2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 103–120. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in

higher dimensions ^{∗}

Pavel Drábek

Abstract

In this paper we study Dirichlet boundary-value problems, for the p-Laplacian, of the form

−Δ_{p}u − λ_{1}|u|^{p−2}u = f in Ω,
u = 0 on ∂Ω,

where Ω ⊂ ℝ^{N} is a bounded domain with smooth boundary ∂Ω, N ≥ 1, p > 1, f ∈ C(Ω̄) and λ_{1} > 0 is the first eigenvalue of Δ_{p}. We study the geometry of the energy functional

E_{p}(u) = \frac{1}{p} ∫_{Ω} |∇u|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u|^{p} − ∫_{Ω} fu

and show the difference between the case 1 < p < 2 and the case p > 2. We also give the characterization of the right hand sides f for which the Dirichlet problem above is solvable and has multiple solutions.

1 Introduction and statement of the results

Our aim is to study the solvability of the Dirichlet boundary-value problem

\begin{matrix}−Δ_{p}u − λ_{1}|u|^{p−2}u = f in Ω, \\ u = 0 on ∂Ω. \end{matrix}  (1.1)

Here p > 1 is a real number, Ω is a bounded domain in ℝ^{N} with sufficiently smooth boundary ∂Ω, Δ_{p}u = div(|∇u|^{p−2}∇u) is the p-Laplacian and f ∈ C(Ω̄). We assume that if N ≥ 2 then ∂Ω is a compact connected manifold of class C^{2}. By λ_{1} we denote the first eigenvalue of the related homogeneous eigenvalue problem

\begin{matrix}−Δ_{p}u − λ|u|^{p−2}u = 0 in Ω, \\ u = 0 on ∂Ω. \end{matrix}  (1.2)

∗Mathematics Subject Classifications: 35J60, 35P30, 35B35, 49N10.

Key words: p-Laplacian, variational methods, PS condition, Fredholm alternative, upper and lower solutions \begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

In this paper, the function u is said to be a (weak) solution of (1.1) if u ∈ W_{0}^{1,2}(Ω) and the integral identity

∫_{Ω} |∇u|^{p−2}∇u · ∇v − λ_{1} ∫_{Ω} |u|^{p−2}uv = ∫_{Ω} fv  (1.3)

holds for all v ∈ W_{0}^{1,p}(Ω).

As for the properties of λ_{1} (see e.g. [2, 17]), let us mention that λ_{1} is positive, simple and isolated and the corresponding eigenfunction φ_{1} (associated with λ_{1}) satisfies φ_{1} > 0 in Ω, \frac{∂φ_{1}}{∂n} < 0 on ∂Ω, where n denotes the exterior unit normal to ∂Ω. One also has φ_{1} ∈ C^{1,ν}(Ω̄) with some ν ∈ (0, 1) (see e.g. [9, Lemma 2.2, p. 115]). Moreover, λ_{1} can be characterized as the best (the greatest) constant C > 0 in the Poincaré inequality

∫_{Ω} |∇u|^{p} ≥ C ∫_{Ω} |u|^{p}  (1.4)

for all u ∈ W_{0}^{1,p}(Ω), where identity

∫_{Ω} |∇u|^{p} − λ_{1} ∫_{Ω} |u|^{p} = 0

holds exactly for the multiples of the first eigenfunction φ_{1}.

Let us recall (see e.g. [9, pp. 114, 115]) that, for every h ∈ L^{∞}(Ω), the problem

\begin{matrix}Δ_{p}u = h in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (1.5)

has a unique solution u ∈ W_{0}^{1,p}(Ω) ∩ C^{1,ν}(Ω̄). Moreover, since C^{1,ν}(Ω̄) is compactly imbedded into C^{1}(Ω̄), we can introduce the compact operator

Δ_{p}^{−1} : L^{∞}(Ω) → C^{1}(Ω̄)

such that u = Δ_{p}^{−1}h is the unique solution of (1.5). In particular, every solution of (1.1) belongs to C_{0}^{1}(Ω̄).

In our further considerations we will use the standard spaces W_{0}^{1,p}(Ω), L^{p}(Ω), C(Ω̄) and C^{1}(Ω̄) (or C_{0}^{1}(Ω̄), respectively), with corresponding norms

∥u∥ = ( ∫_{Ω} |∇u|^{p})^{1/p}, ∥u∥_{L^{p}} = ( ∫_{Ω} |u|^{p})^{1/p},

∥u∥_{C} = max_{x∈Ω} |u(x)|, ∥u∥_{C^{1}} = ∥u∥_{C} + max_{x∈Ω} |∇u(x)|,

respectively, (here |·| denotes the Euclidean norm in ℝ or ℝ^{N}). The subscript 0 indicates that the traces (or values) of functions are equal zero on ∂Ω. Moreover, for the element h of any of the above mentioned space we use the following (L^{2}– orthogonal) decomposition

h(x) = h̃(x) + h̄φ_{1}(x),

and also L^{2}–nonorthogonal decomposition

h(x) = h̃(x) + ĥ,

where h̄, ĥ ∈ ℝ and

∫_{Ω} h̃(x)φ_{1}(x)dx = 0.

The particular subspaces formed by h̃(x) will be denoted by W̃_{0}^{1,p}(Ω), C̃(Ω̄), and C̃_{0}^{1}(Ω̄), respectively.

By B_{X}(v, ρ) we denote the open ball in the space X with the center v and radius ρ, where X = C(Ω̄) or X = C_{0}^{1}(Ω̄). We introduce the energy functional associated with (1.1):

E_{f}(u): = \frac{1}{p} ∫_{Ω} |∇u|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u|^{p} − ∫_{Ω} fu, u ∈ W_{0}^{1,p}(Ω).

This functional is continuously Fréchet differentiable on W_{0}^{1,p}(Ω) and its critical points correspond one–to–one to solutions of (1.1).

Our main results concern the geometry of E_{f} and the structure of the set of its critical points on one hand and the solvability properties of (1.1) on the other hand. They are formulated in theorems below.

Theorem 1.1 Let 1 < p < 2 and 0 ≠ f̃ ∈ C̃(Ω̄). Then there exists ρ = ρ(f̃) > 0 such that for any f ∈ B_{C}(f̃, ρ) the functional E_{f} is unbounded from below and has at least one critical point. Moreover, for f ∈ B_{C}(f̃, ρ) \ C̃(Ω̄) the functional E_{f} has at least two distinct critical points.

Theorem 1.2 Let p > 2 and 0 ≠ f̃ ∈ C̃(Ω̄). Then the functional Ef̃ is bounded from below and has at least one critical point (which is the global minimizer). Moreover, there exists ρ = ρ(f̃) > 0 such that for f ∈ B_{C}(f̃, ρ) \ C̃(Ω̄) the functional E_{f} has at least two distinct critical points.

Theorem 1.3 Let p > 1, p ≠ 2, f̃ ∈ C̃(Ω̄). Then the problem (1.1) has at least one solution if f = f̃. For 0 ≠ f̃ ∈ C̃(Ω̄) there exists ρ = ρ(f̃) > 0 such that (1.1) has at least one solution for any f ∈ B_{C}(f̃, ρ). Moreover, there exist real numbers F_{−} < 0 < F_{+} (see Fig. 1) such that the problem (1.1) with f = f̃+ f̂ has

(i) No solution for f̂ ∉ [F_{−}, F_{+}]

(ii) At least two distinct solutions for f̂ ∈ (F_{−}, 0) ∪ (0, F_{+})

(iii) At least one solution for f̂ ∈ {F_{−}, 0, F_{+}}.

Remark 1.4 Note that standard bootstrap regularity argument implies that any solution from Theorems 1.1–1.3 belongs to L^{∞}(Ω) (cf. Drábek, Kufner, Nicolosi [10]). It follows then from the regularity results of Tolksdorf [23] (see also Di Benedetto [6] and Liebermann [16]) that it belongs to C^{1,ν}(Ω̄) with some ν ∈ (0, 1). In particular, our solution is an element of C_{0}^{1}(Ω̄).

[figure: →; ↑; F_{−}; F_{+}; f̃ = 1; C(Ω̄); f̃; C̃(Ω̄)]

Figure 1: “Slice” of C(Ω̄) containing all constants and one fixed f̃ ∈ C̃(Ω̄).

Remark 1.5 In particular, it follows from our results that the set of f ∈ C(Ω̄) for which (1.1) has at least one solution has a nonempty interior in C(Ω̄).

Remark 1.6 Note that Theorem 1.3 provides necessary and sufficient condition for solvability of the problem (1.1). This condition is in fact of Landesman–Lazer type (see [15], cf. also [11]). Indeed, given f̃ ∈ C̃(Ω̄), f̃ ≠ 0, the problem (1.1) with the right hand side f(x) = f̃(x) + f̂ has a solution if and only if

F_{−}(f̃) ≤ \frac{1}{∥φ_{1}∥_{L^{1}}} ∫_{Ω} f(x)φ_{1}(x)dx ≤ F_{+}(f̃).

However, it should be pointed out that this condition differs from the original condition of Landesman and Lazer due to the fact that F_{−} and F_{+} depend on the component f̃ of the right hand side f and not on the perturbation term (which is actually not present in our problem (1.1)). By homogeneity we have that for any t > 0,

F_{±}(tf̃) = tF_{±}(f̃).

Our proofs rely on the combination of the variational approach and the method of lower and upper solutions. We also use essentially the results obtained by Drábek and Holubová [8], Takáč [21] and Fleckinger–Pellé and Takáč [14]. In fact, Theorem 1.1 was proved already in [8], however, here different approach is used. During the preparation of this manuscript the author received preprint of Takáč [22], where similar result to our Theorem 1.3 is proved. However, the approach used in [22] is very different from ours.

Our objective in this paper is to avoid complicated technical assumptions. For this reason we restrict to rather special domains Ω and right hand sides f. On the other hand, we belive that in our approach the main ideas appear more clearly and that possible generalization of Ω or f will not bring any new insight neither into the geometry of E_{f} nor to the solvability of (1.1).

It should be mentioned that our approach covers also the case N = 1, and completes thus previous results in this direction proved by Del Pino, Drábek and Manásevich [5], Drábek, Girg and Manásevich [7], Manásevich and Takáč [18], Binding, Drábek and Huang [3], Drábek and Takáč [12]. In fact, the first relevant result which led to better understanding of the problem appeared in [5].

Note also that our Theorems 1.1, 1.2 and 1.3 express not only the difference between the linear case p = 2 and the nonlinear case p ≠ 2 but also the striking difference between the case 1 < p < 2 and the case p > 2. The main goal of this paper is actually to emphasize this fact.

2 Auxiliary assertions, survey of known facts

It should be pointed out that E_{f} is continuously differentiable and weakly lower semicontinuous functional on W_{0}^{1,p}(Ω). The following notions are crutial in the study of the geometry of the functional E_{f}.

Definition 2.1 We say that the functional

E_{f}: W_{0}^{1,p}(Ω) → ℝ

has a local saddle point geometry if we can find u, v ∈ W_{0}^{1,p}(Ω) which are separated by W̃_{0}^{1,p}(Ω) in the sense that

E_{f}(u) < inf_{w∈W̃_{0}^{1,p}(Ω)} E_{f}(w), E_{f}(v) < inf_{w∈W̃_{0}^{1,p}(Ω)} E_{f}(w)

and any continuous path from u to v in W_{0}^{1,p}(Ω) has a nonempty intersection with W̃_{0}^{1,p}(Ω).

We say that E_{f} has a local minimizer geometry if we can find open bounded set D ⊂ W_{0}^{1,p}(Ω) such that

inf_{u∈D} E_{f}(u) < inf_{u∈∂D} E_{f}(u).

The following lemma is crutial for application of variational methods. Its proof can be found in [8, Lemma 2.2] (or in [7, Proposition 2.1] in one dimensional case).

Lemma 2.2 Let p > 1, f = f̃ + f̄φ_{1} with f̄ ≠ 0. Then E_{f} satisfies Palais– Smale (PS) condition, i.e. if E_{f}(u_{n}) → c ∈ ℝ, E′_{f}(u_{n}) → 0 then {u_{n}} contains strongly convergent subsequence in W_{0}^{1,p}(Ω).

Note that the assertion of Lemma 2.2 is not true if f̄ = 0 (see [5]). The following assertion deals with the case 1 < p < 2 and provides the information about the geometry of the energy functional E_{f}.

Lemma 2.3 (see [8, Lemma 2.1]) Let 1 < p < 2 and f̃ ∈ C̃(Ω̄), f̃ ≠ 0. Then Ef̃ has a local saddle point geometry. Moreover, there are two sequences {u_{n}}, {v_{n}} ⊂ W_{0}^{1,p}(Ω) such that for any n ∈ ℕ, u_{n} and v_{n} are separated by W̃_{0}^{1,p}(Ω) and

E(u_{n}) → −∞, E(v_{n}) → −∞.

Later, in Section 4, we show that the situation is different if p > 2 and prove that E_{f} has a local minimizer geometry in this case.

The following notions are crutial in the application of the method of lower and upper solutions.

Definition 2.4 A function u_{s} ∈ C^{1}(Ω̄) is an upper solution of (1.1) if

∫_{Ω} |∇u_{s}|^{p−2}∇u_{s} · ∇v − λ_{1} ∫_{Ω} |u_{s}|^{p−2}u_{s}v ≥ ∫_{Ω} fv ∀v ∈ W_{0}^{1,p}(Ω), v ≥ 0,

u_{s} ≥ 0 on ∂Ω.

In an analogous way we define a lower solution u_{l} of (1.1).

Definition 2.5 Let u, v ∈ C^{1}(Ω̄). We say that u ≺ v if u(x) < v(x) on Ω, and for x ∈ ∂Ω either u(x) < v(x), or u(x) = v(x) and (∂u/∂n)(x) > (∂v/∂n)(x).

Definition 2.6 A lower solution u_{l} of (1.1) is said to be strict if every solution u of (1.1) such that u_{l} ≤ u on Ω satisfies u_{l} ≺ u. In an analogous way we define a strict upper solution of (1.1).

For h ∈ C(Ω̄) we define an operator T_{f}: C_{0}^{1}(Ω̄) → C_{0}^{1}(Ω̄) as T_{f}(v) = u where u satisfies

Δ_{p}u = f(x) − λ_{1}|v|^{p−2}v in Ω,
u = 0 on ∂Ω.

The operator T_{f} is compact and its fixed points, i.e. u = T_{f}(u) u ∈ C_{0}^{1}(Ω), correspond to solutions of the original problem (1.1). The following assertions are proved in [8], the idea comes from [4].

Lemma 2.7 (Well–Ordered Lower and Upper Solutions) Let u_{l} and u_{s} be lower and upper solutions, respectively, of (1.1) such that u_{l} ≤ u_{s}. Then the problem (1.1) has at least one solution u satisfying

u_{l} ≤ u ≤ u_{s} in Ω.

If, moreover, u_{l} and u_{s} are strict and satisfy u_{l} ≺ u_{s}, then there exists R_{0} > 0 such that for all R ≥ R_{0}

deg[I − T_{f}; M_{1}, 0] = 1,

where M_{1} = {u ∈ C_{0}^{1}(Ω̄); u_{l} ≺ u ≺ u_{s}} ∩ B_{C_{0}^{1}}(0, R).

Lemma 2.8 (Non–Ordered Lower and Upper Solutions) Let u_{l} and u_{s} be lower and upper solutions, respectively, of (1.1) and u_{l}(x_{0}) > u_{s}(x_{0}) for some x_{0} ∈ Ω. Then (1.1) has at least one solution in the closure (with respect to C^{1}-norm) of the set

S = {u ∈ C_{0}^{1}(Ω̄); x_{1}, x_{2} ∈ Ω: u(x_{1}) < u_{l}(x_{1}), u(x_{2}) > u_{s}(x_{2})}.

Set M_{2} = S ∩ B_{C_{0}^{1}}(0, R) and assume that there is no solution of (1.1) on ∂M_{2}. Then there exists R_{0} > 0 such that for all R ≥ R_{0}

deg[I − T_{f}; M_{2}, 0] = −1.

As an immediate consequence of Lemmas 2.7 and 2.8 we have the following proposition.

Proposition 2.9 Let (1.1) be solvable for f_{1} ∈ C(Ω̄) and f_{2} ∈ C(Ω̄) such that f_{1}(x) ≤ f_{2}(x), x ∈ Ω̄. Then it is also solvable for any f ∈ C(Ω̄) such that f_{1}(x) ≤ f(x) ≤ f_{2}(x), x ∈ Ω̄.

Proof. Let u_{i} be a solution of (1.1) with f_{i}, i = 1, 2. Then u_{l} = u_{1} and u_{s} = u_{2} are lower and upper solutions, respectively, of (1.1) with f. Then either Lemma 2.7 or 2.8 applies to get a solution. □

The following assertion deals with the case p > 2 and helps to get the information about the geometry of the energy functional E_{f}.

Proposition 2.10 ([14, Theorem 1.1]) There exists a positive constant C = C(p, Ω) such that for all u ∈ W_{0}^{1,p}(Ω), u(x) = ũ(x) + ūφ_{1}(x),

∫_{Ω} |∇u|^{p} − λ_{1} ∫_{Ω} |u|^{p} ≥ C(|ū|^{p−2} ∫_{Ω} |∇φ_{1}|^{p−2}|∇ũ|^{2} + ∫_{Ω} |∇ũ|^{p}).

We will need also the following imbedding type inequality (see [21, Lemma 4.2], [14, Lemma 4.2]): Let p > 2, then there exists C̃ > 0 such that for all u ∈ W_{0}^{1,p}(Ω),

( ∫_{Ω} |u|^{2})^{1/2} ≤ C̃( ∫_{Ω} |∇φ_{1}|^{p−2}|∇u|^{2})^{1/2}.  (2.1)

The last assertion of this section is related to the application of the degree argument in the proof of Theorem 1.3.

Proposition 2.11 (see [21, Theorems 2.3 and 2.8]) Let p > 1 and K be a compact set in C(Ω̄) and ∫_{Ω} fφ_{1} ≠ 0 for any f ∈ K. Then there exists a constant C̃_{1} = C̃_{1}(K) > 0 such that

∥u∥_{C_{0}^{1}} ≤ C̃_{1}

for any possible solution u of (1.1) with f ∈ K.

3 Proof of Theorem 1.1

For the case 1 < p < 2, consider the energy functional

Ef̃(u): = \frac{1}{p} ∫_{Ω} |∇u|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u|^{p} − ∫_{Ω} f̃u, u ∈ W_{0}^{1,p}(Ω),

where f̃ ∈ C̃(Ω̄), f̃ ≠ 0. It was proved in Drábek and Holubová [8] that this functional has a local saddle point geometry and, in particular, it is unbounded from below (see also Lemma 2.3). It is also known (see DelPino, Drábek and Manásevich [5]) that Ef̃ does not satisfy (PS) condition in general. So we cannot deduce the existence of critical point of Ef̃ directly.

It follows from [8, proof of Lemma 2.1] that

lim_{|ū|→∞} inf_{ũ∈W̃_{0}^{1,p}(Ω)}\{\frac{1}{p} ∫_{Ω} |ū∇φ_{1} + ∇ũ|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ūφ_{1} + ũ|^{p} − ∫_{Ω} f̃ũ\} = −∞. (3.1)

Moreover, the infimum is achieved for any fixed ū ∈ ℝ at some ũū ∈ W̃_{0}^{1,p}(Ω). Indeed, for fixed ū ∈ ℝ the functional

ũ ↦ \frac{1}{p} ∫_{Ω} |ū∇φ_{1} + ∇ũ|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ūφ_{1} + ũ|^{p} − ∫_{Ω} f̃ũ

is weakly lower semicontinuous and coercive on W̃_{0}^{1,p}(Ω). Weak lower semicontinuity follows from the same property of the norm on W̃_{0}^{1,p}(Ω) and compactness of the imbedding W_{0}^{1,p}(Ω) ↪↪ L^{p}(Ω). Coercivity is proved via contradiction. Assume that there is a sequence {ũ_{n}} ⊂ W̃_{0}^{1,p}(Ω) such that ∥ũ_{n}∥ → ∞, and

\frac{1}{p} ∫_{Ω} |ū∇φ_{1} + ∇ũ_{n}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ūφ_{1} + ũ_{n}|^{p} − ∫_{Ω} f̃ũ_{n} ≤ C

for some constant C > 0 independent of n. Dividing the last inequality by ∥ũ_{n}∥^{p} and passing to the limit for n → ∞, we obtain

lim_{n→∞}\{\frac{1}{p} ∫_{Ω} |\frac{ū∇φ_{1}}{∥ũ_{n}∥} + ∇ˆũ_{n}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |\frac{ūφ_{1}}{∥ũ_{n}∥} + ˆ ũ_{n}|^{p} − ∫_{Ω} f̃\frac{ũ_{n}}{∥ũ_{n}∥^{p}} \} ≤ 0,

where ˆ ũ_{n} = \frac{ũ_{n}}{∥ũ_{n}∥}. The closedness of W̃_{0}^{1,p}(Ω) and the compactness of the imbedding W_{0}^{1,p}(Ω) ↪↪ L^{p}(Ω) imply that there exists ũ_{0} ∈ W̃_{0}^{1,p}(Ω), ∥ũ_{0}∥ = 1, such that

\frac{1}{p} ∫_{Ω} |∇ũ_{0}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ũ_{0}|^{p} = 0.

However, this contradicts the variational characterization and the simplicity of λ_{1}.

Lemma 3.1 Let ũū ∈ W̃_{0}^{1,p}(Ω) be as above. Then ∥ũū∥_{L^{p}} = o(ū) as |ū| → ∞.

Proof. (i) Assume that there exists {ū_{n}} ⊂ ℝ such that ū_{n} → ∞ and

\frac{ū_{n}}{∥ũū_{n}∥} → 0.  (3.2)

Set ˆ ũū_{n} = ũū_{n}/∥ũū_{n}∥. It follows from (3.1) that

lim inf_{ū_{n}→∞} \{\frac{1}{p} ∫_{Ω} |\frac{ū_{n}}{∥ũ_{ū_{n}∥}} ∇φ_{1} + ∇ˆũū_{n}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |\frac{ū_{n}}{∥ũū_{n}∥}φ_{1} + ˆ ũū_{n}|^{p}

− \frac{1}{∥ũū_{n}∥^{p−1}} ∫_{Ω} f̃ˆũū_{n}\} ≤ 0. (3.3)

Passing to a subsequence if necessary we conclude ˆ ũū_{n} ⇀ u_{0} in W_{0}^{1,p}(Ω), ˆ ũū_{n} → u_{0} in L^{p}(Ω) and

∫_{Ω} u_{0}φ_{1} = 0.  (3.4)

At the same time, for large u ∈ ℕ, we have

\frac{1}{p} ∫_{Ω} |\frac{ū_{n}}{∥ũū_{n}∥}∇φ_{1} + ∇ˆũū_{n}|^{p} ≥ ε

with some ε > 0. It follows then from (3.3) that

\frac{λ_{1}}{p} ∫_{Ω} |u_{0}|^{p} ≥ ε

which means that u_{0} ≠ 0. At the same time we get from (3.3) that

\frac{1}{p} ∫_{Ω} |∇u_{0}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u_{0}|^{p} ≤ 0

and so the variational characterization and simplicity of λ_{1} imply that u_{0} = kφ_{1}, k ≠ 0. But this contradicts (3.4).

(ii) Assume that ū_{n} → ∞ and there exist constant C > 0 independent of n such that

\frac{∥ũū_{n}∥}{ū_{n}} ≤ C.  (3.5)

It follows from (3.1) that

lim_{ū_{n}→∞} inf\{\frac{1}{p} ∫_{Ω} |∇φ_{1} + ∇(\frac{ũū_{n}}{ū_{n}})|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |φ_{1} + \frac{ũū_{n}}{ū_{n}} |^{p} − ∫_{Ω} f̃\frac{ũū_{n}}{ū_{n}^{p}} \} ≤ 0. (3.6)

Passing to a subsequence if necessary, we conclude that there is u_{0} ∈ W_{0}^{1,p}(Ω) such that \frac{ũūn}{ū_{n}} ⇀ u_{0} in W_{0}^{1,p}(Ω), \frac{ũūn}{ū_{n}} → u_{0} in L^{p}(Ω) and

∫_{Ω} u_{0}φ_{1} = 0.

Let u_{0} ≠ 0. Then we get from (3.6) that

\frac{1}{p} ∫_{Ω} |∇φ_{1} + ∇u_{0}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |φ_{1} + u_{0}|^{p} ≤ 0,

which contradicts the variational characterization and simplicity of λ_{1}. Hence u_{0} = 0, i.e.

\frac{ũū_{n}}{ū_{n}} → 0 in L^{p}(Ω).  (3.7)

Assume now that the assertion of lemma is not true. Then there is a sequence {ū_{n}} ⊂ ℝ, ū_{n} → ∞, such that for some C̃_{2} > 0 we have

\frac{∥ũū_{n}∥_{L^{p}}}{ū_{n}} ≥ C̃_{2}.

For such a sequence we have that either (3.2) or (3.5) holds. The former case is impossible by (i) the latter case contradicts (3.7). □

As a consequence of Lemma 3.1 we have

min_{ũ∈W̃_{0}^{1,p}(Ω)}\{\frac{1}{p} ∫_{Ω} |ū∇φ_{1} + ∇ũ|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ūφ_{1} + ũ|^{p} − ∫_{Ω} f̃ũ\} = o(ū), |ū| → ∞.

(3.8)

Lemma 3.2 For a given T > 0 there exists R > 0 such that for any ū ∈ [0, T] and ũ ∈ W̃_{0}^{1,p}(Ω), ∥ũ∥ = R, we have

\frac{1}{p} ∫_{Ω} |ū∇φ_{1} + ∇ũ|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ūφ_{1} + ũ|^{p} − ∫_{Ω} f̃ũ ≥ 0.  (3.9)

Proof. Assume that there is T > 0, ū_{n} ∈ [0, T], ∥ũ_{n}∥ → ∞ such that

\frac{1}{p} ∫_{Ω} |ū_{n}∇φ_{1} + ∇ũ_{n}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |ū_{n}φ_{1} + ũ_{n}|^{p} − ∫_{Ω} f̃ũ_{n} < 0.  (3.10)

Set ˆ ũ_{n} = ũ_{n}/∥ũ_{n}∥. Passing to subsequences if necessary we can assume that ˆũ ⇀ u_{0} in W_{0}^{1,p}(Ω), ∫_{Ω} u_{0}φ_{1} = 0, ū_{n} → ū_{0} ∈ [0, T]. At the same time, dividing (3.10) by ∥ũ_{n}∥^{p}, passing to the limit for n → ∞ we derive that u_{0} ≠ 0 and

\frac{1}{p} ∫_{Ω} |∇u_{0}|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u_{0}|^{p} ≤ 0

which contradicts the variational characterization and simplicity of λ_{1}. □

Let ρ > 0 be small enough (to be specified later) and consider f ∈ B_{C}(f̃, ρ)\ C̃(Ω̄). Then f splits as follows:

f(x) = f̃(x) + f̄φ_{1}(x)

↑  → { ∥ũ∥ ≤ R  φ_{1}  T φ_{1} D W̃_{0}^{1,p}(Ω)  W_{0}^{1,p}(Ω)

Figure 2: The set D constructed in the proof of Theorem 1.1

with |f̄| small, f̄ ≠ 0. Then

E_{f}(u) =\frac{1}{p} ∫_{Ω} |∇u|^{p} − \frac{λ_{1}}{p} ∫_{Ω} |u|^{p} − ∫_{Ω} f̃ũ − ū ∫_{Ω} f̄φ_{1}
=Ef̃(u) − ū ∫_{Ω} f̄φ_{1}, u ∈ W_{0}^{1,p}(Ω),

where u = ūφ_{1} + ũ. Let f̄ < 0, so f̄ ∈ (−ρ̄, 0) with small ρ̄ > 0. We shall construct the set

D = {u ∈ W_{0}^{1,p}(Ω): u = ūφ_{1} + ũ, ū ∈ (0, T), ∥ũ∥ < R}

with T > 0 and R > 0 to be specified later. We choose T_{1} > 0 so that

Ef̃(ũ_{T_{1}}) ≤ 2Ef̃(ũ_{0})  (3.11)

(this is possible due to (3.1), remind that ũ_{T_{1}} and ũ_{0} are the points where inf_{ũ∈W̃_{0}^{1,p}(Ω)} Ef̃(ūφ_{1} + ũ) is achieved for ū = T_{1} and ū = 0, respectively). Then take ρ > 0 (and hence ρ̄ > 0) so small that

E_{f}(T_{1}φ_{1} + ũ_{T_{1}}) ≤ \frac{3}{2}Ef̃(ũ_{0})  (3.12)

if f ∈ B_{C}(f̃, ρ) \ C̃(Ω̄). Now we choose T > 0 so that

E_{f}(T φ_{1} + ũ_{T}) ≥ 0  (3.13)

(this is possible due to (3.8) and f̄ < 0). Finally, we choose R = R(T) > 0 according to Lemma 3.2 (see Fig. 2). Then it follows from Lemma 3.2, (3.12) and (3.13) that

inf_{u∈D} E_{f}(u) < inf_{u∈∂D} E_{f}(u).  (3.14)

Since E_{f} is weakly lower semicontinuous functional on D there exists a global minimizer of E_{f} in D. Let u_{D} ∈ D be the point of global minimum, i.e.

E_{f}(u_{D}) = min_{u∈D} E_{f}(u).

Note that E_{f} is unbounded from below. This is easy to see, choosing e.g. u_{n} = ū_{n}φ_{1}, ū_{n} → −∞, we obtain E_{f}(u_{n}) → −∞. So, E_{f} has a Mountain Pass Theorem Geometry. Because E_{f} satisfies also (PS) condition according to Lemma 2.2, we can apply the results of Rabinowitz [20] to derive the existence of u_{0} ∈ W_{0}^{1,p}(Ω), u_{0} ≠ u_{D}, which is also a critical point of E_{f}. To summarize, we proved that for f ∈ B_{C}(f̃, ρ) \ C̃(Ω̄) the functional E_{f} has at least two distinct critical points. The case f̄ > 0 is similar.

It remains to prove that Ef̃ has at least one critical point. This follows from the argument based on the method of upper and lower solutions. It follows from the previous considerations that there is f̄ > 0 small enough such that E_{f̃±f̄φ_{1}} has critical points u_{±} ∈ W_{0}^{1,p}(Ω), i.e.

∫_{Ω} |∇u_{±}|^{p−2}∇u_{±} · ∇v − λ_{1} ∫_{Ω} |u_{±}|^{p−2}u_{±}v = ∫_{Ω} f̃v ± ∫_{Ω} f̄φ_{1}v

holds for any v ∈ W_{0}^{1,p}(Ω). It follows from Proposition 2.9 that there is a solution u ∈ W_{0}^{1,p}(Ω) satisfying

∫_{Ω} |∇u|^{p−2}∇u · ∇v − λ_{1} ∫_{Ω} |u|^{p−2}uv = ∫_{Ω} f̃v

for any v ∈ W_{0}^{1,p}(Ω). This is equivalent to the fact that u is a critical point of Ef̃. This completes the proof of Theorem 1.1.

4 Proof of Theorem 1.2

We consider the case p > 2 and the energy functional Ef̃ with f̃ ∈ C̃(Ω̄), f̃ ≠ 0. Let us choose a function φ ∈ W_{0}^{1,p}(Ω), φ ≥ 0 in Ω and such that

{x ∈ Ω: φ(x) > 0} ⊂ {x ∈ Ω: f̃(x) > 0}

(note that this is possible because the latter set is an open subset of Ω). Then there exists t > 0 (small enough) such that for v = tφ we have

Ef̃(v) < 0.  (4.1)

Making use of Proposition 2.10 the Hölder and Young inequalities we have the following estimate

Ef̃(u) ≥\frac{C}{p} [|ū|^{p−2} ∫_{Ω} |∇φ_{1}|^{p−2}|∇ũ|^{2} + ∫_{Ω} |∇ũ|^{p}] − ( ∫_{Ω} |f̃|^{p′})^{1/p′}( ∫_{Ω} |ũ|^{p})^{1/p}
≥\frac{C}{p} [|ū|^{p−2} ∫_{Ω} |∇φ_{1}|^{p−2}|∇ũ|^{2} + ∫_{Ω} |∇ũ|^{p}] − \frac{C_{1}^{p}ε^{p}}{p} ∥ũ∥^{p} − \frac{1}{ε^{p}p′} ∥f̃∥_{L^{p′}}^{p′} ,

where C_{1} > 0 is the constant of the imbedding W_{0}^{1,p}(Ω) ↪ L^{p}(Ω). Choosing C_{1}^{p}ε^{p} = \frac{C}{2} we arrive at

Ef̃(u) ≥ \frac{C}{2p}∥ũ∥^{p} + \frac{C}{p} |ū|^{p−2} ∫_{Ω} |∇φ_{1}|^{p−2}|∇ũ|^{2} − \frac{(\frac{2}{C})^{\frac{1}{p−1}} C_{1}^{1− \frac{1}{p}}}{p′} ∥f̃∥_{L^{p′}}^{p′} . (4.2)

It follows from here that there exists R = R(f̃) > 0 such that for any u = ūφ_{1} + ũ ∈ W_{0}^{1,p}(Ω) with ∥ũ∥ = R we have

Ef̃(u) > 0.  (4.3)

Let us consider now u = ūφ_{1} + ũ ∈ W_{0}^{1,p}(Ω) for which

\frac{C}{p} |ū|^{p−2} ∫_{Ω} |∇φ_{1}|^{p−2}|∇ũ|^{2} ≤ C_{2}∥f̃∥_{L^{p′}}^{p′}  (4.4)

where we denoted C_{2} = \frac{1}{p′} (\frac{2}{C})^{\frac{1}{p−1}} C_{1}^{1− \frac{1}{p}}. It follows then from the Hölder inequality that Ef̃(u) ≥ −∥f̃∥_{L^{2}}∥ũ∥_{L^{2}}. If we combine this with (2.1) and (4.4) we get

Ef̃(u) ≥ −\frac{C̃p^{1/2}C_{2}^{1/2}∥f̃∥_{L^{2}}∥f∥L^{p′}^{\frac{p′}{2}}}{C^{1/2}|ū|^{\frac{p−2}{2}}} .  (4.5)

Let us define the set

D = {u ∈ W_{0}^{1,p}(Ω): u = ūφ_{1} + ũ, ū ∈ (−T, T), ∥ũ∥ < R}

with R mentioned above and T > 0 to be fixed later (see Fig. 3). It follows from (4.1) that

i: = inf_{u∈D} Ef̃(u) < 0

independently of T ≫ 1. It follows from (4.5) that for u = ±T φ_{1} + ũ satisfying (4.4) we have

Ef̃(u) > i  (4.6)

if T is large enough. On the other hand we have directly from (4.2) that

Ef̃(u) ≥ 0 > i  (4.7)

for u = ±T φ_{1} + ũ which do not satisfy (4.4). Now, if we combine (4.3), (4.6) and (4.7), we get

i < inf_{u∈∂D} Ef̃(u).  (4.8)

Thus Ef̃ has a local minimizer geometry. In particular, it follows also from above considerations that Ef̃ is bounded from below on W_{0}^{1,p}(Ω). Since Ef̃ is weakly lower semicontinuous functional on the bounded, convex and closed set D̄, it has to achieve its minimum there. Due to (4.8) the minimizer is an interior point of D and due to the differentiability of Ef̃ it is a critical point at the same time.

Let ρ > 0 and consider f ∈ B_{C}(f̃, ρ) \ C̃(Ω̄). Then, as in Section 3, split f as follows:

f(x) = f̃(x) + f̄φ_{1}(x)

with f̄ ≠ 0. Then

E_{f}(u) = Ef̃(u) − ū ∫_{Ω} f̄φ_{1}

↑  → { ∥ũ∥ ≤ R  φ_{1}  T φ_{1} −T φ_{1}  D W̃_{0}^{1,p}(Ω)  W_{0}^{1,p}(Ω)

Figure 3: The set D constructed in the proof of Theorem 1.2

and thus E_{f} is unbounded from below (we can use the same reasoning as in the previous section). If ρ is small enough (and so is |f̄|) then inequality (4.8) still holds. This means that E_{f} has a Mountain Pass Theorem Geometry and we proceed exactly as in the previous section to conclude the existence of at least two distinct critical points of Ef̃. This completes the proof of Theorem 1.2.

5 Proof of Theorem 1.3

Let f̃ ∈ C̃(Ω̄). Then it follows from Theorems 1.1 and 1.3 that the problem (1.1) has at least one weak solution. It follows from these theorems that for f̃ ≠ 0 there exists ρ = ρ(f̃) > 0 such that (1.1) has at least one solution for any f ∈ B_{C}(f̃, ρ). So we shall concentrate to the proof of the second part of Theorem 1.3. To this end we shall split f ∈ C(Ω̄) as follows

f(x) = f̃(x) + f̂.  (5.1)

Define

F_{−} = F_{−}(f̃) := inf f̂, F_{+} = F_{+}(f̃) := sup f̂,

where the infimum and the supremum are taken over all f̂ for which (1.1) (with f(x) given above) has a solution. It follows directly from the first part of Theorem 1.3 that F_{−} < 0 < F_{+}. To prove that F_{±} are finite we argue by contradiction. Let us suppose that there exist sequences {f̂_{n}} ⊂ ℝ, {u_{n}} ⊂ C_{0}^{1}(Ω̄), such that f̂_{n} → ∞ and u_{n} is a solution to (1.1) with the right hand side

f_{n}(x) = f̃(x)+f̂_{n}. Dividing the equation in (1.1) by f̂_{n}, setting v_{n}: = f̂_{n}^{− \frac{1}{p−1}} u_{n}, and using the compactness of Δ_{p}^{−1}, we find that v_{n} → v_{0} in C_{0}^{1}(Ω̄) (at least for a subsequence). Moreover, v_{0} satisfies

−Δ_{p}v_{0} − λ_{1}|v_{0}|^{p−2}v_{0} = 1 in Ω,
v_{0} = 0 on ∂Ω.

But this is a contradiction with the nonexistence result proved e.g. in [1, 13]. Hence (1.1) has no solution provided f̂ ∉ [F_{−}, F_{+}] which proves (i).

It follows directly from Proposition 2.9 that (1.1) is solvable for any f̂ ∈ (F_{−}, F_{+}). Let now f̂ = F_{−}. Consider f̂_{n} > F_{−}, f̂_{n} → F_{−} and denote by u_{n} ∈ C_{0}^{1}(Ω̄) corresponding solutions of (1.1) with f(x) = f̃(x) + f̂_{n}. According to Proposition 2.11 the sequence {u_{n}} is bounded in C_{0}^{1}(Ω̄). Compactness of Δ_{p}^{−1} implies the existence of a subsequence (denoted again by {u_{n}}) for which u_{n} → u_{−} in C_{0}^{1}(Ω̄) for some u_{−} ∈ C_{0}^{1}(Ω̄). Moreover, similarly as above, u_{−} satisfies

−Δ_{p}u_{−} − λ_{1}|u_{−}|^{p−2}u_{−} = f̃(x) + F_{−} in Ω
u_{−} = 0 on ∂Ω

Similarly, we prove that (1.1) is solvable for f(x) = f̃(x)+F_{+}. This proves (iii).

It remains to prove the multiplicity result stated in (ii). We proceed via contradiction. To this end we apply the degree theory combined with Lemmas 2.7, 2.8 and Propositions 2.9 and 2.11. Let us assume that f̂ ∈ (0, F_{+}) (the proof in case f̂ ∈ (F_{−}, 0) is similar). Then the problem (1.1) with f(x) = f̃(x) + f̂ has a solution u and there exist 0 < f̂_{1} < f̂ < f̂_{2} < F_{+} such that (1.1) has also solutions u_{i} for f_{i}(x) = f̃(x)+f̂_{i}, i = 1, 2. It is straightforward to verify that u_{1} and u_{2} are lower and upper solutions, respectively, of (1.1) with the right hand side f. We assume that u is unique solution of (1.1) obtained by Proposition 2.9, i.e. it is either u_{1} ≤ u ≤ u_{2} in Ω or u ∈ S̄ (with S defined in Lemma 2.8). Assume that the former case occurs, u_{1}, u_{2} are strict, and u_{1} ≺ u_{2}, i.e. u ∉ ∂M_{1} with R = R_{0} large enough (with M_{1} defined in Lemma 2.7). Then according to Lemma 2.7, we have that

deg[I − T_{f}; M_{1}, 0] = 1.  (5.2)

Let us choose f̂_{3} > F_{+}. It follows from above considerations that (1.1) with f_{3}(x) = f̃(x) + f̂_{3} has no solution. Hence

deg[I − T_{f_{3}}; B_{C_{0}^{1}}(0, R), 0] = 0  (5.3)

for arbitrary R > 0. Consider now the family of functions

f_{t}(x) = f̃(x) + tf̂+ (1 − t)f̂_{3}, t ∈ [0, 1].

Then K = {f_{t} ∈ C(Ω̄): t ∈ [0, 1]} is a compact subset of C(Ω̄) and

H(t, ·) = I − T_{f_{t}}, t ∈ [0, 1],

is a homotopy of compact perturbations of the identity. It follows from Proposition 2.11 that for R = R_{1} > R_{0} large enough we have that

deg[I − T_{f_{t}}; B_{C_{0}^{1}}(0, R_{1}), 0]

is constant for t ∈ [0, 1]. Due to (5.3) we have also

deg[I − T_{f}; B_{C_{0}^{1}}(0, R_{1}), 0] = 0.  (5.4)

Additivity property of the degree and (5.2), (5.4) imply that there is ǔ in B_{C_{0}^{1}}(0, R_{1}) \ M_{1} which is a solution of (1.1) and evidently ǔ ≠ u which is a contradiction with uniqueness of u.

The proof follows the same lines if u ∈ S̄ and u ∉ ∂M_{2} (with M_{2} defined in Lemma 2.8). The only difference consists in substituting (5.2) by

deg[I − T_{f}; M_{2}, 0] = −1.

Assume, now, that unique solution u is obtained by means of Lemma 2.7 but u ∈ ∂M_{1}. Since R_{0} can be chosen large enough this means that u_{1} ⊀ u or u ⊀ u_{2}. Let us assume u_{1} ⊀ u (the other case is similar). This means that either there exists x_{0} ∈ Ω such that u_{1}(x_{0}) = u(x_{0}) or there exists x̌_{0} ∈ ∂Ω such that \frac{∂u_{1}}{∂n} (x̌_{0}) = \frac{∂u}{∂n}(x̌_{0}). We choose δ > 0 small enough (to be specified later) and define u_{1}^{δ}(x) = u_{1}(x) − δ, x ∈ Ω. Then u_{1}^{δ} ∈ C^{1}(Ω̄) and u_{1}^{δ} ≺ u. We prove that for δ small this new function u_{1}^{δ} is lower solution of (1.1). Indeed, since u_{1} ∈ C(Ω̄), there exists a constant C = C(∥u_{1}∥_{C}) > 0 such that for any x ∈ Ω̄,

||u_{1}(x) − δ|^{p−2}(u_{1}(x) − δ) − |u_{1}(x)|^{p−2}u_{1}(x)| ≤ |δ|^{p−1},

for 1 < p < 2, and

||u_{1}(x) − δ|^{p−2}(u_{1}(x) − δ) − |u_{1}(x)|^{p−2}u_{1}(x)| ≤ C|δ|,

for p > 2. In either case, there exists δ_{0} > 0 such that for all 0 < δ < δ_{0} we have

∫_{Ω} ||u_{1}^{δ}(x)|^{p−2}u_{1}^{δ}(x) − |u_{1}(x)|^{p−2}u_{1}(x)| ψ(x)dx ≤ \frac{f̂− f̂_{1}}{2λ_{1}} ∫_{Ω} ψ(x)dx (5.5)

for all ψ ≥ 0, ψ ∈ W_{0}^{1,p}(Ω). Since ∇u_{1}^{δ}(x) = ∇u_{1}(x), x ∈ Ω, it follows from (5.5) that

∫_{Ω} |∇u_{1}^{δ}|^{p−2}∇u_{1}^{δ} · ∇ψ − λ_{1} ∫_{Ω} |u_{1}^{δ}|^{p−2}u_{1}^{δ}ψ ≤ ∫_{Ω} f̃ψ + f̄∫_{Ω} ψ,

for any ψ ≥ 0, ψ ∈ W_{0}^{1,p}(Ω), i.e. u_{1}^{δ} is a lower solution of (1.1).

Similarly we can define an upper solution u_{2}^{δ} = u_{2} + δ such that u ≺ u_{2}^{δ}. We define then a new set M_{1}^{δ} by means of u_{1}^{δ}, u_{2}^{δ}, with u_{1}^{δ} ≺ u_{2}^{δ}, and since u ∉ ∂M_{1}^{δ}, we proceed as above to get a contradiction with the uniqueness of u.

Assume, now, that unique solution u is obtained by means of Lemma 2.8 but u ∈ ∂M_{2}. Since R_{0} can be chosen large enough this means that we have two similar possibilities (which can occur simultaneously):

(i) either u(x) ≥ u_{1}(x), x ∈ Ω, and there exists x_{0}^{l} ∈ Ω such that u(x_{0}^{l}) = u_{1}(x_{0}^{l}) or there exists x̌_{0}^{l} ∈ ∂Ω such that \frac{∂u_{1}}{∂n} (x̌_{0}^{l}) = \frac{∂u}{∂n}(x̌_{0}^{l}), (ii) either u(x) ≤ u_{2}(x), x ∈ Ω, and there exists x_{0}^{s} ∈ Ω such that u(x_{0}^{s}) = u_{2}(x_{0}^{s}) or there exists x̌_{0}^{s} ∈ ∂Ω such that \frac{∂u_{2}}{∂n} (x̌_{0}^{s}) = \frac{∂u}{∂n}(x̌_{0}^{s}).

Let us assume that the first possibility (i) occurs. Then for δ small we define a function u_{1}^{δ} = u_{1} − δ. If the second possibility (ii) occurs then we define u_{2}^{δ} = u_{2} + δ. By the same reason as above, u_{1}^{δ} and u_{2}^{δ} are lower and upper solutions of (1.1), respectively, and they are still non-ordered if δ is small enough. Moreover, for M_{2}^{δ} defined by means of u_{1}^{δ}, u_{2}^{δ}, we have that u ∉ \overline{M}_{2}^{δ}. By Lemma 2.8 there must be ǔ ∈ \overline{M}_{2}^{δ} which is a solution of (1.1) and ǔ ≠ u. This contradicts again the uniqueness of u.

The proof of multiplicity result stated in Theorem 1.3 (ii) is thus proved and so the whole proof is finished. □

Acknowledgements. This research was partially supported by grant number 201/00/0376 from the Grant Agency of the Czech Republic and Ministry of Education of Czech Republic, number MSM 235200001. The results in this article were presented in September 2001 at the Workshop “Quasilinear Elliptic and Parabolic Equations and Systems” held in C.I.R.M., Luminy, Marseille, France.

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Pavel Drábek Centre of Applied Mathematics University of West Bohemia P. O. Box 314, 306 14 Plzeň, Czech Republic e-mail: pdrabek@kma.zcu.cz