INFINITELY MANY RADIAL SOLUTIONS FOR A SUPER-CUBIC KIRCHHOFF TYPE PROBLEM IN A BALL

. We prove the existence of inﬁnitely many radial solutions to a Kirchhoﬀ type problem in a ball with a super-cubic nonlinearity. Our methods rely on bifurcation analysis and energy estimates.

That is, f has super-cubic and subcritical growth. Let λ 1 < λ 2 < · · · < λ k < · · · → +∞ denote the eigenvalues of u rr + N − 1 r u r + λu(r) = 0 r ∈ (0, 1] Our main result is the following theorem. Remark 1.2. Unlike other results on the existence of infinitely many solutions, we do not assume f to be odd. Our assumption f (0) > 0 plays a crucial role in the bifurcation analysis of radial solutions to be used in this paper.
For Ω a general smooth bounded domain, the existence and multiplicity of solutions for (1.1) has been extensively studied. In [14], Perera and Zhang proved the existence of a nontrivial solution using the Yang index and critical groups for f asymptotically cubic and not resonant with the nonlinear spectrum. In [24], they revisited (1.1) via invariant sets of descent flow and found the existence of a positive, a negative solution and a sign changing solution for f sub-cubic, asymptotically cubic and super-cubic. In [16], Song, Tang and Chen proved the existence of three solutions for f nearly resonant to the first nonlinear eigenvalue from below based on Ekeland's variational principle and the mountain pass lemma. In [15] the same authors proved the existence of solutions for f resonant to higher nonlinear eigenvalues.
The existence of infinitely many solutions for problem (1.1) in general bounded domains and f odd can be found in [8,18,22,21]. In [21] the existence of infinitely many sign-changing solutions was proved using a combination of invariant sets of descent flow and Ljusternik-Schnirelman type minimax method for f (u) = |u| p−2 u, p ∈ (2, 2 * ). In [8] infinitely many large energy solutions were found via the fountain theorem under Ambrosetti-Rabinowitz's 4-super quadratic condition or general 4-super quadratic at infinity with the global monotonicity condition: f (x, t) t 3 is an increasing function of t ≥ 0 for every x ∈ Ω. (1.5) These results were extended in [18,22]. In [18], (1.5) was replaced by the following condition: There exists θ ≥ 1 such that θG(x, t) ≥ G(x, st) for all (x, t) ∈ Ω × R and s ∈ [0, 1], has been extensively studied. See, [2,6,9,10,13,1,5,11,17,19,23,7,12,20,25]. Since the Sobolev embedding H 1 (R N ) → L s (R N )(2 ≤ s ≤ 2 * ) is not compact, it is usually difficult to prove the Palais-Smale condition for the problem in R N . In order to overcome this difficulty, some conditions have been imposed on the potential function V . For V constant or radial, see [2,6,9,10,13]; for V bounded from below, see [2,1,5,11,17,19,23] and for V 's such that Palais-Smale sequences converge while the the corresponding Sobolev embedding may not be compact, see [7,12,20,25]. It is worth pointing out that assuming f to be odd is a key ingredient in the aforementioned references on the existence of infinitely many solutions for (1.1) for both Ω bounded and R N .
We base our arguments on the fact that if u is a solution to the singular ordinary differential equation then u is a solution to (1.1). We investigate the solutions to (1.6) by considering the initial value problem and the bifurcation properties of (1.6).
2. Bifurcation analysis of radial solutions. Since each eigenvalue λ k of (1.4) is simple, f (0) = 0, f (0) = 1, by Theorem 1.7 of [4], for each positive integer k there exists a continuum of solutions to (1.8) bifurcating from (λ k , 0) with u(0) > 0. Let such a continuum be Γ k . By uniqueness of solutions to initial value problems, if (λ, u) ∈ Γ k then u (x) = 0 for u(x) = 0. This and the connectedness of Γ k imply that if (λ, u) ∈ Γ k then u has exactly k zeros in (0, 1]. Hence Γ k ∩ Γ j is empty for k = j. Thus, by global bifurcation theory (see Theorem 8.2, [3]), Γ k is unbounded. Since p < (N + 2)/(N − 2), a priori estimates for elliptic equations imply that if {(λ j , u j )} j is a sequence in Γ k and { u j } j converges to +∞ then λ j converges to zero. Similarly, for each positive integer k, there exists an unbounded continuumΓ k of solutions to (1.8) bifurcating from (λ k , 0) with u(0) < 0. Figure 1 below provides a sketch of the above analysis.
Thus, by inequality above and (3.9) we have .
In particular, taking r = s 1 in inequality above we have .