Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 753–766

Global Bifurcation Results for a Semilinear Biharmonic Equation on all of IR^{N}

N. M. Stavrakakis and N. Zographopoulos

Abstract. We prove existence of positive solutions for the semilinear problem

(−Δ)^{2}u = λg(x)f(u),  u(x) > 0 (x ∈ ℝ^{N}),  lim_{|x|→+∞}u(x) = 0

under the main hypothesis N > 4 and g ∈ L^{N/4}(ℝ^{N}). First, we employ classical spectral analysis for the existence of a simple positive principal eigenvalue for the linearized problem. Next, we prove the existence of a global continuum of positive solutions for the problem above, branching out from the first eigenvalue of the differential equation in the case that f(u) = u. This fact is achieved by applying standard local and global bifurcation theory. It was possible to carry out these methods by working between certain equivalent weighted and homogeneous Sobolev spaces. Keywords: Biharmonic equations, nonlinear eigenvalue problems, local and global bifurcation

theory, maximum principle, indefinite weights

AMS subject classification: 35 B 32, 35 B 40, 35 J 40, 35 J 65, 35 J 70, 35 H 12

1. Introduction

In this paper we study existence and properties of solutions of the semilinear biharmonic eigenvalue problem

(−Δ)^{2}u = λg(x)f(u)  (x ∈ ℝ^{N})  (1.1)_{λ}
u(x) > 0 (x ∈ ℝ^{N}), lim_{|x|→+∞}u(x) = 0,  (1.2)

where λ ∈ ℝ and N > 4. The general hypotheses, which will be assumed throughout the paper, are the following ones:

(G) g is a smooth function, at least of type C^{1,α}(ℝ^{N}), for some α ∈ (0, 1), such that g ∈ L^{N/4}(ℝ^{N}) ∩ L^{∞}(ℝ^{N}) and g(x) ≥ 0.

(F) f : ℝ ⟼ [0, ∞) is a smooth function such that f(0) = 0, f′(0) > 0 and f(u) > 0 for all u ≠ 0. Also, f′, f′′ ∈ L^{∞} and there is k^{∗} > 0 such that |f(s)| ≤ k^{∗}|s| for all s ∈ ℝ.

Both authors: Nat. Techn. Univ., Dept. Math., Zografos Campus, 157 80 Athens, Greece e-mail: nikolas@central.ntua.gr

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

The literature for the problem in the bounded domain case is quite complete. For example, among others, we mention the papers of Ph. Clément et al. [6], R. Dalmasso [8], D. E. Edmunds et al. [10] and Y.-G. Gu [16]. We also mention the existence and non-existence results in the papers of E. Mitidieri [20], L. Peletier and R. C. A. M. van der Vorst [23], and P. Pucci and J. Serrin [25] on the subject.

In general, in the unbounded domain case the problem becomes more complicate; among other reasons

(i) compact operators are not expected and (ii) it is not clear a priori in which function spaces solutions of (1.1)_{λ} might lie.

In general, we need spaces which will control the asymptotic behavior of the solutions (and their derivatives) at infinity and such spaces, as we will see in our case, are weighted or/and homogeneous Sobolev spaces.

Recently many authors have studied nonlinear polyharmonic problems in unbounded domains. We refer to more representative of them in the problem: in the radial case the works of Y. Furusho and T. Kusano [13], T. Kusano et al. [17], E. S. Noussair et al. [21], and C. A. Swanson and L. S. Yu [32, 33], in the non-radial sub- (super-) linear case the works of W. Allegretto and L. S. Yu [2], F. Bernis [3], C. A. Swanson [31] as well as the results on the one-dimensional problem by L. Peletier et al. [24]. Also, the fixed point theory is used in several cases as in the paper of T. Kusano et al. [18] (see also the references therein). Maximum principle results for the biharmonic equation in unbounded domains are obtained recently by N. M. Stavrakakis and G. Sweers [30]. Uniqueness questions for the radial case are studied recently in the work of C. A. Swanson [32]. Let also notice that the study in the above mentioned papers [2, 3, 9, 11, 21, 31 - 33] is based in homogeneous Sobolev spaces of type similar to the one used here.

In all these papers the weight function is non-negative. To our knowledge the only works, were the eigenvalue problem for the linear polyharmonic problem with indefinite weight function is discussed, are those published by J. Fleckinger and her co-workers (see, for example, A. Djellit [9], J. Fleckinger and M. L. Lapidus [11] and the references therein). However, their weight function is of a certain fractional type at infinity. Finally, for more general weights of L^{P} -type, in the semilinear case for the Laplace operator we refer to K. J. Brown and N. M. Stavrakakis [4] and for a quasilinear eigenvalue problem to J. Fleckinger et al. [12].

To be able to carry out our study and especially to apply the bifurcation methods, we introduce certain equivalent weighted and homogeneous Sobolev spaces. This is done in Section 2. We construct two function spaces which will form the base to develop our theory for both the linear and semilinear problem. These spaces are, on the one hand, the Hilbert space V_{2}, i.e. the closure of the C_{0}^{∞}(ℝ^{N})-functions with respect to the norm

|||u|||_{2} = \{∫_{ℝ^{N}} |Δu|^{2}dx − \frac{α}{2} ∫_{ℝ^{N}} g|u|^{2}dx\}

for an appropriate positive constant α to be chosen later and, on the other hand, the standard “energy” space D^{2,2}, i.e. the closure of the C_{0}^{∞}(ℝ)-functions with respect to the norm

||u||_{D^{2,2}}^{2} = ∫_{ℝ} |Δu|^{2}dx.

Also, we briefly state results, to be used later, concerning the existence, regularity and the asymptotic behavior of the first positive principal eigenvalue λ_{1} of the linearized problem (2.1)_{λ}.

In Section 3, we study the existence of a local continuum of positive solutions branching out from the first eigenvalue of the semilinear biharmonic problem (1.1)_{λ}- (1.2).

Finally, we get the global character of this continuum in Section 4. This fact is achieved by applying standard global bifurcation theory in the homogeneous Sobolev space D^{2,2}.

Notation: We denote by B_{r}(z) the open ball with center z ∈ ℝ^{N} and radius r. For simplicity we use the symbol || · ||_{p} for the norm || · ||_{L^{p}(ℝ)} and L^{p} for the space L^{p}(ℝ^{N}) (1 ≤ p ≤ ∞). The end of proofs is marked by the sign

2. Space setting - the linearized problem

In this section we shall discuss the existence of non-zero simple principal eigenvalues for the linearized biharmonic problem

(−Δ)^{2}u = λg(x)u  (x ∈ ℝ^{N})  (2.1)_{λ}
u(x) > 0 (x ∈ ℝ^{N}), lim_{|x|→+∞}u(x) = 0.  (2.2)

To simplify notation but without loss of generality we shall assume that f′(0) = 1, so that equation (2.1)_{λ} becomes the linearization of equation (1.1)_{λ}. The proofs of all results presented in this section are given in detail by L. Peletier and N. M. Stavrakakis in [22]. Also, the main results where announced in [28].

The natural space setting for the eigenfunctions of this problem, as we show next, will be the space D^{2,2}(ℝ^{N}), i.e. the closure of the C_{0}^{∞}(ℝ^{N})-functions with respect to the norm

||u||_{D^{2,2}} = (∫_{ℝ^{N}} |Δu|^{2}dx)^{1/2} .

It can be shown (see, for example, in C. A. Swanson [30]) that

D^{2,2} = \{u ∈ L^{\frac{2N}{N−4}} (ℝ^{N}) : |∇^{2}u| ∈ L^{2}(ℝ^{N})\}

and that there exists K > 0 such that for all u ∈ D^{2,2}

||u||_{\frac{2N}{N−4}} ≤ K ||u||_{D^{2,2}}.

So D^{2,2}(ℝ^{N}) is a reflexive Banach space.

Our approach is based on the following inequality of generalized Poincaré type.

Lemma 2.1. Suppose g ∈ L^{N/4}(ℝ^{N}). Then there exists α = 1/K||g||_{N/4}^{1/2} > 0 such that

∫_{ℝ^{N}} |Δu|^{2}dx ≥ α ∫_{ℝ^{N}} |g| |u|^{2}dx,

for all u ∈ C_{0}^{∞}(ℝ^{N}).

Thus if g ∈ L^{N/4}(ℝ^{N}) and α > 0 is as in Lemma 2.1, we can define an inner product on C_{0}^{∞}(ℝ^{N}) by

⟨u, v⟩_{2} = ∫_{ℝ^{N}} ΔuΔv dx − \frac{α}{2} ∫_{ℝ^{N}} guv dx.

Next we define V_{2} to be the completion of C_{0}^{∞}(ℝ^{N}) with respect to the above product. The space V_{2} depends on the function g; it is natural to expect that V_{2} grows as |g| becomes smaller. However, under condition (G) we prove that V_{2} is independent of this function. In fact, the space V_{2} is characterized by the following

Lemma 2.2. Suppose g ∈ L^{N/4}(ℝ^{N}). Then V_{2} = D^{2,2}.

Thus we may henceforth suppose that ||| · |||_{2}, the norm in V_{2}, coincides with the norm in D^{2,2} and that the inner product in V_{2} is given by

⟨u, v⟩ = ∫_{ℝ^{N}} ΔuΔv dx.

Proceeding as for example in [4], we define a bilinear form by

β(u, v) = ∫_{ℝ^{N}} guv dx (u, v ∈ V_{2}).

By the fact that V_{2} ⊆ L^{\frac{2N}{N−4}} (ℝ^{N}) we obtain that β is bounded in V_{2}. Hence by the Riesz Representation Theorem we can define a bounded linear operator M such that

β(u, v) = ⟨Mu, v⟩ (u, v ∈ V_{2}).

It is standard to check the following result.

Lemma 2.3. Suppose g ∈ L^{N/4}(ℝ^{N}). Then the operator M is selfadjoint and compact.

By means of classical spectral methods we have proved the existence, positivity and principality of the first eigenvalue. These results are given by the following

Theorem 2.4. Equation (2.1)_{λ} admits a positive first eigenvalue λ_{1} given by

λ_{1} = inf_{⟨Mu,u⟩=1}||u||_{D^{2,2}}^{2}.

The associated eigenfunction ϕ belongs to D^{2,2}(ℝ^{N}).

For the weak formulation of the problem the following result is necessary.

Theorem 2.5. Suppose that g satisfies hypothesis (G) and u ∈ D^{2,2}. Then there exists a sequence {R_{n}} ⊂ ℝ with R_{n} → ∞ as n → +∞ such that

lim_{n→∞} ∫_{∂B_{Rn}} ∇u\frac{∂∇u}{∂n} dS = 0 = lim_{n→∞} ∫_{∂B_{Rn}} u\frac{∂Δu}{∂n} dS = 0.

Having in mind the application of bifurcation theory to the study of problem (2.1)_{λ}, information concerning the dimension of the eigenspace associated to the principal eigenvalues of the linearized biharmonic problem (2.1)_{λ} are of basic importance. The main results in this direction, needed in the rest of the paper, can be stated as follows.

Theorem 2.6. Let g satisfy hypothesis (G). Then we have:

(i) The eigenspace corresponding to the principal eigenvalue λ_{1} is of dimension 1. (ii) λ_{1} is the only eigenvalue of (2.1)_{λ} to which a positive eigenfunction corresponds.

The proof of this theorem is long and technical. We again refer to [21] for the detailed proof.

Remark 2.7. The algebraic and the geometric multiplicities of the eigenvalues of the problem under discussion are equal since the operator M is compact and selfadjoint (see Lemma 2.3.) (see E. Zeidler [34]).

3. Local bifurcation results

In this section we shall obtain results on the existence of solutions for the nonlinear problem (1.1)_{λ}-(1.2), close to (λ_{1}, 0), by considering local bifurcation of solutions from the zero solution. First, we state a general asymptotic and regularity result for the solutions of this problem. Using Agmon’s theorem (see [1: Theorem 6.1]) and Serrin’s estimates from [27] (see also [15: Theorems 8.17 and 9.19]) we can prove the L^{σ}-character and the C_{loc}^{4,α}-regularity as well as the asymptotic properties of the solutions of (1.1)_{λ}. For these technics we refer also to S. Luckhaus [19]. The main results of this and the next section were announced in the paper [29].

Lemma 3.1. Suppose that u ∈ D^{2,2} is a solution of equation (1.1)_{λ}. Then:

(i) u is a classical solution, i.e. u ∈ C_{loc}^{4,α}(ℝ^{N}). Moreover, for any x_{0} ∈ ℝ^{N} and p > 1 we have

||u||_{W^{4,p}(B_{1}(x_{0}))} ≤ C||u||_{L^{\frac{2N}{N−4}} (B_{2}(x_{0}))},  (3.1)

where C = C(x_{0}, λ, N, m, ||g||_{∞}, |||u|||_{2}).

(ii) D^{β}u(x) decay uniformly to zero as |x| → +∞, for all |β| ≤ 3.

Proof. For the detailed proof we refer to [2: Lemma 5]

To apply local bifurcation theory we introduce the nonlinear operator P : ℝ×V_{2} → V_{2} through the relation

⟨P (λ, u), ϕ⟩ = ∫_{ℝ^{N}} ΔuΔϕ dx − λ ∫_{ℝ^{N}} gf(u)ϕ dx  (3.2),

for all ϕ ∈ V_{2}, where ⟨·, ·⟩ denotes the inner product in D^{2,2}.

758 N. M. Stavrakakis and N. Zographopoulos

Lemma 3.2. The operator P is well defined by (3.2). Proof. For fixed u ∈ D^{2,2} we define the functional

F (ϕ) = ∫_{ℝ^{N}} ΔuΔϕ dx − λ ∫_{ℝ^{N}} gf(u)ϕ dx,

for ϕ ∈ D^{2,2}. Since f satisfies hypothesis (F), then f ∈ L^{\frac{2N}{N−4}} (ℝ^{N}) and so for some constant K_{1}

\begin{aligned}|F (ϕ)| & ≤ ||Δu||_{2} ||Δϕ||_{2} + |λ| ||g||_{N/4} ||f(u)||_{\frac{2N}{N−4}} ||ϕ||_{\frac{2N}{N−4}} \\ & ≤ K_{1}(||Δu||_{2} + |λ| ||g||_{N/4} ||f(u)||_{\frac{2N}{N−4}} )||ϕ||_{V_{2}}. \end{aligned}

So F is a bounded linear functional. Hence by the Riesz Representation Theorem we may define P as in (3.2)

Lemma 3.3. The operator P defined by (3.2) is continuous and for N = 5, 6, …, 12 Fréchet differentiable with continuous Fréchet derivatives given by

\begin{aligned}⟨P_{u}(λ, u)ϕ, ψ⟩ & = ∫_{ℝ^{N}} ΔϕΔψ dx − λ ∫_{ℝ^{N}} gf′(u)ϕψ dx, \\ ⟨P_{λ}(λ, u), ϕ⟩ & = − ∫_{ℝ^{N}} gf(u)ϕ dx, \\ ⟨P_{λu}(λ, u)ϕ, ψ⟩ & = − ∫_{ℝ^{N}} gf′(u)ϕψ dx, \end{aligned}

for all ϕ, ψ ∈ D^{2,2}.

Proof. For completeness we juct sketch the proof.

(i) To prove the continuity of P, we have

⟨P (λ, u) − P (μ, w), v⟩ = ⟨P (λ, u) − P (λ, w), v⟩ + ⟨P (λ, w) − P (μ, w), v⟩

= |∫_{ℝ^{N}} (Δu − Δw)Δv dx|

+ λ |∫_{ℝ^{N}} g(f(u) − f(w))v dx| + |λ − μ| |∫_{ℝ^{N}} gf(w)v dx|

≤ {1 + λ sup_{τ∈(0,1)} ||g||_{N/4} ||f′(u + τw)||_{∞}}||u − w||_{D^{2,2}}||v||_{D^{2,2}}

+ c|λ − μ| ||g||_{N/4}||w||_{D^{2,2}}||v||_{D^{2,2}}.

So P is continuous at any (λ, u) ∈ ℝ × D^{2,2}.

(ii) Let u, h ∈ D^{2,2} and

P (λ, u, h) = P (λ, u + h) − P (λ, u) − P_{u}(λ, u)h.

Then we have

∥P (λ, u + h) − P (λ, u) − P_{u}(λ, u)h∥_{D^{2,2}}^{2}

≤ |λ| |∫_{ℝ^{N}} g(f(u + h) − f(u) − f′(u)h) P (λ, u, h)dx|

≤ |λ| sup_{t∈(0,1)} ||f′′(u + th)||_{∞} ||g||_{q} ||h||_{p}^{2} ||P (λ, u, h)||_{p},

where p = \frac{2N}{N−4} and \frac{1}{q} + \frac{3}{p} = 1, i.e. q = \frac{2N}{12−N}. Hence for N = 5, 6, …, 12 we get that q ≥ \frac{N}{4} and so g ∈ L^{q}(ℝ^{N}). Hence we have

∥P (λ, u + h) − P (λ, u) − P_{u}(λ, u)h∥_{D^{2,2}}

≤ |λ| sup_{t∈(0,1)} ||f′′(u + th)||_{∞} ||g||_{q} ||h||_{D^{2,2}}^{2}

= o(||h||_{D^{2,2}}).

So P is Fréchet differentiable in ℝ × D^{2,2}.

(iii) Let u, v, ϕ ∈ D^{2,2}. Then we have

|⟨P_{u}(λ, u)h − P_{u}(μ, v)h, ϕ⟩|

≤ |⟨P_{u}(λ, u)h − P_{u}(λ, v)h, ϕ⟩ + ⟨P_{u}(λ, v)h − P_{u}(μ, v)h, ϕ⟩|

≤ |λ| |∫_{ℝ^{N}} g(f′(u) − f′(v))hϕ dx| + |λ − μ| |∫_{ℝ^{N}} gf′(v)ϕ dx|

≤ |λ| sup_{t∈(0,1)} ||f′′(tu + (1 − t)v)||_{∞} ||g||_{q} ||u − q||_{D^{2,2}} ||h||_{D^{2,2}} ||ϕ||_{D^{2,2}}

+ |λ − μ| ||g||_{N/4} ||f′(v)||_{∞} ||h||_{p} ||ϕ||_{p},

where p = \frac{2N}{N−4} and q = \frac{2N}{12−N} for N = 5, 6, …, 12. Hence we have that P_{u} is continuous in ℝ × D^{2,2}

Next we have the following local result.

Theorem 3.4 (Local Bifurcation). There exists ε_{0} > 0 and continuous functions η : (−ε_{0}, ε_{0}) → ℝ and ψ : (−ε_{0}, ε_{0}) → [ϕ]^{⊥} such that η(0) = λ_{1}, ψ(0) = 0 and every non-trivial solution of P (λ, u) = 0 in a small neighbourhood of (λ_{1}, 0) is of the form (λ_{ε}, u_{ε}) = (η(ε), εϕ + εψ(ε)).

Proof. We shall prove that the operator P satisfies all the hypotheses of the local bifurcation theorem in M. Crandall and P. Rabinowitz [7].

(i) The operator P_{u}(λ_{1}, 0) is linear, compact, selfadjoint and P_{u}(λ_{1}, 0)ϕ = 0 if and only if ϕ ∈ V_{2} is a solution of equation (2.1)_{λ_{1}}. Therefore N(P_{u}(λ_{1}, 0)) = [ϕ] where ϕ is the principal eigenfunction of (2.1)_{λ_{1}}. So ψ ∈ R(P_{u}(λ_{1}, 0)) if and only if there exists w ∈ D^{2,2}(ℝ^{N}) and ⟨ψ, ϕ⟩ = ⟨P_{u}(λ_{1}, 0)w, ϕ⟩. But selfadjointness of P_{u}(λ_{1}, 0) implies that

⟨P_{u}(λ_{1}, 0)w, ϕ⟩ = ⟨w, P_{u}(λ_{1}, 0)ϕ⟩ = ⟨w, 0⟩ = 0.

Hence R(P_{u}(λ_{1}, 0)) = [ϕ]^{⊥}.

(ii) Let w ∈ N(P_{u}(λ_{1}, 0)) ∩ R(P_{u}(λ_{1}, 0)). Then P_{u}(λ_{1}, 0)w = 0 and there exists ψ ∈ D^{2,2}(ℝ^{N}) such that ⟨P_{u}(λ_{1}, 0)ψ, w⟩ = ⟨w, w⟩. Again selfadjointness implies that 0 = ⟨ψ, P_{u}(λ_{1}, 0)w⟩ = ⟨w, w⟩. So

⟨w, w⟩ = ||w||_{D^{2,2}} = (∫_{ℝ^{N}} |Δw|^{2} dx)^{1/2} = 0.

Then using Lemma 3.1/(ii) we obtain that w ≡ 0 in ℝ^{N}, i.e.

N(P_{u}(λ_{1}, 0)) ∩ R(P_{u}(λ_{1}, 0)) = {0}.

Also, it is easy to see that

N(P_{u}(λ_{1}, 0)) ⊕ R(P_{u}(λ_{1}, 0)) = D^{2,2}(ℝ^{N}).

(iii) Finally, we have that P_{λu}(λ_{1}, 0)ϕ ∉ R(P_{u}(λ_{1}, 0)) (transversality condition) since

⟨P_{λu}(λ_{1}, 0)ϕ, ϕ⟩ = − ∫_{ℝ^{N}} gϕ^{2}dx < 0

and the proof is completed

The last statement of this section describes the sign of solutions of equation (1.1)_{λ} close to the bifurcation point.

Theorem 3.5. Let (λ_{ε}, u_{ε}) be solutions of equation (1.1)_{λ} given by Theorem 3.4, ε > 0. Then there exists ε_{0} > 0 such that u_{ε} > 0 in ℝ^{N} whenever 0 < ε ≤ ε_{0}.

Proof. From the above theorem we have that u_{ε} = εϕ+εψ(ε) satisfies (1.1)_{λ} where λ = η(ε). We have

(−Δ)^{2}ψ(ε) = η(ε)g \frac{f(εϕ + εψ(ε))}{ε} − λ_{1}gϕ,

which implies that

(−Δ)^{2}ψ(ε) − ρ(x)ψ(ε) = (η(ε) − λ_{1})gϕ + \frac{1}{2}η(ε)gϕf′′(ξ(ε, x))u_{ε},

where ρ(x) = η(ε)g[1 + \frac{1}{2}f′′(ξ(ε, x))u_{ε}] and ξ(ε, x) lies between 0 and u_{ε}(x). Whenever 0 < ε ≤ ε_{0}, it is easy to prove that, for some positive constant C, |u_{ε}(x)| ≤ C for all x ∈ ℝ^{N}, and so ρ is uniformly bounded (i.e. ||ρ||_{∞} < +∞). Moreover,

||gϕ||_{\frac{2N}{N−4}} ≤ ||g||_{∞} ||ϕ||_{\frac{2N}{N−4}} ≤ ||g||_{∞} ||ϕ||_{V_{2}}

and

||gϕu_{ε}||_{\frac{2N}{N−4}} ≤ ||g||_{∞} ||ϕ||_{∞} ||u_{ε}||_{\frac{2N}{N−4}} ≤ ||g||_{∞} ||ϕ||_{∞} ||u_{ε}||_{V_{2}}.

If N = 5, 6, 7 and \frac{s}{2} = \frac{2N}{N−4}, then s > N. Hence by Lemma 3.1 with p = \frac{s}{2} = \frac{2N}{N−4}, there exists a constant K > 0 (independent of ε) such that

sup_{|x|≤R_{0}} |ψ(ε)(x)| ≤ K||g||_{∞}{(η(ε) − λ_{1})||ϕ||_{V_{2}} + ||ϕ||_{∞} ||u_{ε}||_{V_{2}}}.

Since ϕ(x) > 0 for all x in the compact set B_{R_{0}} = {x ∈ ℝ^{N} : |x| ≤ R_{0}}, it follows that there exists ε_{1} > 0 such that ϕ(x) + ψ(ε)(x) > 0 for all x ≤ R_{0} provided that 0 < ε < ε_{1}. Suppose 0 < ε < ε_{1} and that u_{ε}(x_{0}) < 0 for some x_{0} ∈ ℝ^{N}. We put equation (1.1)_{λ} in form of the system

\begin{matrix}−Δu_{ε} & = v_{ε}, \\ −Δv_{ε} & = λg(x)f(u_{ε}) \end{matrix} \}  (3.3)

where we know by Lemma 3.1 that lim_{|x|→+∞} u_{ε}(x) = lim_{|x|→+∞} v_{ε}(x) = 0. Since g(x)f(u) ≥ 0 in ℝ^{N} × ℝ^{+}, for any x_{0} ∈ ℝ^{N} we may apply the maximum principle on the ball B_{r}(x_{0}), for r large enough to obtain v_{ε}(x_{0}) ≥ inf_{|x|=r} v_{ε}(x). Letting r → +∞ we get that −Δu ≥ 0 so u_{ε} ≥ 0. Therefore applying the maximum principle again (see [15: Theorem 8.18]) on the non-trivial solution u_{ε} we get that u_{ε} > 0.

Since lim_{|x|→∞} u_{ε}(x) = 0, it follows that there must exist x_{1}, |x_{1}| > R, such that u_{ε} attains a negative minimum at x_{1}. But then

−Δu_{ε}(x_{1}) = λg(x_{1})f(u_{ε}(x_{1})) > 0,

which is impossible. Hence u_{ε}(x) > 0 for all x ∈ ℝ^{N} whenever 0 < ε < ε_{1}

4. Global continuation of the branch

To discuss the global nature of the continuum of solutions bifurcating from (λ_{1}, 0), we write the operator P in the form P (λ, u) = u − λR(u), where

⟨R(u), ϕ⟩ = ∫_{ℝ^{N}} g(x)f(u(x))ϕ(x) dx (ϕ ∈ D^{2,2}).

Also, we assume that R satisfies the relation

R(u) = Mu + H(u),

where M denotes the same linear operator as in Section 2, i.e.

⟨Mu, v⟩ = ∫_{ℝ^{N}} guv dx (u, v ∈ D^{2,2})

and H(u) = O(|||u|||_{2}^{β}) as |||u|||_{2} → 0, for some β > 1.

It is shown in Section 2 that λ_{1} is an eigenvalue of L and by Theorem 2.6 the eigenspace associated with λ_{1} has algebraic and geometric (see Remark 2.7) multiplicity 1. Also, as f is a Lipschitz function, it can be proved by modifying slightly the proof of Lemma 2.3 that R is a compact operator (see in [22]). Thus we can apply the classical global bifurcation theorem (see P. Rabinowitz [26]) to obtain the following

Theoprem 4.1. There exists a continuum C of non-zero solutions of problem (1.1)_{λ} − (1.2) bifurcating from (λ_{1}, 0), which is either

(i) unbounded or

(ii) contains a point (λ, 0), where λ ≠ λ_{1} is an eigenvalue of (2.1)_{λ}.

Moreover, C has a connected subset C^{+} ⊂ C − {(η(ε), u_{ε}) : −ε_{0} ≤ ε ≤ 0} for some ε_{0} > 0 such that C^{+} also satisfies one of the above alternatives. Close to the bifurcation point (λ_{1}, 0), C^{+} consists of the curve ε → (η(ε), u_{ε}) (0 < ε ≤ ε_{0}).

The following lemmas describe the nature of solutions lying on C^{+}.

Lemma 4.2. There exists λ_{∗} > 0 such that λ > λ_{∗} whenever (λ, u) ∈ C^{+}.

Proof. Suppose u ∈ V_{2} is a solution of problem (1.1)_{λ}−(1.2). Multiplying equation (1.1)_{λ} by u, integrating over B_{R}, letting R → ∞ and using Theorem 2.5 we obtain

|||u|||_{2}^{2} = ∫_{ℝ^{N}} |∇u|^{2}dx

= λ ∫_{ℝ^{N}} gf(u)u dx

≤ λk^{∗}||g||_{N/4} ||u||2_{\frac{2N}{N−4}} (where |f(u)| ≤ k^{∗}u for all u)

≤ λK_{1}||g||_{N/4} |||u|||_{2}^{2},

where K_{1} is a constant and the result follows. Therefore we may get that λ_{∗} = 1/K_{1}||g||_{N/4} > 0

In order to proceed further we must investigate the L^{∞}-closeness of solutions which are close in ℝ × D^{2,2}; since D^{2,2} does not embed into L^{∞}(ℝ^{N}), this is not immediately obvious. Actually, we can prove

Lemma 4.3. Let N < 8. Then there exist constants K_{1} > 0 and K_{2} > 0 such that

|u_{λ}(x) − u_{μ}(x)| ≤ K_{1}|λ − μ| + K_{2}||u_{λ} − u_{μ}||_{D^{2,2}}, for all x ∈ ℝ^{N},

whenever μ is close to λ and u_{λ}, u_{μ} ∈ D^{2,2} are solutions of equations (1.1)_{λ} and (1.1)_{μ}, respectively.

Proof. Indeed, it is easy to see that

(−Δ)^{2}(u_{λ} − u_{μ}) = g{λf(u_{λ}) − μf(u_{μ})}.

Hence by Lemma 3.1 there exists C > 0 such that

|u_{λ}(x) − u_{μ}(x)|

≤ sup_{y∈B_{1}(x)} |u_{λ}(y) − u_{μ}(y)|

≤ C{||u_{λ} − u_{μ}||_{L_{p}(B_{2}(x))} + ||g [λf(u_{λ}) − μf(u_{μ})]||_{p}}

(where p = \frac{2N}{N−4} and by Lemma 3.1 2p = \frac{4N}{N−4} > N, so N < 8)

≤ C_{1}|||u_{λ} − u_{μ}|||_{2} + C_{2}||g||_{∞}{||(λ − μ)f(u_{λ})||_{p} + ||μ[f(u_{λ}) − f(u_{μ})]||_{p}}

≤ C_{1}|||u_{λ} − u_{μ}|||_{2} + C_{2}|λ − μ| ||g||_{∞} |||u_{λ}|||_{2} + C_{3}|μ| ||g||_{∞} |||u_{λ} − u_{μ}|||_{2}, where C_{1}, C_{2} and C_{3} are constants and the result follows

Theorem 4.4. C^{+} contains no points of the form (λ, 0), where λ ≠ λ_{1}.

Proof. Suppose that there exists (λ, u) ∈ C^{+} such that u(x_{0}) < 0 for some x_{0} ∈ ℝ^{N}. By Theorem 3.5, u(x) > 0 for all x ∈ ℝ^{N} whenever (λ, u) ∈ C^{+} is close to (λ_{1}, 0). Moreover, by Lemma 4.3, points in C^{+} which are close in ℝ × V must also be close in ℝ × L^{∞}(ℝ^{N}). Hence there must exist (λ_{0}, u_{0}) ∈ C^{+} such that u_{0}(x) ≥ 0 for all x ∈ ℝ^{N} but u_{0}(x_{0}) = 0 for some x_{0} ∈ ℝ^{N}, and in any neighbourhood of (λ_{0}, u_{0}) we can find a point (λ̂, û) ∈ C^{+} with û(x) < 0 for some x ∈ ℝ^{N}. Let B denote any open ball containing x_{0}. Then

\begin{matrix}−Δu_{0}(x) − λg(x)\frac{f(u_{0}(x))}{u_{0}(x)}u_{0}(x) & = 0  on B \\ u_{0}(x) & ≥ 0  on ∂B. \end{matrix} \}

It follows from the Serrin Maximum principle (see [14]) that u_{0} ≡ 0 on B. Hence u_{0} ≡ 0 on ℝ^{N}.

Thus we can construct a sequence {(λ_{n}, u_{n})} ⊆ C^{+} such that u_{n}(x) > 0 for all n ∈ ℕ and x ∈ ℝ^{N}, u_{n} → 0 in V_{∈} and λ_{n} → λ_{0}. Let v_{n} = \frac{u_{n}}{|||u_{n}|||_{2}} . Since u_{n} = λ_{n}M(u_{n}) + λ_{n}H(u_{n}) we have

v_{n} = λ_{n}M(v_{n}) + λ_{n} \frac{H(u_{n})}{||u_{n}||_{V}} .

Since M is compact, there exists a subsequence of {v_{n}} (which we again denote by \{v_{n}\}) such that \{M(v_{n})\} is convergent. Since lim_{n→∞} \frac{H(u_{n})}{|||u_{n}|||_{2}} = 0, \{v_{n}\} is convergent to v_{0}, say, and v_{0} = λ_{0}M(v_{0}). Since v_{n} ≥ 0 for all n ∈ ℕ, v_{0} ≥ 0. Since by Theorem 2.6 λ_{1} is the only positive eigenvalue corresponding to a positive eigenfunction, it follows that λ_{1} = λ_{0}. Thus (λ_{0}, u_{0}) = (λ_{1}, 0) and this contradicts the fact that every neighbourhood of (λ_{0}, u_{0}) must contain a solution (λ̂, û) ∈ C^{+} with û(x) < 0, for some x ∈ ℝ^{N}. Hence u(x) > 0 for all x ∈ ℝ^{N} whenever (λ, u) ∈ C^{+}

Corollary 4.5. C^{+} must connect (λ_{1}, 0) to ∞ in ℝ × D^{2,2}.

The next theorem shows that C^{+} cannot become unbounded at a finite value of λ; in order to obtain this result we must impose some further restrictions on f and g.

Theorem 4.6. Suppose that, for some γ ∈ (0, 1), f(s) ∼ |s|^{γ} at infinity and g ∈ L^{p}(ℝ^{N}), where p < \frac{2N}{2N−(γ+1)(N−4)}. Then there exists a continuous function K : ℝ^{+} → ℝ^{+} such that |||u|||_{2} ≤ K(λ) whenever (λ, u) ∈ C^{+}.

Proof. As in Lemma 4.2 we obtain that if u satisfies problem (1.1)_{λ} − (1.2), then

|||u|||_{2}^{2} = λ ∫_{ℝ^{N}} gf(u)u dx ≤ λK ∫_{ℝ^{N}} |g|u^{1+γ}dx ≤ λK||g||_{p}^{1/p}||u||_{(1+γ)q}^{1/q}

where q is such that \frac{1}{p} + \frac{1}{q} = 1. We set (1 + γ)q = \frac{2N}{N−4}. Then we get \frac{1}{q} = \frac{(1+γ)(N−4)}{2N} =

β < 2. We obtain ||u||_{2}^{2} ≤ λK||g||_{p}^{1/p}||u||_{2}^{β}. So ||u||_{2} ≤ λK||g||p^{\frac{1}{2−β}} ≡ K(λ) and the proof

is complete

As an immediate consequence of the previous results we can give the following complete description of the continuum C^{+}.

Theorem 4.7 (Global Bifurcation). Suppose that N = 5, 6, 7, for some γ ∈ (0, 1), Sf(s) ∼ |s|^{γ} at infinity and g ≥ 0, g ∈ L^{p}(ℝ^{N}) ∩ L^{∞}(ℝ^{N}), where 1 < p < \frac{2N}{2N−(γ+1)(N−4)}. Then there exists a continuum C^{+} ⊆ ℝ × D^{2,2} of solutions of problem (1.1)_{λ} − (1.2) bifurcating from the zero solution at (λ_{1}, 0) such that:

(i) If (λ, u) ∈ C^{+}, then λ > λ_{∗} > 0, u ∈ L^{∞} and u(x) > 0 for all x ∈ ℝ^{N}.

(ii) {λ : (λ, u) ∈ C^{+} for some u ∈ D^{2,2}} ⊇ (λ_{1}, ∞].

In particular, problem (1.1)_{λ} −(1.2) has a non-trivial solution u ∈ D^{2,2} such that u(x) > 0 for all x ∈ ℝ^{N} whenever λ > λ_{1}.

Remark 4.7. The restriction on the dimension seems to be relared, via the Sobolev embeddings, to the certain technique used. For example, in the case of the Laplacian problem the authors of the work [4] have proved that the restriction on the dimension appearing there, based on the bifurcation technique, may be removed by an other method (super- and sub-solutions) as it is proved in the work [5]. However, in the work [5] we got a general existence result, instead of the more geometric inside in the problem we gained by applying bifurcation methods. We guess that the dimension restriction is the price we have to pay for the geometric clarity of the solution branches.

Acknowledgments. Part of this work was made possible by financial support from grant number ERBCHRXCT 930409 under the EC Human Capital and Mobility Scheme.

References

[1] Agmon, S.: The L_{p} approach to the Dirichlet problem. Ann. Scuola Norm. Sup. di Pisa 13 (1959), 405 – 448.

[2] Allegretto, W. and L. S. Yu: Decaying solutions of 2m^{th} order elliptic problems. Can. J. Math. 43 (1991), 449 – 460.

[3] Bernis, F.: Elliptic and parabolic semilinear problems without conditions at infinity. Arch. Rat. Mech. Anal. 106 (1989), 217 – 241.

[4] Brown, K. J. and N. M. Stavrakakis: Global bifurcation results for a semilinear elliptic equation on all of ℝ^{N}. Duke Math. J. 85 (1996), 77 – 94.

[5] Brown, K. J. and N. M. Stavrakakis: On the construction of super and subsolutions for a semilinear elliptic equation on all of ℝ^{N}. Nonlin. Anal. TMA 32 (1998), 87 – 95.

[6] Clément Ph., de Figueiredo, D. G. and E. Mitidieri: Positive solutions of semilinear elliptic systems. Comm. Part. Diff. Equ. 17 (1992), 923 – 940.

[7] Crandall, M. and P. H. Rabinowitz: Bifurcation from simple eigenvalues. J. Func. Anal. 8 (1971), 321 – 340.

[8] Dalmasso, R.: Elliptic equations of order 2m in annular domains. Trans. Amer. Math. Soc. 347 (1995), 3575 – 3585.

[9] Djellit, A.: Valeurs propres de problèmes elliptiques indèfinis sur des ouverts non-bornés. Ph.D. Thesis. Toulouse: Université Paul Sabatier 1992.

Global Bifurcation Results 765

[10] Edmunds, D. E., Fortunato, D. and E. Jannelli: Critical exponents, critical dimensions and the biharmonic operator. Arch. Rat. Mech. Anal. 112 (1992), 269 – 289.

[11] Fleckinger, J. and M. L. Lapidus: Eigenvalues for elliptic boundary value problems with an indefinite weight function. Trans. Amer. Math. Soc. 295 (1986), 305 – 324.

[12] Fleckinger, J., Manasevich, R., Stavrakakis, N. M. and F. de Thelin: Principal eigenvalues for some quasilinear elliptic systems on ℝ^{N}. Adv. Diff. Equ. Vol 2 (6) (1997), 981–1003.

[13] Furusho, Y. and T. Kusano: Existence of positive entire solutions for higher order quasilinear elliptic equations. J. Math. Soc. Japan 46 (1994), 449 – 465.

[14] B. Gidas, Ni, W.-M. and L. Nirenberg: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209 – 243.

[15] Gilbarg, G. and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order (2nd Edition). Berlin: Springer-Verlag 1983.

[16] Gu Yong-Geng: The eigenvalue problems of elliptic equations of higher order. Acta Math. Scientia 11 (1991), 361 – 368.

[17] Kusano, T., Naito, M. and C. A. Swanson: Radial entire solutions to even order semilinear elliptic equations in the plane. Proc. Roy. Soc. Edinb. 105A (1987), 275 – 287.

[18] Kusano, T., Naito, M. and C. A. Swanson: Radial Entire solutions of even order semilinear elliptic equations. Can. J. Math. 195 (1988), 1281 – 1300.

[19] Luckhaus, S.: Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order. J. Reine Angew. Math. 306 (1979), 192 – 207.

[20] Mitidieri, E.: A Rellich type identity and applications. Comm. Part. Diff. Equ. 18 (1993), 125 – 151.

[21] Noussair, E. S., Swanson, C. A. and Y. Yianfu: Critical semilinear biharmonic equations in ℝ^{N}. Proc. Roy. Soc. Edinb. 121A (1992), 139 – 148.

[22] Peletier, L. A. and N. M. Stavrakakis: Existence and simplicity of principal eigenvalues for a linear polyharmonic problem on ℝ^{N}., in progress.

[23] Peletier, L. A. and R. C. A. M. van der Vorst: Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation. Diff. Int. Equ. 5 (1992), 747 – 767.

[24] Peletier, L. A., Troy, W. C. and R. C. A. M. van der Vorst: Stationary solutions of a fourth order nonlinear diffusion equation. Preprint 1995.

[25] Pucci, P. and J. Serrin: Critical exponents and critical dimensions for polyharmonic operators. J. Math. Pur. Appl. 69 (1990), 55 – 83.

[26] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7 (1971), 487 – 513.

[27] Serrin, J.: Local behavior of solutions of quasilinear equations. Acta. Math. 111 (1964), 247 – 302.

[28] Stavrakakis, N. M.: Eigenvalue problem for indefinite-weight linear elliptic equations in ℝ^{N}. In: ICIAM/GAMM 95 Proceedings (eds.: O. Mahrenholtz and R. Mennicken). Z. Ang. Math. Mech. (ZAMM) Special Issue 2 (1996), pp. 683 – 685.

[29] Stavrakakis. N. M.: Semilinear biharmonic problems on ℝ^{N}. In: Reaction Diffusion Systems (eds.: G. Caristi and E. Miditieri). New York: Marcel Dekker Inc. 1997, pp. 365 – 376.

[30] Stavrakakis, N. M. and G. Sweers: Positivity for a noncooperative system of elliptic equations on all of ℝ^{N}. Adv. Diff. Equ. 4 (1999), 115-136.

[31] Swanson, C. A.: The best Sobolev constant. Appl. Anal. 47 (1992), 227 – 239.

[32] Swanson, C. A.: Uniqueness for semilinear polyharmonic problems. Nonlin. Anal. 25 (1995), 1055 – 1062.

[33] Swanson, C. A. and L. S. Yu: Radial polyharmonic problems in ℝ^{N}. J. Math. Anal. Appl. 174 (1993), 461 – 466.

[34] Zeidler, E.: Nonlinear Functional Analysis and its Applications. Berlin: Springer-Verlag 1990.

Received 29.12.1997; in revised form 31.03.1999