.3). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M46">View MathML</a>. Clearly the solutions of problem (1.3) are the critical points of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M47">View MathML</a>

(4.1)

where | · | denotes the norm in L2(Ω), and therefore, J C1(H, ℝ). We denote by Mj the eigenspace corresponding to the eigenvalue λj. If m ≥ 0 is an integer number, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M48">View MathML</a>

Clearly H+(m) ∩ H-(m) = Mm.

Proposition 4.1 [1] If (g1), (g2) hold, then the functional J defined by (4.1) satisfies the condition (C) in ]0, +∞[.

Proof of Theorem 1.1 If G(0) = 0, then by (g3), G takes its minimum at 0, so that g(0) = 0 and 0 is a solution of (1.3). We assume that G(0) > 0. Similar to the proof as for the case in [1], there exists R, γ > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M49">View MathML</a>

Let ∂B = H-(k) ∩ SR, A = H+(k + 1), then by Example 3.1 we get that ∂B and A link, and J is bounded on B = H-(k) ∩ BR. Moreover, by Proposition 4.1, J satisfies condition (C) in ]0, +∞[. Therefore, the conclusion of Theorem 1.1 follows by Theorem 3.1.   ■

Remark 4.1 If J(0) = 0, then the solutions obtained in Theorem 1.1 are sign-changing ones.

Proof of Theorem 1.2 Since g(0) = 0, u(x) = 0 is a solution of (1.3). In this case, we are interested in finding the existence of sign-changing solutions to problem (1.3). The case g(t) = 0, ∀t ∈ ℝ is trivial. We assume that g(t) ≠ 0 for some t. Then, it is easy to see that (g2), (g3) and (1.4) imply g'(0) > 0. Similar to the proof as for Theorem 5.1 [1], each of the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M50">View MathML</a>

(4.2)

where λk λ1 and there exists λh σ(-Δ) with λ2 λh λk such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/18/mathml/M51">View MathML</a>

(4.3)

Under (4.1), there exist three positive constants ρ < R, γ such that

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Since J(0) = G(0) · |Ω| ≥ 0 (|Ω| is the Lebesgue measure of Ω), we have

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Fix e M1 Sρ, set

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Then, by Example 3.1, A and ∂B link and J is bounded on B. Moreover, by Proposition 4.1, J satisfies condition (C) in ]0, +∞[. Then, by Theorem 3.1, J possesses a critical point u0 such that J(u0) ≥ J(0) + γ. So u0 is a sign-changing solution to problem (1.3).

Under (4.3) with similar arguments as given above, we get

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where B(h, R) = {u + te : u H-(h - 1) ∩ BR, e Mh S1, 0 ≤ t R}. Set

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Then, by Example 3.2, A and ∂B link and J is bounded on B. Moreover, by Proposition 4.1, J satisfies condition (C). Using Theorem 3.1, we can conclude that J possesses a sign-changing critical point u0 with J(u0) ≥ J(0) + γ.   ■

Remark 4.2 If g'(0) = 0, i.e., resonance at 0 is allowed, then by using an argument similar to that in the proof of Theorem 1.2, problem (1.3) still has at least a sign-changing solution under these conditions: Let g(0) = 0. Assume that (g1), (g2) hold and

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Moreover, suppose that either of the following holds:

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Proof of Theorem 1.3 By Proposition 3.1 and Lemma 5.3 [1], the assumptions of Theorem 3.2 are satisfied with

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Thus, there exist at least

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distinct pairs of sign-changing solutions of problem (1.3).   ■

Remark 4.3 We also allow resonance at zero in problem (1.3). By using Theorem 3.2 and Lemma 5.4 [1], we have assumed that g is odd and that (g1)(g2) are satisfied. Suppose in addition

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Then, the problem (1.3) possesses at least dim Mk - 1 distinct pairs of sign-changing solutions. (Mk denotes the eigenspace corresponding to λk with k ≥ 2)

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the anonymous referee for his or her suggestions. This study was supported by the Chinese National Science Foundation (11001151,10726003), the National Science Foundation of Shandong (Q2008A03) and the Science Foundation of China Postdoctoral (201000481301) and Shandong Postdoctoral.

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