Abstract
By means of critical point and index theories, we obtain the existence and multiplicity of signchanging 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 condition1 Introduction
In this article, we consider the following equation,
where Ω is a bounded domain in ℝ^{n }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., exists. If we set
then we can write
with
We denote λ_{1 }< λ_{2 }< ⋯ < λ_{j }< ⋯ 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 λ_{k}. The situation
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:
We assume that g is a smooth function satisfying the following conditions.
(g_{1}) g(t) · t → 0 as t → ∞.
(g_{2}) the real function is well defined and G(t) → 0 as t → +∞.
(g_{3}) G(t) ≥ 0, ∀t ∈ ℝ.
Theorem 1.1 If (g_{1})  (g_{3}) 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 (g_{1})  (g_{3}) hold, and
then problem (1.3) has at least one signchanging solution.
Theorem 1.3 Assume that (g_{1})(g_{3}) hold, g is odd, and G(0) ≥ 0. Moreover, suppose that there exists an eigenvalue λ_{h }< λ_{k }s.t.
Then, problem (1.3) possess at least m = dim(M_{h }⊕ ⋯ ⊕ M_{k})  1 distinct pairs of signchanging solutions (M_{j }denotes the eigenspace corresponding to λ_{j}).
Remark 1.2 In the article [1], they only show the existence of solutions to problem (1.3), while we obtain its signchanging solutions under the same conditions.
The resonance problem has been widely studied by many authors using various methodssee [16] and the references therein. We will use critical point and pseudoindex theories to obtain the signchanging 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 signchanging solutions.
2 Preliminaries
We denote by X a real Banach space. B_{R }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 ∈ C^{1}(X, ℝ).
In the literature, deformation theorems have been proved under the assumption that J ∈ C^{1}(X, ℝ) satisfies the wellknown PalaisSmale 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 ∈ C^{1}(X, ℝ) satisfies the condition (C) in ]c_{1}, c_{2}[ (∞ ≤ c_{1 }< c_{2 }≤ +∞) if
(i) every bounded sequence {u_{k}} ⊂ J^{1 }(]c_{1}, c_{2}[), for which {J(u_{k})} is bounded and J'(u_{k}) → 0, possesses a convergent subsequence, and
(ii) ∀c ∈] c_{1}, c_{2}[, ∃σ, R, α > 0 s.t. [c  σ, c + σ] ⊂] c_{1}, c_{2}[ 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
Proposition 2.2 [1] Let X be a real Banach space, and let J ∈ C^{1}(X, ℝ) satisfy the condition (C) in ]c_{1}, c_{2}[. If c ∈]c_{1}, c_{2}[ and N is any neighborhood of K_{c}, then there exists a bounded homeomorphism η of X onto X and constants , s.t. satisfying the following properties:
(i) η(A_{c}_{+}_{ε}\N) ⊂ A_{c}_{}_{ε}.
(ii) η(A_{c}_{+}_{ε}) ⊂ A_{c}_{}_{ε}, if K_{c }= ∅.
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 Gequivariant, if the functional J is Ginvariant. 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 linkingtype 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 H_{1 }and H_{2 }be two closed subspaces of H such that
Hence, if A = H_{1}, B = B_{R }∩ H_{2}, then, A and ∂B link.
Example 3.2 Let H_{1 }and H_{2 }be two closed subspaces of H such that H = H_{1 }⊕ H_{2}, dim H_{2 }< ∞, and consider e ∈ H_{1}, e = 1, 0 < ρ < R_{1}, R_{2}, set
Then, A and ∂B link.
Let X ⊂ H be a Banach space densely embedded in H. Assume that H has a closed convex cone P_{H }and that P := P_{H }∩ X has interior points in X. Let J ∈ C^{1}(H, ℝ). In the article [10], those authors construct the pseudogradient 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 u_{n }= σ(t_{n}, v) → u in H as t_{n }→ ∞ for some v ∉ W and u ∈ K, then u_{n }→ u in X; (c) If u_{n }∈ K ∩ W is such that u_{n }→ u in H, then u_{n }→ u in X; (d) For any u ∈ ∂W\K, we have 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
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 ∈ C^{1}(H, ℝ) such that
(J_{1})J satisfies condition (C) in ]0, +∞[;
(J_{2}) 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.
(b) A and ∂B link;
Then, a* defines below is a critical value of J
Furthermore, assume 0 ∉ K_{a}*, then K_{a}_{* }∩ S ≠ ∅, if a* > b_{0 }and K_{a}_{* }∩ A ≠ ∅, if a* = b_{0}.
In this article, we shall consider the symmetry given by a ℤ_{2 }action, more precisely even functionals.
Theorem 3.2 Suppose J ∈ C^{1}(H, ℝ) and the positive cone P is an admissible invariant for J, K_{c }∩ ∂P = ∅, for c > 0, such that
(J_{1}) J satisfies condition (C) in ]0, +∞[, and J(0) ≥ 0;
(J_{2}) There exist two closed subspace H^{+}, H^{ }of H, with codim H^{+ }< +∞ and two constants c_{∞ }> c_{0 }> J(0) satisfying
(J_{3}) 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 [c_{0}, c_{∞}].
Remark 3.1 The above theorem locates the critical points more precisely than Theorem 3.3 in [10].
We shall use pseudoindex theory to prove Theorem 3.2. First, we need the notation of genus and its properties, see [10,12]. Let
with more preciseness, we denote i_{X}(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) i_{X}(A) = 0 if and only if A = ∅;
(ii) A ⊂ B ⇒ i_{X}(A) ≤ i_{X}(B) (monotonicity);
(iii) i_{X}(A ∪ B) ≤ i_{X}(A) + i_{X}(B) (subadditivity);
(v) if A is a compact set, then i_{X}(A) < +∞ and there exists δ > 0 s.t. i_{X}(N_{δ}(A)) = i_{X}(A), where N_{δ}(A) denotes the closed δ  neighborhood of A (continuity);
(vi) if i_{X}(A) > k, V is a kdimensional subspace of X, then A ∩ V^{⊥ }≠ ∅;
(vii) if W is a finite dimensional subspace of X, then i_{X}(h(S_{ρ}) ∩ 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
The proposition is still true when we replace ∑_{X }by ∑_{H }with obvious modification.
Proposition 3.3 [10,11] If A ∈ ∑_{X }with 2 ≤ i_{X}(A) < ∞, then A ∩ S ≠ ∅.
Proposition 3.4 Let A ∈ ∑_{H}, then A ∩ X ∈ ∑_{X }and i_{H}(A) ≥ i_{X}(A ∩ X).
Now, we shall discuss about the notion of pseudoindex.
Definition 3.2 [1] Let be an index theory on H related to a group G, and B ∈ ∑. We call a pseudoindex theory (related to B and I) a triplet
where is a group of homeomorphism on H, and i* : ∑ → ℕ ∪ {+∞} is the map defined by
Proof of Theorem 3.2 Consider the genus and the pseudoindex theory relate to I and B = S_{ρ }∩ H^{+}, , where
Obviously, conditions (a_{1})(a_{2}) of Theorem 2.9 [1] are satisfied with a = 0, b = +∞ and b = S_{ρ }∩ H^{+}. Now, we prove the condition that (a_{3}) is satisfied with . It is obvious that , and by property (iv) of genus, we have
Now, by (viii) of Proposition 3.2, we have
Therefore we get
Then, by Theorem 2.9 in [11] and Proposition 3.3 above, the numbers
are critical values of J and
If for every k, c_{k }≠ c_{k}_{+1}, then we get the conclusion of Theorem 3.2. Assume now that
Then, similar to the proof of Theorem 2.9 [11], where K_{c }is replaced by K_{c}∩S and A by A ∩ S, we have
Now, from Proposition 3.3 and (3.1), we deduce that
Since a finite set (not containing 0) has genus 1, we deduce from (3.2) and (3.3) that K_{c }above contains infinitely many signchanging critical points. Therefore, J has at least m := dim H^{ }codim H^{+ }1 distinct pairs of signchanging critical points in X\P ∪ (P) with critical values belonging to [c_{0}, c_{∞}].
If codim H^{+ }= 0, then we consider c_{j }for j ≥ 2. As per the above arguments, and if c := c_{j }= ⋯ = c_{j}_{+}_{l }for 2 ≤ j ≤ j + l ≤ dim H^{ }with l ≥ 1, then i(K_{c }∩ S) ≥ l + 1 ≥ 2.
Therefore, J has at least dim H^{ }1 pairs of signchanging critical points with values belong to [c_{0}, c_{∞}]. ■
Remark 3.2 Theorem 3.1 above can also be proved by the pseudoindex theory in the same way as Theorem 3.2.
4 Proof of Theorems 1.11.3
We shall apply the abstract results of Section 3 to problem (1.3). Let , . Clearly the solutions of problem (1.3) are the critical points of the functional
where  ·  denotes the norm in L^{2}(Ω), and therefore, J ∈ C^{1}(H, ℝ). We denote by M_{j }the eigenspace corresponding to the eigenvalue λ_{j}. If m ≥ 0 is an integer number, set
Clearly H^{+}(m) ∩ H^{}(m) = M_{m}.
Proposition 4.1 [1] If (g_{1}), (g_{2}) 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 (g_{3}), 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
Let ∂B = H^{}(k) ∩ S_{R}, A = H^{+}(k + 1), then by Example 3.1 we get that ∂B and A link, and J is bounded on B = H^{}(k) ∩ B_{R}. 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 signchanging 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 signchanging 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 (g_{2}), (g_{3}) and (1.4) imply g'(0) > 0. Similar to the proof as for Theorem 5.1 [1], each of the following holds:
where λ_{k }≠ λ_{1 }and there exists λ_{h }∈ σ(Δ) with λ_{2 }≤ λ_{h }≤ λ_{k }such that
Under (4.1), there exist three positive constants ρ < R, γ such that
Since J(0) = G(0) · Ω ≥ 0 (Ω is the Lebesgue measure of Ω), we have
Fix e ∈ M_{1 }∩ S_{ρ}, set
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 u_{0 }such that J(u_{0}) ≥ J(0) + γ. So u_{0 }is a signchanging solution to problem (1.3).
Under (4.3) with similar arguments as given above, we get
where B(h, R) = {u + te : u ∈ H^{}(h  1) ∩ B_{R}, e ∈ M_{h }∩ S_{1}, 0 ≤ t ≤ R}. Set
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 signchanging critical point u_{0 }with J(u_{0}) ≥ 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 signchanging solution under these conditions: Let g(0) = 0. Assume that (g_{1}), (g_{2}) hold and
Moreover, suppose that either of the following holds:
Proof of Theorem 1.3 By Proposition 3.1 and Lemma 5.3 [1], the assumptions of Theorem 3.2 are satisfied with
Thus, there exist at least
distinct pairs of signchanging 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 (g_{1})(g_{2}) are satisfied. Suppose in addition
Then, the problem (1.3) possesses at least dim M_{k } 1 distinct pairs of signchanging solutions. (M_{k }denotes the eigenspace corresponding to λ_{k }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.
References

Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983). Publisher Full Text

Bartsch, T, Li, SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419–441 (1997). Publisher Full Text

Li, SJ, Liu, JQ: Computations of critical groups at degenerate ciritical point and applications to nonlinear differential equations with resonace. Houson J Math. 25, 563–582 (1999)

Molle, R, Passaseo, D: Nonlinear elliptic equations with large supercritical exponents. Calc Var. 26, 201–225 (2006). Publisher Full Text

Qian, AX: Neumann problem of elliptic equation with strong resonance. Nonlinear Anal TMA. 66, 1885–1898 (2007). Publisher Full Text

Su, JB, Zhao, LG: An elliptic resonant problem with multiple solutions. J Math Anal Appl. 319, 604–616 (2006). Publisher Full Text

Schechter, M, Wang, ZQ, Zou, WM: New linking theorem and signchanging solutions. Commun Partial Equ. 29, 471–488 (2004). Publisher Full Text

Schechter, M: Resonance problems with respect to the Fučik spectrum. Trans Am Math Soc. 352, 4195–4205 (2000). Publisher Full Text

Qian, AX, Li, SJ: Multiple nodal soltuions for elliptic equations. Nonlinear Anal. 37, 615–632 (2004)

Li, SJ, Wang, ZQ: LjusternikSchnirelman theory in partially ordered Hilbert spaces. Trans Am Math Soc. 354, 3207–3227 (2002). Publisher Full Text

Rabinowitz, P: Minimax methods in critical point theory with applications to differential equations. CBMS Reg Cof Ser Math. 65, (1986)