The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity^{∗}

Alberto Ferrero Dipartimento di Mathematica Università di Pisa Largo Bruno Pontecorvo 5 56127 Pisa (Italy)

Hans-Christoph Grunau Fakultät für Mathematik Otto-von-Guericke-Universität Postfach 4120 39016 Magdeburg (Germany)

Abstract

For a semilinear biharmonic Dirichlet problem in the ball with supercritical power-type nonlinearity, we study existence/nonexistence, regularity and stability of radial positive minimal solutions. Moreover, qualitative properties, and in particular the precise asymptotic behaviour near x = 0 for (possibly existing) singular radial solutions, are deduced. Dynamical systems arguments and a suitable Lyapunov (energy) function are employed.

1 Introduction

Many papers have studied second order elliptic boundary value problems with supercritical growth. We only mention the work by Brezis, Cazenave, Martel, Ramiandrisoa and Brezis, Vazquez [5, 6], where the role of singular solutions, the change in the bifurcation diagrams in dependence on the space dimension and the nonlinearity as well as many other interesting features were highlighted. In these works, also many references to previous related important work can be found.

According to [14, Section 4.2 (c)], it is an important task to gain also a deeper understanding for related higher order problems. Due to the lack of a general maximum principle and many other strong tools typical for second order equations, up to now only relatively limited results have been available for this case.

In [2] biharmonic Dirichlet problems with exponential nonlinearities could be studied to some extent, while important questions concerning e.g. the existence of singular solutions are in general still open. A first investigation of biharmonic Dirichlet problems with power-type nonlinearities was done in [3]. It turned out, however, that the power case is technically much more involved than the exponential case, where advantage could be taken of the extreme convexity properties of the nonlinearity. It is the goal of the present paper to develop tools which enable us to prove analogues for the results in [2] also for power-type nonlinearities. Moreover, we improve on some of the achievements for the exponential case.

Let us now describe the scope of the present paper in some detail. We study the Dirichlet problem

\{ \begin{matrix}Δ^{2}u = λ(1 + u)^{p}, & in B, \\ u > 0 & in B, \\ u = |∇u| = 0 & on ∂B, \end{matrix}  (1)

∗Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.

where B ⊂ ℝ^{n} is the unit ball, λ > 0 is an eigenvalue parameter, n ≥ 5 and p > \frac{n+4}{n−4}. The subcritical case p < \frac{n+4}{n−4} is by now “folklore”, where existence and multiplicity results are easily established by means of variational methods. For the critical case p = \frac{n+4}{n−4} (under Navier boundary conditions), we refer to [3].

According to related work on second order equations and on the biharmonic Dirichlet problem with exponential nonlinearity, we address questions concerning existence/nonexistence, smoothness and stability of positive minimal solutions, characterization of radial singular solutions in terms of critical points of associated dynamical systems, and some further qualitative properties of (possibly existing) singular solutions. In the present work, a singular solution is always understood to be singular at the origin x = 0.

The first of our results, a precise formulation of which is given in the next section, concerns existence of “minimal” solutions. It is shown that there exists a limiting parameter λ^{∗} = λ^{∗}(n, p) such that one has existence of stable regular minimal solutions to (1) for λ ∈ (0, λ^{∗}), while for λ > λ^{∗}, not even singular solutions exist. In second order problems this immediate switch from existence of regular to nonexistence even of singular solutions is established by using suitable functions of possibly existing singular solutions as bounded supersolutions, see [5]. Such techniques seem to fail completely for fourth and higher order problems. Here, we employ dynamical systems arguments, and this is one reason why we have to formulate our results in the ball. A second reason for this restriction is the need to apply comparison principles which are known to fail in general domains but to hold in the ball.

Further results concern qualitative properties of radial singular solutions. The precise blow-up rate ∼ C|x|^{−4/(p−1)} at x = 0 is determined and an explicit estimate from below is deduced. For this purpose, in Section 4, we transform the differential equation in (1) into an autonomous system of ordinary differential equations and apply subtle energy estimates. This technique has proved to be very powerful for studying the precise asymptotic behaviour of entire solutions in ℝ^{n} in [9]. Moreover, a characterization of singular (resp. regular) radial solutions to (1) in terms of the corresponding dynamical system is given. This system is shown to have two critical points, the unstable manifolds of which are related to singular (resp. regular) radial solutions.

As in the case of an exponential nonlinearity, we cannot yet provide an analytical proof for the existence of singular solutions. In that case, see [2], a computer assisted proof was given for dimensions n = 5, …, 16. We are convinced that for selected values of n and p, the same should also work in the present situation. However, since we have to consider not only countably many values of n but also uncountably many of p, a computer assisted proof may not be suitable here and we formulate the existence of singular solutions to (1) with a suitable parameter λ_{s} as an important and presumably difficult open problem. Once the existence of singular solutions has been established, further interesting questions concerning the singular parameter λ_{s} and the extremal parameter λ^{∗} arise: In which dimensions n and for which exponents p does one have λ_{s} = λ^{∗} so that one has a singular extremal solution?

“Large” solutions of the differential equation Δ^{2}u = λ(1+u)^{p} are studied in a recent work of Diaz, Lazzo, Schmidt [8]. The focus of their work, however, is different from ours, since they classify those solutions of the differential equation, which converge to ∞ at ∂B.

The paper is organized as follows: In the next section, a precise formulation of our results is given. In Section 3, the partial differential equation in (1) applied to radial functions is transformed into an autonomous system of ordinary differential equations. A phase space analysis is performed in Section 4 in order to characterize singular and regular radial solutions to (1). Section 5 is devoted to proving existence of regular minimal solutions, the stability of which is studied in Section 7. A refined use of Lyapunov or energy functions in Section 6 will yield precise information of qualitative properties of radial singular solutions.

2 Main results

Definition 1. We say that u ∈ L^{p}(B) is a solution of (1) if u ≥ 0 and if for all φ ∈ C^{4}(\overline{B}) with φ|∂B = |∇φ| |∂B = 0 one has

∫_{B} uΔ^{2}φ dx = λ ∫_{B}(1 + u)^{p}φ dx.

We call u singular if u ∉ L^{∞}(B), and regular if u ∈ L^{∞}(B).

A radial singular solution u = u(r) of (1) is called weakly singular if lim_{r→0} r^{4/(p−1)}u(r) ∈ [0, ∞] exists.

Weakly singular solutions display a somehow specified asymptotic behaviour at the origin. Below we shall prove that any radial singular solution is weakly singular. Note that by standard regularity theory for the biharmonic operator (see [1]), any regular solution u of (1) satisfies u ∈ C^{∞}(\overline{B}). Note also that by the positivity preserving property of Δ^{2} in the ball [4] any solution of (1) is positive, see also [2, Lemmas 16 and 18] for a generalized statement. This property is known to fail in general domains. For this reason, we restrict ourselves to balls also in Theorems 1 and 2. Hence, the sub- and supersolution method applies as well as monotone iterative procedures.

We also need the notion of minimal solution:

Definition 2. We call a solution u of (1) minimal if u ≤ v a.e. in B for any further solution v of (1).

In order to state our results, we denote by λ_{1} > 0 the first eigenvalue for the biharmonic operator with Dirichlet boundary conditions

\{ \begin{matrix}Δ^{2}φ_{1} = λ_{1}φ_{1} & in B \\ φ_{1} = \frac{∂φ_{1}}{∂n} = 0 & on ∂B. \end{matrix}  (2)

It is known from the positivity preserving property and Jentzsch’s (or Krein–Rutman’s) theorem that λ_{1} is isolated and simple and that the corresponding eigenfunctions φ_{1} do not change sign.

Define Λ := {λ > 0 : (1) admits a solution}; λ^{∗} := sup Λ.

It is well known that Λ is a bounded interval (see [2, 3]). Our first main result states that on the open interior of this interval, i.e. for any 0 < λ < λ^{∗} = sup Λ < ∞, problem (1) admits a regular minimal solution u. Similarly as in second order problems, where much stronger tools are available, we are able to prove the immediate switch from existence of a regular minimal solution to nonexistence even of singular solutions. For biharmonic supercritical equations this was proved earlier [2, Theorem 1] only for an exponential nonlinearity, where advantage could be taken of the extreme convexity of the exponential. Here, much more refined arguments are needed.

Theorem 1. We have:

(i) For λ ∈ (0, λ^{∗}) problem (1) admits a minimal regular solution. This solution is radially symmetric and strictly decreasing in r = |x|.

(ii) For λ = λ^{∗} problem (1) admits at least one not necessarily bounded solution.

(iii) For λ > λ^{∗} problem (1) admits no (not even singular) solutions.

Moreover

λ^{∗} ∈ [K_{0}, \frac{λ_{1}}{p} ),  (3)

where

K_{0} = \frac{8}{(p − 1)^{4}} ((n − 2)(n − 4)(p − 1)^{3} + 2(n^{2} − 10n + 20)(p − 1)^{2} − 16(n − 4)(p − 1) + 32) . (4)

For the proof see Section 5.

The regular minimal solution is stable:

Theorem 2. Assume λ ∈ (0, λ^{∗}). Let u_{λ} be the corresponding minimal solution of (1). Denote by μ_{1}(λ) the first eigenvalue of the linearized operator Δ^{2} − λp(1 + u_{λ})^{p−1}. Then, μ_{1}(λ) > 0.

We remark that, obviously, this Theorem also holds in the subcritical and critical range, i.e. for any p > 1. For a proof, see Section 7.

The notion of weakly singular radial solution is motivated from a somehow technical point of view, because within this class, by definition, the asymptotic behaviour at the origin is in some sense specified. Consequently, here and also in the previous work [2], a number of results can be proven much more easily in this restricted class of singular solutions.

However, our next main result, which is proved using a suitable energy functional in Section 6, states that:

Theorem 3. Any radial singular solution of (1) is weakly singular.

That means that in what follows we need no longer distinguish between weakly singular and general radial singular solutions. The corresponding question for the Dirichlet problem with exponential nonlinearity had to be left open in [2].

In Section 3 below we shall transform the differential equation in (1) for radial functions into an autonomous system (17) having precisely two critical points O and P. With the help of these critical points we can give a precise characterization of regular and singular solutions of (1).

Theorem 4. Let u = u(r) be a radial solution of (1) and let

W (t) = (w_{1}(t), w_{2}(t), w_{3}(t), w_{4}(t))

be the corresponding trajectory relative to (17). Then:

(i) u is regular (i.e. u ∈ L^{∞}(B)) if and only if

lim_{t→−∞} W (t) = O.

(ii) u is singular if and only if

lim_{t→−∞} W (t) = P.

This result is proved in Section 4.

Again, by means of energy considerations, in Section 6 we can give an explicit estimate from below for radial singular solutions and the corresponding singular parameter:

Theorem 5. Assume that u_{s} is a singular radial solution of (1) with parameter λ_{s}. Then, λ_{s} > K_{0} and

u_{s}(x) > (\frac{K_{0}}{λ_{s}} )^{1/(p−1)} |x|^{−4/(p−1)} − 1.  (5)

In particular, any radial solution to (1) for λ ≤ K_{0} is regular.

3 An autonomous system

In radial coordinates r = |x|, the differential equation in (1) reads

u^{(4)}(r)+ \frac{2(n − 1)}{r}u′′′(r)+ \frac{(n − 1)(n − 3)}{r^{2}}u′′(r)− \frac{(n − 1)(n − 3)}{r^{3}}u′(r) = λ(1+u(r))^{p}  r ∈ [0, 1]. (6)

In order to make the results and techniques from [9] accessible to our present work, as there we put first

U(x) := (u + 1)(x/\sqrt[4]{λ}) (x ∈ B\sqrt[4]{λ}(0)), u(x) = U(\sqrt[4]{λ}x) − 1 (x ∈ B).

For x ∈ B\sqrt[4]{λ}(0) one has

Δ^{2}U(x) = U(x)^{p}.  (7)

Our purpose here is to transform (7) first into an autonomous equation and, subsequently, into an autonomous system. For some of the estimates which follow, it is convenient to rewrite the original assumption p > \frac{n+4}{n−4} as

(n − 4)(p − 1) > 8.  (8)

Inspired by the proof of [22, Proposition 3.7] (see also [10, 13]) we set as in [9]

U(r) = r^{−4/(p−1)} v(log r), r ∈ (0, \sqrt[4]{λ}),  v(t) = e^{4t/(p−1)} U (e^{t}) , t ∈ (−∞, \frac{1}{4} log λ). (9)

We take from [9] that, after the change (9), equation (7) may be rewritten as

v^{(4)}(t) + K_{3}v′′′(t) + K_{2}v′′(t) + K_{1}v′(t) + K_{0}v(t) = v^{p}(t)  (t < \frac{1}{4} log λ),  (10)

where the constants K_{i} = K_{i}(n, p) (i = 0, …, 3) are given by

K_{0} = \frac{8}{(p − 1)^{4}} [(n − 2)(n − 4)(p − 1)^{3} + 2(n^{2} − 10n + 20)(p − 1)^{2} − 16(n − 4)(p − 1) + 32] ,

K_{1} = −\frac{2}{(p − 1)^{3}} [(n − 2)(n − 4)(p − 1)^{3} + 4(n^{2} − 10n + 20)(p − 1)^{2} − 48(n − 4)(p − 1) + 128] ,

K_{2} = \frac{1}{(p − 1)^{2}} [(n^{2} − 10n + 20)(p − 1)^{2} − 24(n − 4)(p − 1) + 96] ,

K_{3} = \frac{2}{p − 1} [(n − 4)(p − 1) − 8] .

By using (8), it is not difficult to show that K_{1} = K_{3} = 0 if p = \frac{n+4}{n−4} and that for n ≥ 5, p > \frac{n+4}{n−4}

K_{0} > 0 ,  K_{1} < 0 ,  K_{3} > 0.  (11)

On the other hand, the sign of K_{2} depends on n and p. We emphasize that the sign of K_{1} and K_{3} is due to assumption (8).

Finally, we put

z(t) := v(−t),  t > −\frac{1}{4} log λ.  (12)

For z, we have the differential equation analogous to (10):

z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) + K_{0}z(t) = z^{p}(t)  (t > −\frac{1}{4} log λ).  (13)

In order to study the possibly singular behaviour of u near r = 0, we have to investigate the behaviour of z for t → ∞. Equation (13) has two equilibrium points, namely 0 and K_{0}^{1/(p−1)}. First we show that once the solution converges to an equilibrium point, then all derivatives converge to 0 as t → ∞.

Proposition 1. Assume that z : [T_{0}, ∞) → ℝ exists for some T_{0} and solves a constant coefficient fourth order equation

z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) = f(z(t))  (t > −T_{0}),  (14)

where f ∈ C^{1}(ℝ) and where the coefficients may be considered as arbitrary real numbers K_{j} ∈ ℝ. Moreover, let z_{0} be such that f(z_{0}) = 0 and assume that z satisfies lim_{t→∞} z(t) = z_{0}. Then, for k = 1, …, 4, one also has:

lim_{t→∞} z^{(k)}(t) = 0.  (15)

If f ∈ C^{k_{0}+1} in a neighbourhood of {z_{0}}, then (15) holds true for all k < k_{0} + 4.

Proof. By assumption, we have for any q > 1 that

lim_{t→∞} ∫_{t−2}^{t+3} |f(z(τ))|^{q} dτ = 0,  lim_{t→∞} ∥z( .) − z_{0}∥_{C^{0}([t−2,t+3])} = 0.

We consider (14) as a fourth order “elliptic” equation and apply local L^{q}–estimates, which could of course be directly obtained in a much easier way for the ordinary differential equation (14), and conclude

lim_{t→∞} ∥z( .) − z_{0}∥_{W^{4,q}(t−1,t+2)} = 0.

By combining now Sobolev embedding and classical local Schauder estimates we have that

lim_{t→∞} ∥z( .) − z_{0}∥_{C^{4,α}(t,t+1)} = 0.

The differential equation (14) finally shows the claim for any k in the given range. □

We now write (10) as a system in ℝ^{4}. We obtain from (9)

\frac{U′(r)}{r^{3}} = r^{−4p/(p−1)} [v′(t) − \frac{4}{p − 1}v(t)]  (16)

so that

U′(r) = 0 ⟺ v′(t) = \frac{4}{p − 1}v(t) .

This fact suggests the definition

w_{1}(t) = v(t) , w_{2}(t) = v′(t) − \frac{4}{p − 1}v(t) , w_{3}(t) = v′′(t) − \frac{4}{p − 1}v′(t) , w_{4}(t) = v′′′(t) − \frac{4}{p − 1}v′′(t)

so that (10) becomes

\{ \begin{matrix}w′_{1}(t) & = & \frac{4}{p−1}w_{1}(t) + w_{2}(t) \\ w′_{2}(t) & = & w_{3}(t) \\ w′_{3}(t) & = & w_{4}(t) \\ w′_{4}(t) & = & C_{2}w_{2}(t) + C_{3}w_{3}(t) + C_{4}w_{4}(t) + w_{1}^{p}(t) , \end{matrix}  (17)

where

C_{m} = − ∑_{k=m−1}^{4} \frac{K_{k}4^{k+1−m}}{(p − 1)^{k+1−m}}  for m = 1, 2, 3, 4  with K_{4} = 1 .  (18)

This gives first that C_{1} = 0 so that the term C_{1}w_{1}(t) does not appear in the last equation of (17). Moreover, we have the explicit formulae:

C_{2} = \frac{2}{(p−1)^{3}} [(n − 2)(n − 4)(p − 1)^{3} + 2(n^{2} − 10n + 20)(p − 1)^{2} − 16(n − 4)(p − 1) + 32] = \frac{p−1}{4}K_{0},

C_{3} = −\frac{1}{(p−1)^{2}} [(n^{2} − 10n + 20)(p − 1)^{2} − 16(n − 4)(p − 1) + 48],
C_{4} = −\frac{2}{p−1} [(n − 4)(p − 1) − 6].

We recall the phase space analysis performed in [9]. System (17) has the two stationary points (corresponding to v_{0} := 0 and v_{s} := K_{0}^{1/(p−1)})

O(0, 0, 0, 0)  and  P (K_{0}^{1/(p−1)}, −\frac{4}{p − 1}K_{0}^{1/(p−1)}, 0, 0) .  (19)

Let us consider first the “regular point” O. The linearized matrix at O is

M_{O} = ( \begin{matrix}\frac{4}{p−1} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & C_{2} & C_{3} & C_{4} \end{matrix} )

and the characteristic polynomial is

λ ↦ λ^{4} + K_{3}λ^{3} + K_{2}λ^{2} + K_{1}λ + K_{0} .

Then, according to MAPLE^{TM}, the eigenvalues are given by

λ_{1} = 2\frac{p + 1}{p − 1}, λ_{2} = \frac{4}{p − 1}, λ_{3} = \frac{4p}{p − 1} − n, λ_{4} = 2\frac{p + 1}{p − 1} − n.

Since we assume that p > \frac{n+4}{n−4} > \frac{n}{n−4} > \frac{n+2}{n−2}, we have

λ_{1} > λ_{2} > 0 > λ_{3} > λ_{4}.

This means that O is a hyperbolic point and that both the stable and the unstable manifolds are two-dimensional.

Around the “singular point” P the linearized matrix of the system (17) is given by

M_{P} = ( \begin{matrix}\frac{4}{p−1} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ pK_{0} & C_{2} & C_{3} & C_{4} \end{matrix} ) .  (20)

The corresponding characteristic polynomial is

ν ↦ ν^{4} + K_{3} ν^{3} + K_{2} ν^{2} + K_{1}ν + (1 − p)K_{0}

and the eigenvalues are given by

ν_{1} = \frac{N_{1} + \sqrt{N_{2} + 4\sqrt{N_{3}}}}{2(p − 1)} ,  ν_{2} = \frac{N_{1} − \sqrt{N_{2} + 4\sqrt{N_{3}}}}{2(p − 1)} ,
ν_{3} = \frac{N_{1} + \sqrt{N_{2} − 4\sqrt{N_{3}}}}{2(p − 1)} ,  ν_{4} = \frac{N_{1} − \sqrt{N_{2} − 4\sqrt{N_{3}}}}{2(p − 1)} ,

where

N_{1} := −(n − 4)(p − 1) + 8, N_{2} := (n^{2} − 4n + 8)(p − 1)^{2},
N_{3} := (9n − 34)(n − 2) (p − 1)^{4} + 8(3n − 8)(n − 6) (p − 1)^{3}
+(16n^{2} − 288n + 832) (p − 1)^{2} − 128(n − 6)(p − 1) + 256.

The stability of the stationary point P is described by the following

Proposition 2. Assume that p > \frac{n+4}{n−4}.

(i) We have ν_{1}, ν_{2} ∈ ℝ and ν_{2} < 0 < ν_{1}. (ii) For 5 ≤ n ≤ 12 we have ν_{3}, ν_{4} ∉ ℝ and Re ν_{3} = Re ν_{4} < 0. (iii) For n ≥ 13 there exists p_{c} > \frac{n+4}{n−4} such that:

if p < p_{c}, then ν_{3}, ν_{4} ∉ ℝ and Re ν_{3} = Re ν_{4} < 0.

if p = p_{c}, then ν_{3}, ν_{4} ∈ ℝ and ν_{4} = ν_{3} < 0.

if p > p_{c}, then ν_{3}, ν_{4} ∈ ℝ and ν_{4} < ν_{3} < 0. The number p_{c} is the unique value of p > \frac{n+4}{n−4} such that

\begin{matrix}−(n − 4)(n^{3} − 4n^{2} − 128n + 256)(p − 1)^{4} + 128(3n − 8)(n − 6)(p − 1)^{3} \\ +256(n^{2} − 18n + 52)(p − 1)^{2} − 2048(n − 6)(p − 1) + 4096 = 0. \end{matrix}

The function n ↦ p_{c} is strictly decreasing and approaches 1 as n → ∞.

According to Proposition 2, in all cases we have

ν_{1} > 0,  ν_{2} < 0,  Re ν_{3} = Re ν_{4} < 0.

This means that P has a three dimensional stable manifold and a one dimensional unstable manifold (as in the exponential case, see [2, Sect. 3.1]).

4 Characterization of regular and weakly singular solutions

Now, we are in position to give a precise formulation of Theorem 4. According to Theorem 3, to be proved below, we may restrict ourselves to weakly singular solutions.

Theorem 6. Let u = u(r) be a radial solution of (1) and let

W (t) = (w_{1}(t), w_{2}(t), w_{3}(t), w_{4}(t))

be the corresponding trajectory relative to (17). Then:

(i) u is regular (i.e. u ∈ L^{∞}(B)) if and only if

lim_{t→−∞} W (t) = O.

(ii) u is weakly singular if and only if

lim_{t→−∞} W (t) = P.

As a first step in proving this theorem we show that there are only a few possible values for lim_{t→−∞} v(t), provided the limit exists. The following proposition holds independently of the signs of the coefficients K_{i}.

Proposition 3. Let v be a positive solution of (10) on (−∞, \frac{1}{4} log λ) and assume that there exists L ∈ [0, +∞] such that

lim_{t→−∞} v(t) = L .

Then, L ∈ {0, K_{0}^{1/(p−1)}}.

Proof. This is almost the same as in [9, Proposition 5]. We remark that there the arguments are not affected by reversing the time or – equivalently – changing the sign of the coefficients K_{1} and K_{3}. For the reader’s convenience, we give the proof in Section A of the appendix. □

Proof of Theorem 6. (i) Assuming that W corresponds to a regular solution, it is obvious that lim_{t→−∞} W (t) = O. Let us now conversely assume that lim_{t→−∞} W (t) = O; we have to prove that the corresponding solution u of (1) is regular. We calculate the eigenvectors of M_{O} corresponding to the positive eigenvalues i.e. spanning the unstable manifold. These are

W⃗_{1} = (1, 2, 4\frac{p + 1}{p − 1}, 8 (\frac{p + 1}{p − 1})^{2}) for λ_{1} = 2\frac{p + 1}{p − 1};

W⃗_{2} = (1, 0, 0, 0) for λ_{2} = \frac{4}{p − 1}.

Since λ_{1} > λ_{2}, all trajectories approaching O as t → −∞ are tangent to W⃗_{2} except one, which is tangent to W⃗_{1} (see Theorem IX.6.2 in [12]).

But for a solution to (1), the latter case cannot occur, since one always has u(r) > 0, u′(r) ≤ 0, i.e. w_{1} > 0, w_{2} ≤ 0. So, for any solution of (1), we may conclude that ru′(r) = o(u(r)) for r ↘ 0. That means that for any ε > 0 and r > 0 close enough to 0 we have

−ε < \frac{ru′(r)}{u(r)} ≤ 0.

Integration yields that for r ↘ 0

0 ≤ u(r) ≤ Cr^{−ε}.

Using this information, the differential equation, and lim_{t→∞} W (t) = 0, i.e.

r^{4/(p−1)}u(r) → 0, r^{1+4/(p−1)}u′(r) → 0, r^{2+4/(p−1)}Δu(r) → 0, r^{3+4/(p−1)} (Δu)′ (r) → 0,

successive integration of (1) shows that

(Δu)′ (r) = O(1), Δu(r) = O(1), u′(r) = O(1), u(r) = O(1)

for r ↘ 0. This shows that u is regular. (ii) Let W belong to a weakly singular solution. Then, by definition, part (i), and Propositions 1 and 3, we see that lim_{t→−∞} W (t) = P. The converse conclusion is obvious. □

5 Regular minimal solutions

The main goal here is to prove the most difficult part of Theorem 1, namely the immediate switch from existence of regular minimal solutions to nonexistence even of singular solutions.

Theorem 7. Assume that u_{s} is a solution of (1) with parameter λ_{s}. Then, for any λ ∈ (0, λ_{s}), the Dirichlet problem (1) has a regular radially decreasing minimal solution.

We start by proving the following

Lemma 1. Let u be a radial solution of the Dirichlet problem (1), and define the corresponding funtions U = U(r) and v = v(t) according to (7) and (9) respectively for r ∈ (0, \sqrt[4]{λ}) and t ∈ (−∞, \frac{1}{4} log λ). Then, v is bounded.

Proof. For contradiction, assume that v is not bounded. In view of Proposition 3 we may exclude that the limit as t → −∞ exists and equals +∞. Hence we assume that

0 ≤ lim inf_{t→−∞} v(t) < lim sup_{t→−∞} v(t) = +∞.

This shows that there exists a sequence t_{k} → −∞ of local maxima for v such that for all k

lim_{k→+∞} v(t_{k}) = +∞, v′(t_{k}) = 0.  (21)

Define

λ_{k} = v^{p−1}(t_{k})  (22)

so that

lim_{k→+∞}λ_{k} = +∞.

Since (10) is an autonomous equation the translated function

ṽ_{k}(t) = v(t + t_{k} − \frac{1}{4} log λ_{k}), t ∈ (−∞, \frac{1}{4} log λ − t_{k} + \frac{1}{4} log λ_{k})

also solves (10). In particular, the function

Ũ_{k}(r) = r^{−4/(p−1)}ṽ_{k}(log r)

is a radial solution of the equation (7) which satisfies the conditions

Ũ_{k}(\sqrt[4]{λ_{k}}) = λ_{k}^{−1/(p−1)}ṽ_{k}(\frac{1}{4} log λ_{k}) = λ_{k}^{−1/(p−1)}v(t_{k}) = 1  (23)

and by (16), (21), (22)

Ũ′_{k}(\sqrt[4]{λ_{k}}) = −\frac{4}{p − 1}λ_{k}^{−1/4} < 0.  (24)

Next, we define the radial function

u_{k}(r) = Ũ_{k}(\sqrt[4]{λ_{k}}r) − 1 = λ_{k}^{−1/(p−1)}e^{4t_{k}/(p−1)}U (re^{t_{k}}) − 1  (25)

so that by (23) and (24) we have

Δ^{2}u_{k} = λ_{k}(1 + u_{k})^{p}, u_{k} > 0 in B,
u_{k} = 0  on ∂B,
−\frac{∂u_{k}}{∂n} = \frac{4}{p − 1} > 0  on ∂B.

This boundary value problem is solved in a weak sense, since U is a weak solution of (7). One should observe that one also has a comparison principle in B with respect to the boundary datum −\frac{∂u}{∂n}, see [11].

This shows that u_{k} is a weak supersolution for the problem

\{ \begin{matrix}Δ^{2}u = λ_{k}(1 + u)^{p}, u > 0 & in B, \\ u = |∇u| = 0 & on ∂B. \end{matrix}  (26)

By standard arguments, see for example Lemma 3.3 in [3], we infer that for any λ_{k} problem (26) admits a weak solution. Since λ_{k} → +∞ this contradicts the nonexistence of solutions of (1) for large λ (see the proof of Theorem 3 in [2] for more details). This completes the proof of the lemma. □

Proof of Theorem 7. Suppose now that u_{s} is a solution of (1) corresponding to λ = λ_{s}. After possibly replacing u_{s} by the minimal solution of (1) corresponding to λ = λ_{s}, we may assume that u_{s} is radial. We look for a regular solution of (1) for a fixed λ ∈ (0, λ_{s}). Put u_{0} = u_{s} and define u_{1} = \frac{λ}{λ_{s}} u_{0} so that u_{1} solves

∫_{B} u_{1}Δ^{2}φ dx = λ ∫_{B}(1 + u_{0})^{p}φ dx ∀φ ∈ C^{4}(\overline{B}) ∩ H_{0}^{2}(B).

We define by iteration u_{k} as the unique solution of

∫_{B} u_{k}Δ^{2}φ dx = λ ∫_{B}(1 + u_{k−1})^{p}φ dx ∀φ ∈ C^{4}(\overline{B}) ∩ H_{0}^{2}(B).  (27)

By the weak comparison principle (see Lemma 16 in [2]) we deduce that

0 < u_{min} ≤ u_{k} ≤ u_{k−1} ∀k ≥ 1,  (28)

where u_{min} denotes the minimal solution of (1) with respect to the parameter λ. By monotone convergence it follows that there exists u ∈ L^{p}(B) such that u_{k} → u in L^{p}(B) as k → ∞, u ≥ u_{min}. Moreover, passing to the limit in (27) we have

∫_{B} uΔ^{2}φ dx = λ ∫_{B}(1 + u)^{p}φ dx ∀φ ∈ C^{4}(\overline{B}) ∩ H_{0}^{2}(B).

Fix \overline{ϑ} ∈ (\frac{λ}{λ_{s}} , 1) and introduce a strictly increasing sequence \{ϑ_{k}\} with \frac{λ}{λ_{s}} < ϑ_{k} < \overline{ϑ} for any k ≥ 1. Note that for any α > 0 and for any β > α there exists γ > 0 such that for all s ≥ 0

(1 + αs)^{p} ≤ β^{p}(1 + s)^{p} + γ.  (29)

By (29) there exists C_{1} > 0 such that for all φ ∈ C^{4}(\overline{B}), φ ≥ 0, φ = |∇φ| = 0 on ∂B

∫_{B} u_{2}Δ^{2}φ dx = λ ∫_{B}(1 + u_{1})^{p}φ dx = λ ∫_{B} (1 + \frac{λ}{λ_{s}} u_{0})^{p} φ dx

≤ ∫_{B} λ [ϑ_{1}^{p}(1 + u_{0})^{p} + C_{1}] φ dx = ∫_{B}(ϑ_{1}^{p}u_{1} + λC_{1}ψ)Δ^{2}φ dx,

where ψ is the unique solution of the Dirichlet problem

\{ \begin{matrix}Δ^{2}ψ = 1 & in B, \\ ψ = |∇ψ| = 0 & on ∂B. \end{matrix}

The weak comparison principle yields

u_{2} ≤ ϑ_{1}^{p}u_{1} + λC_{1}ψ ≤ ϑ_{1}u_{1} + λC_{1}ψ.

Iterating this procedure we prove that for any k ≥ 1 there exists C_{k} > 0 such that

u_{k+1} ≤ ϑ_{k}u_{k} + λC_{k}ψ.  (30)

Since we chose \frac{λ}{λ_{s}} < ϑ_{k} < \overline{ϑ} < 1 for any k ≥ 1, by (30) it follows that

u_{k} ≤ (\overline{ϑ})^{k}u_{0} + D_{k} ∀k ≥ 1  (31)

for a suitable D_{k} > 0. Therefore, for any ε > 0 there exists \overline{k} such that (\overline{ϑ})\overline{k} < ε and hence by (28) and (31) we have

0 ≤ u ≤ u\overline{k} ≤ εu_{0} + D\overline{k}.  (32)

Making use of (9), (32), and Lemma 1 we deduce that for any ε > 0

0 ≤ lim sup_{r→0^{+}} r^{4/(p−1)}u(r) ≤ lim sup_{r→0^{+}}(εr^{4/(p−1)}u_{0}(r) + r^{4/(p−1)}D\overline{k}) = εL,

where L = lim sup_{r→0^{+}} r^{4/(p−1)}u_{0}(r) < +∞. This proves that

lim_{r→0^{+}}r^{4/(p−1)}u(r) = 0.

Finally by (9), Proposition 1 and Theorem 4 we conclude that u ∈ L^{∞}(B).

The minimal solution u_{min} may now be obtained by means of an iterative procedure starting with 0. Radial symmetry is so obvious. For monotonicity we refer to [7, Lemma 2.2] or [21, Proposition 1].

□

Proof of Theorem 1. First, we remark that (i) and (iii) are proved by Theorem 7. As for (ii), i.e. existence of a possibly singular solution for the extremal parameter λ^{∗}, we can proceed as outlined in [2, Lemma 22]. By means of a generalized Pohožaev identity one can obtain uniform bounds for the minimal regular solutions to (1) (λ ∈ (0, λ^{∗})) in H_{0}^{2}(B) ∩ L^{p+1}(B), which allow to perform a monotone limit as λ ↗ λ^{∗}.

Alternatively, one may refer to [3, Proposition 3.6] and Theorem 2 the proof of which requires only Theorem 7. In [3], Navier instead of Dirichlet boundary conditions are considered. However, since here we are working in the ball, no changes in the argument are needed.

It remains to prove the estimate (3) for λ^{∗}. The explicit singular solution of the differential equation

u(x) := |x|^{−4/(p−1)} − 1

is also a weak supersolution for (1) with parameter λ = K_{0}. To see this one observes that u is only weakly singular near the origin, and that also for biharmonic equations, one has a kind of Hopf lemma for the boundary data, see [11]. This shows λ^{∗} ≥ K_{0}.

In order to show λ^{∗} < λ_{1}/p, we multiply (1) by the positive first eigenfunction φ_{1} of (2) and obtain

λ_{1} ∫_{B} uφ_{1} dx = ∫_{B} uΔ^{2}φ_{1} dx = λ ∫_{B}(1 + u)^{p}φ_{1} dx > pλ ∫_{B} uφ_{1} dx,

thereby proving the desired inequality. □

6 Energy considerations

Let u be a radial singular solution of (1), and let v = v(t) be the corresponding function defined in (9). Let z(t) = v(−t) so that z(t) solves the equation (13) for t > −\frac{1}{4} log λ.

The following energy functional

E(t) := \frac{1}{p + 1}z(t)^{p+1} − \frac{K_{0}}{2} z(t)^{2} − \frac{K_{2}}{2} |z′(t)|^{2} + \frac{1}{2}|z′′(t)|^{2}  (33)

will help to show that every singular solution is weakly singular, i.e. to prove Theorem 3. Moreover, in the second part of this section, we shall specify the asymptotic behaviour of any (weakly) singular solution near r = 0, i.e. of z(t) for t → ∞.

The first result is analogous to Proposition 1.

Lemma 2. Let z : (−\frac{1}{4} log λ, ∞) → ℝ be the solution of (13) corresponding to a radial singular solution of (1). Then, for k = 1, …, 4 the functions z and z^{(k)} are bounded in (−\frac{1}{4} log λ, ∞).

Proof. By Lemma 1 it follows immediately that z(t) = v(−t) is bounded in (−\frac{1}{4} log λ, ∞). Put I = (−\frac{1}{4} log λ, ∞) and t_{0} = −\frac{1}{4} log λ. Then, z^{p}(t) − K_{0}z(t) is bounded in I and hence, by local L^{q}–estimates for fourth order elliptic equations, we infer that for any q > 1 there exists a constant C_{q} > 0 such that for any t > t_{0} + 1 we have

∥z( .)∥_{W^{4,q}(t−1,t+2)} ≤ C_{q} ∥z∥_{L^{∞}(I)} .

By combining Sobolev embeddings and local Schauder estimates we conclude that there exists a positive constant independent of t, still denoted by C_{q}, such that

∥z( .)∥_{C^{4,α}(t,t+1)} ≤ C_{q} ∥z∥_{L^{∞}(I)} .
□

Arguing as in [9], in the next four lemmas we prove some summability properties for the function z and its derivatives.

Lemma 3. Let t_{0} = −\frac{1}{4} log λ. Then

∫_{t_{0}}^{∞} |z′(s)|^{2}ds + ∫_{t_{0}}^{∞} |z′′(s)|^{2}ds < ∞.

Proof. Let E(t) be the function defined in (33). For any t > t_{0} we obtain by integration by parts and exploiting (13)

E(t) − E(t_{0}) = ∫_{t_{0}}^{t} E′(s) ds = ∫_{t_{0}}^{t} (z^{p}z′ − K_{0}zz′ − K_{2}z′z′′ + z′′z′′′) ds
= z′(t)z′′′(t) − z′(t_{0})z′′′(t_{0}) + ∫_{t_{0}}^{t} z′ (z^{p} − K_{0}z − K_{2}z′′ − z^{(4)}) ds
= z′(t)z′′′(t) − z′(t_{0})z′′′(t_{0}) + ∫_{t_{0}}^{t} z′ (−K_{3}z′′′ − K_{1}z′) ds
= z′(t)z′′′(t) − z′(t_{0})z′′′(t_{0}) − K_{3}z′(t)z′′(t) + K_{3}z′(t_{0})z′′(t_{0})
+ ∫_{t_{0}}^{t} (K_{3}z′′(s)^{2} − K_{1}z′(s)^{2}) ds.  (34)

By Lemma 2 it follows that E(t) and the functions z′(t), z′′(t), z′′′(t) are bounded in I = (t_{0}, ∞), while around t_{0}, they are obviously smooth. This together with (34) and the fact that K_{3} > 0, K_{1} < 0 proves the claim. □

Lemma 4. We have

∫_{t_{0}}^{∞} |z′′′(s)|^{2}ds < ∞.

Proof. We multiply the equation (13) by z′′ and integrate over (t_{0}, t) to obtain

∫_{t_{0}}^{t} (z^{(4)}(s) − K_{3}z′′′(s) + K_{2}z′′(s) − K_{1}z′(s) + K_{0}z(s)) z′′(s) ds = ∫_{t_{0}}^{t} z^{p}(s)z′′(s) ds.  (35)

First, we prove that all the lower order terms in the integral identity (35) are bounded. By Lemmas 2,3, and integration by parts we have

|∫_{t_{0}}^{t} z(s)z′′(s) ds| ≤ |z(t)z′(t)| + |z(t_{0})z′(t_{0})| + ∫_{t_{0}}^{t} |z′(s)|^{2}ds = O(1) as t → ∞.  (36)

By Lemma 3 and Hölder’s inequality we have

|∫_{t_{0}}^{t} z′(s)z′′(s) ds| ≤ (∫_{t_{0}}^{t} |z′(s)|^{2}ds)^{1/2} (∫_{t_{0}}^{t} |z′′(s)|^{2}ds)^{1/2} = O(1) as t → ∞.  (37)

By Lemma 2, integration by parts, and arguing as in (37), we obtain

|∫_{t_{0}}^{t} z^{p}(s)z′′(s) ds| ≤ |z^{p}(t)z′(t)| + |z^{p}(t_{0})z′(t_{0})| + |∫_{t_{0}}^{t} pz^{p−1}(s)z′(s)^{2} ds| = O(1) as t → ∞. (38)

Using again Lemma 2 we conclude that

|∫_{t_{0}}^{t} z′′′(s)z′′(s) ds| ≤ \frac{1}{2}|z′′(t)|^{2} + \frac{1}{2}|z′′(t_{0})|^{2} = O(1) as t → ∞.  (39)

Finally, after integration by parts we infer that

∫_{t_{0}}^{t} |z′′′(s)|^{2}ds = z′′′(t)z′′(t) − z′′′(t_{0})z′(t_{0}) − ∫_{t_{0}}^{t} z^{(4)}(s)z′′(s) ds = O(1) as t → ∞

in view of Lemmas 2, 3, and (35)-(39). This completes the proof of the lemma. □

Lemma 5. We have

∫_{t_{0}}^{∞} |z^{(4)}(s)|^{2}ds < ∞.

Proof. We multiply the equation (13) by z^{(4)} and integrate over (t_{0}, t) to obtain

∫_{t_{0}}^{t} |z^{(4)}(s)|^{2}ds = ∫_{t_{0}}^{t} (z^{p}(s) − K_{0}z(s) + K_{1}z′(s) − K_{2}z′′(s) + K_{3}z′′′(s)) z^{(4)}(s) ds.  (40)

Arguing as in the proof of Lemma 4 one can easily prove that the right hand side of (40) remains bounded as t → ∞. This completes the proof of the lemma. □

Lemma 6. We have

∫_{t_{0}}^{∞} z^{2}(s)|z^{p−1}(s) − K_{0}|^{2}ds < ∞.

Proof. Using the differential equation (13) we obtain

[z^{(4)}(s) − K_{3}z′′′(s) + K_{2}z′′(s) − K_{1}z′(s)]^{2} = z^{2}(s)|z^{p−1}(s) − K_{0}|^{2}.

The proof of the lemma follows immediately from Lemmas 3-5. □

Proof of Theorem 3. Let W = (w_{1}, w_{2}, w_{3}, w_{4}) be the solution of the dynamical system (17) corresponding to a radial singular solution u of (1), and let P and O be the stationary points introduced in (19). In view of Lemmas 3-6 we infer that at least one of the two alternatives holds true

∃ {σ_{k}} s.t. σ_{k+1} < σ_{k}, lim_{k→∞}σ_{k} = −∞, lim_{k→∞}W (σ_{k}) = P;  (41)

or

∃ {σ_{k}} s.t. σ_{k+1} < σ_{k}, lim_{k→∞}σ_{k} = −∞, lim_{k→∞}W (σ_{k}) = O.  (42)

Arguing as in Proposition 7 in [9] we conclude that

lim_{t→−∞}W (t) = P

or

lim_{t→−∞}W (t) = O

respectively in the cases (41) and (42). In view of Theorem 6 we may exclude the second case since it would imply that u is a regular solution. Therefore, only the first case may occur and hence, Theorem 6 implies that u is a weakly singular solution. □

The energy functional defined above in (33) will also help to specify the behaviour of (weakly) singular solutions of (1) near r = 0. To this end we may assume that

lim_{t→∞} z(t) = K_{0}^{1/(p−1)}.  (43)

Lemma 7. Let u_{s} be a weakly singular solution of (1) with parameter λ_{s} and z(t) : (−\frac{1}{4} log λ_{s}, ∞) → (0, ∞) the corresponding solution of (13). Then, it cannot happen that z′(t_{0}) = 0 for some t_{0}.

Proof. Assume for contradiction that z′(t_{0}) = 0. Then, by (16), we have that z′(−\frac{1}{4} log λ_{s}) ≠ 0 and hence, z is not a constant. For any t > t_{0} we obtain by arguing as in Lemma 3

E(t) − E(t_{0}) = z′(t)z′′′(t) − K_{3}z′(t)z′′(t) + ∫_{t_{0}}^{t} (K_{3}z′′(s)^{2} − K_{1}z′(s)^{2}) ds.

Letting t → ∞ and observing Proposition 1 yields

E(∞) − E(t_{0}) = ∫_{t_{0}}^{∞} (K_{3}z′′(s)^{2} − K_{1}z′(s)^{2}) ds > 0;
⇒ −\frac{p − 1}{2(p + 1)}K_{0}^{(p+1)/(p−1)} > \frac{1}{p + 1}z(t_{0})^{p+1} − \frac{K_{0}}{2} z(t_{0})^{2} + \frac{1}{2}|z′′(t_{0})|^{2}
≥ min_{ζ≥0} (\frac{ζ^{p+1}}{p + 1} − \frac{K_{0}}{2} ζ^{2}) = −\frac{p − 1}{2(p + 1)}K_{0}^{(p+1)/(p−1)},

a contradiction. □

Proof of Theorem 5. We know from (16) that

v′(\frac{1}{4} log λ_{s}) = \frac{4}{p − 1}v(\frac{1}{4} log λ_{s}) > 0

so that

z′(−\frac{1}{4} log λ_{s}) < 0.

The previous lemma then shows that for all t ≥ −\frac{1}{4} log λ_{s}

z′(t) < 0 ⇒ z(t) > K_{0}^{1/(p−1)}

and

U(x) > K_{0}^{1/(p−1)}|x|^{−4/(p−1)}

so that

u_{s}(x) > (\frac{K_{0}}{λ_{s}} )^{1/(p−1)} |x|^{−4/(p−1)} − 1.

Moreover, we have in particular 0 = u_{s}(1) so that

λ_{s} > K_{0}.
□

Remark. With a completely analogous proof, one can show for any (weakly) singular radial solution u_{s} of the Dirichlet problem for the biharmonic equation Δ^{2}u_{s} = λ_{s} exp(u_{s}) that λ_{s} > 8(n − 2)(n − 4) and that

u_{s}(x) > −4 log |x| + log \frac{8(n − 2)(n − 4)}{λ_{s}}.

This complements [2, Theorem 4].

7 Stability of the minimal regular solution

In this section we shall give the proof of Theorem 2.

Let λ ∈ (0, λ^{∗}), and let u_{λ} be the corresponding minimal solution. By Theorem 7 we know that u_{λ} is a regular solution. Consider the following weighted η-eigenvalue problem

\{ \begin{matrix}Δ^{2}ψ = ηλp(1 + u_{λ})^{p−1}ψ & in B, \\ ψ = |∇ψ| = 0 & on ∂B \end{matrix}  (44)

and let

η_{1}(λ) = inf_{ψ∈H_{0}^{2}(B)\\{0\}} \frac{∫_{B} |Δψ|^{2} dx}{λ ∫_{B} p(1 + u_{λ})^{p−1}ψ^{2} dx}  (45)

be the corresponding first eigenvalue. Since u_{λ} ∈ L^{∞}(B), by compactness of the embedding H_{0}^{2}(B) ⊂ L^{2}(B) we infer that the minimum in (45) is achieved. Note that by the Lagrange multiplier method any minimizer ψ_{1} of η_{1}(λ) solves (44) with η = η_{1}(λ).

Since u_{λ} is a regular solution of (1), by L^{q}–estimates for fourth order elliptic equations and Schauder estimates we infer that both u_{λ} and ψ_{1} are classical solutions of (1) and (44), respectively. In the next lemma we show that ψ_{1} does not change sign in B.

Lemma 8. Let ψ_{1} be a minimizer for η_{1}(λ). Then, ψ_{1} > 0 in B up to a constant multiple.

Proof. Assume for contradiction that ψ_{1} is a sign changing minimizer for η_{1}(λ) and consider the problem

\{ \begin{matrix}Δ^{2}w = η_{1}(λ)λp(1 + u_{λ})^{p−1}|ψ_{1}| & in B, \\ w = |∇w| = 0 & on ∂B. \end{matrix}  (46)

By Boggio’s maximum principle [4], we deduce that w > |ψ_{1}| in B and hence

∫_{B} |Δw|^{2} dx = η_{1}(λ) ∫_{B} λp(1 + u_{λ})^{p−1}|ψ_{1}|w dx < η_{1}(λ) ∫_{B} λp(1 + u_{λ})^{p−1}w^{2} dx

so that

\frac{∫_{B} |Δw|^{2} dx}{λ ∫_{B} p(1 + u_{λ})^{p−1}w^{2} dx} < η_{1}(λ).

This contradicts the definition of η_{1}(λ). Therefore, ψ_{1} is a function of constant sign and hence, up to a constant multiple, we may assume that ψ_{1} ≥ 0 in B. The strict positivity of ψ_{1} follows from (44) and Boggio’s maximum principle [4]. □

Lemma 9. Let η_{1}(λ) be the first eigenvalue of (44). Then η_{1}(λ) > 1.

Proof. Fix \overline{λ} ∈ (λ, λ^{∗}) and consider the corresponding minimal solution u\overline{λ} of (1). Since u_{λ}, u\overline{λ} are minimal solutions for the respective problems we have that u_{λ} ≤ u\overline{λ} in B. Boggio’s maximum principle yields u_{λ} < u\overline{λ} in B. By Lemma 8 we may fix a positive minimizer ψ_{1} of (45). Convexity of s ↦ (1+s)^{p} yields

η_{1}(λ) ∫_{B}(u\overline{λ} − u_{λ})λp(1 + u_{λ})^{p−1}ψ_{1} dx = ∫_{B}(u\overline{λ} − u_{λ})Δ^{2}ψ_{1} dx = ∫_{B} [\overline{λ}(1 + u\overline{λ})^{p} − λ(1 + u_{λ})^{p}] ψ_{1} dx

> λ ∫_{B} [(1 + u\overline{λ})^{p} − (1 + u_{λ})^{p}] ψ_{1} dx ≥ λ ∫_{B} p(1 + u_{λ})^{p−1}(u\overline{λ} − u_{λ})ψ_{1} dx.

This proves that η_{1}(λ) > 1. □

Proof of Theorem 2. Consider now the first eigenvalue μ_{1}(λ) for the linearized operator Δ^{2} − λp(1 + u_{λ})^{p−1}. We have

μ_{1}(λ) = inf_{w∈H_{0}^{2}(B)\\{0\}} \frac{∫_{B} |Δw|^{2} dx − λ ∫_{B} p(1 + u_{λ})^{p−1}w^{2} dx}{∫_{B} w^{2} dx}.

For any w ∈ H_{0}^{2}(B) we have

∫_{B} |Δw|^{2}dx − λ ∫_{B} p(1 + u_{λ})^{p−1}w^{2} dx ≥ (1 − \frac{1}{η_{1}(λ)}) ∫_{B} |Δw|^{2} dx ≥ λ_{1} (1 − \frac{1}{η_{1}(λ)}) ∫_{B} w^{2} dx,

where λ_{1} denotes the first eigenvalue of (2). Lemma 9 now yields

μ_{1}(λ) ≥ λ_{1} (1 − \frac{1}{η_{1}(λ)}) > 0.

This completes the proof of the theorem. □

For 1 < p ≤ \frac{n+4}{n−4}, we may define the action functional J_{λ} associated with the Euler-Lagrange equation (1)

J_{λ}(u) = \frac{1}{2} ∫_{B} |Δu|^{2} dx − \frac{λ}{p + 1} ∫_{B} |1 + u|^{p+1} dx ∀u ∈ H_{0}^{2}(B).

Since for u ∈ H_{0}^{2}(B)

J′′_{λ}(u) = Δ^{2} − λp|1 + u|^{p−1} in L(H_{0}^{2}(B); H^{−2}(B)),

by Theorem 2 we immediately obtain

Corollary 1. Let 1 < p ≤ \frac{n+4}{n−4} and λ ∈ (0, λ^{∗}). Then, the corresponding minimal solution u_{λ} is a local minimum for the functional J_{λ}.

By Theorem 2 and [3, Proposition 3.6] we immediately obtain that Theorem 1 holds true for any superlinear exponent p > 1. Moreover, for subcritical and critical 1 < p ≤ \frac{n+4}{n−4}, according to the related result [3, Theorem 2.2], we expect the existence of two distinct solutions for λ ∈ (0, λ^{∗}).

Appendix

A Proof of Proposition 3

In order to avoid confusion with respect to the time direction we switch to the solution z of (13):

z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) + K_{0}z(t) = z^{p}(t) (t > −\frac{1}{4} log λ).

For contradiction, assume first that L is finite and L ∉ {0, K_{0}^{1/(p−1)}}. Then, z^{p}(t) − K_{0}z(t) → α := L^{p} − K_{0}L ≠ 0 and for all ε > 0 there exists T > 0 such that

α − ε ≤ z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) ≤ α + ε  ∀t ≥ T .  (47)

Take ε < |α| so that α − ε and α + ε have the same sign and let

δ := sup_{t≥T} |z(t) − z(T)| < ∞.

Integrating (47) over [T, t] for any t ≥ T yields

(α − ε)(t − T) + C − |K_{1}|δ ≤ z′′′(t) − K_{3}z′′(t) + K_{2}z′(t) ≤ (α + ε)(t − T) + C + |K_{1}|δ ∀t ≥ T ,

where C = C(T) is a constant containing all the terms z(T), z′(T), z′′(T) and z′′′(T). Repeating twice more this procedure gives

\frac{α − ε}{6}(t − T)^{3} + O(t^{2}) ≤ z′(t) ≤ \frac{α + ε}{6}(t − T)^{3} + O(t^{2}) as t → ∞ .

This contradicts the assumption that z admits a finite limit as t → +∞.

Next, we exclude the case L = +∞. For contradiction, assume that

lim_{t→+∞} z(t) = +∞.  (48)

Then, there exists T ∈ ℝ such that

z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) ≥ \frac{z^{p}(t)}{2} ∀t ≥ T .

Moreover, by integrating this inequality over [T, t] (for t ≥ T), we get

z′′′(t) − K_{3}z′′(t) + K_{2}z′(t) − K_{1}z(t) ≥ \frac{1}{2} ∫_{T}^{t} z^{p}(s)ds + C  ∀t ≥ T ,  (49)

where C = C(T) is a constant containing all the terms z(T), z′(T), z′′(T) and z′′′(T). From (48) and (49) we deduce that there exists T′ ≥ T such that α := z′′′(T′) − K_{3}z′′(T′) + K_{2}z′(T′) − K_{1}z(T′) > 0. Since (10) is autonomous, we may assume that T′ = 0. Therefore, we have

z^{(4)}(t) − K_{3}z′′′(t) + K_{2}z′′(t) − K_{1}z′(t) ≥ \frac{z^{p}(t)}{2}  ∀t ≥ 0 ,  (50)
z′′′(0) − K_{3}z′′(0) + K_{2}z′(0) − K_{1}z(0) = α > 0 .  (51)

We may now apply the test function method developed by Mitidieri-Pohožaev [16]. More precisely, fix T_{1} > T > 0 and a nonnegative function ϕ ∈ C_{c}^{4}[0, ∞) such that

ϕ(t) = \{ \begin{matrix}1 & for t ∈ [0, T] \\ 0 & for t ≥ T_{1} . \end{matrix}

In particular, these properties imply that ϕ(T_{1}) = ϕ′(T_{1}) = ϕ′′(T_{1}) = ϕ′′′(T_{1}) = 0. Hence, multiplying inequality (50) by ϕ(t), integrating by parts, and recalling (51) yields

∫_{0}^{T_{1}}[ϕ^{(4)}(t) + K_{3}ϕ′′′(t) + K_{2}ϕ′′(t) + K_{1}ϕ′(t)]z(t)dt ≥ \frac{1}{2} ∫_{0}^{T_{1}} z^{p}(t)ϕ(t)dt + α .  (52)

We now apply Young’s inequality in the following form: for any ε > 0 there exists C(ε) > 0 such that

zϕ^{(i)} = zϕ^{1/p} \frac{ϕ^{(i)}}{ϕ^{1/p}} ≤ εz^{p}ϕ + C(ε)\frac{|ϕ^{(i)}|^{p/(p−1)}}{ϕ^{1/(p−1)}} ,  ϕ^{(i)} = \frac{d^{i}ϕ}{dt^{i}}  (i = 1, 2, 3, 4).

Then, provided ε is chosen sufficiently small, (52) becomes

C ∑_{i=1}^{4} ∫_{0}^{T_{1}} \frac{|ϕ^{(i)}(t)|^{p/(p−1)}}{ϕ^{1/(p−1)}(t)} dt ≥ \frac{1}{4} ∫_{0}^{T} z^{p}(t)dt + α  (53)

where C = C(ε, K_{i}) > 0. We now choose ϕ(t) = ϕ_{0}(\frac{t}{T}), where ϕ_{0} ∈ C_{c}^{4}([0, ∞), ϕ_{0} ≥ 0 and

ϕ_{0}(τ) = \{ \begin{matrix}1 & for τ ∈ [0, 1] \\ 0 & for τ ≥ τ_{1} > 1 . \end{matrix}

As noticed in [16], there exists a function ϕ_{0} in such class satisfying moreover

∫_{0}^{τ_{1}} \frac{|ϕ_{0}^{(i)}(τ)|^{p/(p−1)}}{ϕ_{0}^{1/(p−1)}(τ)} dτ =: A_{i} < ∞ (i = 1, 2, 3, 4).

Then, thanks to a change of variables in the integrals, (53) becomes

C ∑_{i=1}^{4} A_{i}T^{1−ip/(p−1)} ≥ \frac{1}{4} ∫_{0}^{T} z^{p}(t)dt + α ∀T > 0.

Letting T → ∞, the previous inequality contradicts (48). □

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