USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 131–140. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Three solutions for quasilinear equations in ℝ^{n} near resonance ^{∗}

Pablo De Nápoli & María Cristina Mariani

Abstract

We use minimax methods to prove the existence of at least three solutions for a quasilinear elliptic equation in ℝ^{n} near resonance.

1 Introduction

J. Mawhin and K. Smichtt [7], proved the existence of at least three solutions for the two-point boundary value problem

−u′′ − u + εu = f(x, u) + h(x)
u(0) = u(π) = 0

for ε > 0 small enough, h orthogonal to sin x and f bounded satisfying the sign condition uf(x, u) > 0. In [9], To Fu Ma and L. Sanchez considered the problem

−Δ_{p}u − λ_{1}|u|^{p−2}u + ε|u|^{p−2}u = f(x, u) + h(x)  (1.1)

in W_{0}^{1,p}(Ω) with Ω ⊂ ℝ^{n} a bounded domain, and λ_{1} the first eigenvalue of

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

They proved the following result.

Theorem 1.1 Suppose that p ≥ 2 and that the following two conditions hold:

(H1) f : \overline{Ω} × ℝ^{n} → ℝ^{n} is a continuous function and there exist θ > \frac{1}{p} such that θsf(x, s) − F (x, s) → −∞ as |s| → ∞

(H2) There exists R > 0 such that sf(x, s) > 0 for all x ∈ Ω, |s| ≥ R

Then for every h ∈ L^{p′}(Ω) with ∫_{Ω} h(x)φ_{1}(x)dx = 0, where φ_{1} is the first eigenfunction of (1.2), the equation (1.1) has at least three solutions for ε > 0 small enough.

∗Mathematics Subject Classifications: 35J20, 35J60.

Key words: p-Laplacian, resonance, nonlinear eigenvalue problem.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

We recall that the assumptions on f imply the growth condition

|f(x, s)| ≤ c_{1} + c_{2}|s|^{σ}

with σ = \frac{1}{θ} < p.

These problems have been studied for several authors, see [3, 4, 5, 8].

The functional setting

Our aim is to extend this result to equations in ℝ^{n}. As W^{1,p}(ℝ^{n}) is no longer compactly imbedded into L^{p}(ℝ^{n}), we shall work in the space D^{1,p}, the closure of C_{0}^{1}(ℝ^{n}) with the norm

∥u∥_{1,p} = ( ∫_{ℝ^{n}} |∇u(x)|^{p}dx)^{1/p}

By the Sobolev inequality we have: D^{1,p} ⊂ L^{p^{∗}}(ℝ^{n}) with p^{∗} = \frac{Np}{N−p}, this imbedding is not compact, however in proposition 2.1 we prove that the imbedding D^{1,p} ⊂ L_{g}^{p}(ℝ^{n}) is compact for g ∈ L^{N/p} ∩ L_{loc}^{N/p+ε}.

Simplicity of the first eigenvalue

We recall the simplicity of the first eigenvalue of the p-laplacian that is proved in [4]. They studied the problem:

−Δ_{p}u = g(x)|u|^{p−2}u x ∈ ℝ^{n}  (1.3)
0 < u in ℝ^{n}, lim_{|x|→+∞} u(x) = 0 ,

where 1 < p < n. They proved the theorem below, assuming the following conditions:

(G) g is a smooth function, at least C_{loc}^{0,γ}(ℝ^{n}) for some γ ∈ (0, 1), such that g ∈ L^{N/p}(ℝ^{n}) ∩ L^{∞}(ℝ^{n}) and g(x) > 0 in Ω^{+} with |Ω^{+}| > 0. Also g satisfies one the following two conditions

(G^{+}) g(x) ≥ 0 a.e. in ℝ^{n}

(G^{−}) g(x) < 0 for x ∈ Ω^{−}, with |Ω^{−}| > 0.

Theorem 1.2 1. Let g satisfy (G) and (G^{+}). Then equation (1.3) admits a positive first eigenvalue,

λ_{1} = inf_{B(u)=1} ∥u∥_{D^{1,p}}^{p}  (1.4)

with B(u) = ∫_{ℝ^{n}} |u(x)|^{p}g(x) dx.

2. Let g satisfy (G) and (G^{−}). Then problem (1.3) admits two first eigenvalues of opposite sign:

λ_{1}^{+} = inf_{B(u)=1} ∥u∥_{D^{1,p}}^{p} λ_{1}^{−} = − inf_{B(u)=−1} ∥u∥_{D^{1,p}}^{p}

In both cases the associated eigenfunctions φ_{1}^{+}, φ_{1}^{−} belong to D^{1,p} ∩ L^{∞}.

3. The set of eigenvectors corresponding to λ_{1} is a one dimensional subspace.

Remark 1.3 The first eigenfunction φ_{1} does not change its sign in Ω, so we may assume φ_{1} ≥ 0.

Proof. Taking φ^{−} as a test function in (1.3) with λ = λ_{1} we see that

∫_{ℝ^{n}} |∇(φ^{−})|^{p} = λ_{1} ∫_{ℝ^{n}} |φ_{1}^{−}|^{p}g(x)dx

It follows that φ^{−} = 0 (and φ ≥ 0 ), or φ_{1}^{−} is also a solution of the minimization problem (1.4). In the last case, from the simplicity of the first eigenvalue φ_{1}^{−} = cφ_{1}. It follows that φ^{−} = −φ_{1}, so φ_{1} ≤ 0. ♢

Existence of multiple solutions

In this paper we study quasilinear elliptic equation

−Δ_{p}u = (λ_{1} − ε)g(x)|u|^{p−2}u + f(x, u) + h(x)  (1.5)

in ℝ^{n}. We assume the following:

1. 1 < p < n and ε > 0

2. On the weight g we make the assumptions (G) and (G^{+}) of [4]

3. h ∈ L^{p^{∗′}} and ∫_{ℝ^{n}} hφ_{1}dx = 0

4. We assume that the non linearity f : ℝ^{n} × ℝ → ℝ is continuous and satisfies

(H0) Growth condition.

|f(x, s)| ≤ c_{1}(x) + c_{2}(x)|s|^{σ−1}

with σ < p and c_{1} ∈ L^{(p^{∗})′}, c_{2} ∈ L^{(p^{∗}/σ)′} ∩ L_{loc}^{(p/σ)′+η} for some η > 0. (H1) If F (x, s) = ∫_{0}^{s} f(x, t)dt then \frac{1}{p}sf(x, s)−F (x, s) → −∞ as |s| → ∞.

(H2) Sign condition. There exists R > 0 such that: sf(x, s) > 0 for all x ∈ ℝ^{n}, |s| ≥ R.

For example we may take f(x, s) = c_{2}(x)|s|^{σ−1}s · sgn s where c_{2}(x) satisfies the conditions above, c_{2}(x) > 0, and σ < p.

Note that integrating on condition (H0) we get

F (x, s) ≤ c_{1}(x)|s| + c_{2}(x)\frac{|s|^{σ}}{σ} .

In the next section we will see that for the functional C(u) = ∫_{ℝ^{n}} F (x, u)dx to be of class C^{1}(D^{1,p}(ℝ^{n})), condition (H0) is the natural choice.

Our main result is the following theorem:

Theorem 1.4 Under the assumptions above, problem (1.5) has at least three solutions for ε > 0 small enough.

2 Technical Lemmas

For the proof of theorem 1.4 we will need the following results:

A compactness result in weighted L^{p} spaces

If u ∈ D^{1,p}, 1 ≤ q ≤ p^{∗}, \frac{1}{r} + \frac{q}{p^{∗}} = 1 and g ∈ L^{r}, g ≥ 0, then from Hölder and Sobolev inequalities, we have that

∫_{ℝ^{n}} |u|^{q}g ≤ C ∫_{ℝ^{n}} |∇u|^{p}  (2.1)

and it follows that D^{1,p} ⊂ L_{g}^{q}. The following result proves that under appropriate conditions, this imbedding is also compact. (Other previous results can be found in [6]).

Proposition 2.1 Let 1 ≤ q < p^{∗}, \frac{1}{r} + \frac{q}{p^{∗}} = 1, g ∈ L^{r} ∩ L_{loc}^{r+ε} for some ε > 0. Then the imbedding

D^{1,p} ⊂ L_{g}^{q}(ℝ^{n})

is compact.

Proof. Let (u_{n}) ⊂ D^{1,p} be a bounded sequence:

∥u_{n}∥_{1,p} ≤ C

Then, as D^{1,p} is reflexive, we may extract a weakly convergent subsequence (u_{n_{k}}). For simplicity we assume that u_{n} ⇀ u. We want to prove that in fact u_{n} → u strongly. From Hölder and Sobolev inequalities we have:

∫_{|x|>R} g|u−u_{n}|^{q} ≤ ( ∫_{|x|>R} |g|^{r})^{1/r}( ∫_{|x|>R} |u_{n}−u|^{p^{∗}})^{p/p^{∗}} ≤ C( ∫_{|x|>R} |g|^{r})^{1/r}

Given ε > 0, as g ∈ L^{r} we can choose R > 0 verifying

∫_{|x|>R} g|u − u_{n}|^{q} ≤ \frac{ε}{2}

Now D^{1,p}(ℝ^{n}) ⊂ W_{loc}^{1,p}(ℝ^{n}) continously and by the Rellich-Kondrachov theorem

u_{n} → u strongly in L^{t}(B_{R})

if 1 ≤ t < p^{∗}. We choose s > 1 such that s′ = r + ε, then s < \frac{p^{∗}}{q}, and

∫_{|x|≤R} g|u_{n} − u|^{q} ≤ ( ∫_{|x|≤R} |g|^{s′})^{1/s′}( ∫_{|x|<R} |u − u_{n}|^{qs})^{1/s} ≤ \frac{ε}{2}

if n ≥ n_{0}(ε). So u_{n} → u in L_{g}^{p}(ℝ^{n}). ♢

Some results about the Associated Functional

Under the same assumptions of theorem 1.4, we have the following results:

Lemma 2.2 Let C : D^{1,p}(ℝ^{n}) → ℝ given by C(u) = ∫_{ℝ^{n}} F (x, u)dx. Then C ∈ C^{1}(D^{1,p}(ℝ^{n})) and C′(u)(h) = ∫_{ℝ^{n}} f(x, u)h

Proof. From the Hölder inequality we have that

|C(u)| ≤ ∫_{ℝ^{n}} c_{1}(x)|u| + c_{2}(x)\frac{|u|^{σ}}{σ} dx ≤ ∥c_{1}∥_{(p^{∗})′} ∥u∥_{p^{∗}} + \frac{1}{σ} ∥c_{2}∥_{(p^{∗}/σ)′} ∥u∥_{p^{∗}}^{σ}

From the imbedding D^{1,p} ⊂ L^{p^{∗}} we conclude that C(u) is well defined. In a similar way,

| ∫_{ℝ^{n}} f(x, u)h| ≤ ∫_{ℝ^{n}} c_{1}(x)|h| + c_{2}|u|^{σ−1}|h|
≤ ∥c_{1}∥_{(p^{∗})′} ∥h∥_{p^{∗}} + ∥c_{2}∥_{(p^{∗}/σ)′} ∥u∥_{p^{∗}}^{σ−1} ∥h∥_{p^{∗}}

and we have that ∫_{ℝ^{n}} f(x, u)h is also well defined. Using a similar argument as in [8], we conclude the proof. ♢

Lemma 2.3 Assume that f(x, y) is a Caratheodory function, verifying that

|f(x, u)| ≤ c_{1}(x) + c_{2}(x)|u|^{σ−1}

where 1 ≤ σ < p^{∗}, c_{1} ∈ L^{s_{1}}(ℝ^{n}) with s_{1} = p^{∗}′, and c_{2} ∈ L^{s_{2}} ∩ L_{loc}^{s_{2}+ε} with s_{2} = \frac{p^{∗}}{p^{∗}−σ}. Then the Nemitski operator N_{f} : D^{1,p}(ℝ^{n}) → L^{p^{^{∗}′}} (ℝ^{n}) given by N_{f}(u) = f(x, u) is compact.

Proof. Let (u_{n}) be a sequence in D^{1,p} such that u_{n} ⇀ u weakly in D^{1,p}. We may assume, passing to a subsequence, that u_{n} → u a.e..

As σ < p^{∗}, we apply proposition 2.1 with q = (σ − 1)s_{1} < p^{∗}, g = c_{2}^{s_{1}}. We note that g ∈ L^{r} ∩ L_{loc}^{r+ε′} with r = \frac{p^{∗}−1}{p^{∗}−σ}. We get, passing to a subsequence, that u_{n} → u in L_{g}^{q}.

From theorem IV.9 in [2], we obtain, after passing again to a subsequence, a function m ∈ L_{g}^{q}(ℝ^{n}) such that

|u_{n}(x)| ≤ m(x)

a.e. with respect to the measure g(x)dx. Then, from condition (H0) we deduce that

|f(x, u) − f(x, u_{n})|^{s_{1}} ≤ 2^{s_{1}}[|f(x, u)|^{s_{1}} + |f(x, u_{n})|^{s_{2}}]
≤ 2^{s_{1}+1}[c_{1}(x)^{s_{1}} + c_{2}(x)^{s_{1}}|m|^{(σ−1)s_{1}}] .

Applying the bounded convergence theorem to ∫_{ℝ^{n}} |f(x, u) − f(x, u_{n})|^{s_{1}}dx we obtain that f(x, u_{n}) → f(x, u) in L^{s_{1}}(ℝ^{n}). ♢ Remark 2.4 The weak solutions of equation (1.5) are the critical points in D_{0}^{1,p} of the functional

J_{ϵ}(u) = \frac{1}{p} ∫_{ℝ^{n}} |∇u|^{p}dx − \frac{λ_{1} − ε}{p} ∫_{ℝ^{n}} |u|^{p}g(x)dx − ∫_{ℝ^{n}}(F (x, u) + h(x)u)

Under the previous assumptions it is easy to check that J_{ε} ∈ C^{1}(D^{1,p}).

Let

W = {w ∈ D^{1,p} : ∫_{ℝ^{n}} g(x)|φ_{1}|^{p−2}φ_{1}w = 0}

We recall that as a consequence of proposition 2.1 W is a weakly closed linear subspace.

Lemma 2.5 If ε < λ_{1} , J_{ϵ} is coercive in D^{1,p}, and there exist m > 0 such that inf_{u∈W} J_{ϵ}(u) ≥ −m.

Proof. We suppose 0 < ε < λ_{1}, then

J_{ε}(u) ≥ \frac{1}{p} (1 − \frac{λ_{1} − ϵ}{λ_{1}}) ∫_{ℝ^{n}} |∇u|^{p} − ∫_{ℝ^{n}}(F (x, u) + hu)

and

J_{ε}(u) ≥ \frac{ϵ}{pλ_{1}} ∥u∥_{1,p}^{p} − C_{1} − C_{2} ∥u∥_{1,p}^{σ} − ∥h∥_{(p^{∗})′} ∥u∥_{p^{∗}}

As σ < p, it follows that J_{ε} is coercive.

We define

λ_{W} = inf {∫_{ℝ^{n}} |∇w|^{2} : w ∈ W, ∫_{ℝ^{n}} g(x)|w(x)|^{p} = 1}

We claim that λ_{W} > λ_{1}. In fact if λ_{1} = λ_{W} then we would have w ∈ W verifying

∫_{ℝ^{n}} |w|^{p} = λ_{1}, ∫_{ℝ^{n}} |w|^{p}g(x)dx = 1

So by the simplicity of the first eigenvalue, w = cφ_{1} but this contradicts the definition of W.

Then, for u ∈ W we have

J_{ε}(u) ≥ \frac{λ_{W} − λ_{1}}{pλ_{W}} ∥u∥_{1,p}^{p} − C_{1} − C_{2} ∥u∥_{1,p}^{σ} − ∥h∥_{(p^{∗})′} ∥u∥_{p^{∗}}

Then J_{ε} is uniformly coercive in W respect to ε, and in particular is uniformly bounded from below. ♢

For stating the next result we need the two open sets:

O^{+} = {w ∈ D^{1,p} : ∫_{ℝ^{n}} g(x)|φ_{1}|^{p−2}φ_{1}w > 0},
O^{−} = {w ∈ D^{1,p} : ∫_{ℝ^{n}} g(x)|φ_{1}|^{p−2}φ_{1}w < 0}

The next condition is a variant of the Palais-Smale condition (PS).

We will say that a functional ϕ : D^{1,p} → ℝ verifies the (P S)_{O^{±},c} condition if any sequence (u_{n}) in O^{+} (respectively in O^{−}) with ϕ(u_{n}) → c, ϕ′(u_{n}) → 0, has a subsequence (u_{n_{k}}) → u ∈ O^{+}.

Proposition 2.6 The operator −Δ_{p} : D^{1,p} → (D^{1,p})^{∗} satisfies the (S_{+}) condition: if u_{n} ⇀ u (weakly in D^{1,p}(ℝ^{n}) ) and lim sup_{n→∞} ⟨−Δ_{p}u_{n}, u_{n} − u⟩ ≤ 0, then u_{n} → u (strongly in D^{1,p} )

Proof. This follows from the uniform convexity of D^{1,p}(ℝ^{n}) (see [3])

Lemma 2.7 J_{ϵ} satisfies the (P S) condition, and it verifies (P S)_{O^{±},c} if c < −m.

Proof. Let (u_{n}) ⊂ D^{1,p} be a (P S) sequence such that

J_{ε}(u_{n}) → c, J′_{ε}(u_{n}) → 0

Since J_{ε} is coercive, it follows that (u_{n}) is bounded in D^{1,p}, which is reflexive, so (after passing to a subsequence) we may assume that u_{n} → u weakly. We want to show that in fact, u_{n} → u strongly. We have that

J′_{ε}(u_{n})(u_{n} − u) = ∫ |∇u_{n}|^{p−2}∇u_{n} · ∇(u_{n} − u)
−(λ_{1} − ε) ∫ |u_{n}|^{p−2}u_{n}(u_{n} − u)g(x)dx
− ∫ h(u_{n} − u) − ∫ f(x, u_{n})(u_{n} − u)

Clearly ∫ h(u_{n} − u) → 0 since u_{n} ⇀ u weakly. Then u_{n} → u strongly in L_{g}^{p}(ℝ^{n}) since the imbedding D^{1,p} ⊂ L_{g}^{p} is compact. It follows that: ∫ |u_{n}|^{p−2}u_{n}(u_{n} − u)g(x)dx → 0

From proposition 2.3 and the Hölder inequality

∫ f(x, u_{n})(u_{n}−u)dx = ∫ [f(x, u)−f(x, u_{n})](u_{n}−u)dx+∫ f(x, u)(u_{n}−u) → 0 .

Since J′_{ε}(u_{n})(u_{n} − u) → 0, it follows that

∫ |∇u_{n}|^{p−2}∇u_{n} · ∇(u_{n} − u)dx → 0

or equivalently, ⟨−Δ_{p}u_{n}, u_{n} − u⟩ → 0. By the S_{+} condition, this implies that u_{n} → u strongly in D^{1,p}.

To prove that J_{ϵ} satisfies (P S)_{O^{±},c} for c < −m, consider (u_{n}) ⊂ O^{±} be a (P S)_{c} sequence. There exists a convergent subsequence: u_{n_{k}} → u, and it is enough to prove that u ∈ O^{±}, but if u ∈ ∂O^{±} = W, then c = J(u) ≥ −m, a contradiction. ♢

Lemma 2.8 If ε > 0 is small enough, there exists two numbers, t^{−} < 0 < t^{+}, such that J_{ε}(t^{±}φ_{1}) < −m.

Proof. From ∫ h(x)φ_{1}(x)dx = 0, we have that

J_{ε}(tφ_{1}) = \frac{1}{p} ∫_{ℝ^{n}} 94εt^{p}φ_{1}^{p}g(x) − ∫_{ℝ^{n}} F (x, tφ_{1}(x))dx .

Since φ_{1} ∈ L^{∞}, we can assume that 0 ≤ φ_{1}(x) ≤ 1 for all x ∈ ℝ^{n}.

First, since φ_{1}^{p}g ∈ L^{1}, we can choose ρ big enough, such that:

\frac{1}{p} ∫_{|x|>ρ} φ_{1}^{p}gdx < \frac{m}{2}

and we split the integral J_{ε} in two parts: J_{ε} = J_{ε}^{1} + J_{ε}^{2}, where J_{ε}^{1} is the integral over |x| ≤ ρ, and J_{ε}^{2} is the integral over |x| > ρ.

We define

A(t) = {x : |x| ≤ ρ : φ_{1}(x) > R/t}
B(t) = {x : |x| ≤ ρ : φ_{1}(x) ≤ R/t}

Then

∫_{B(t)}[\frac{ε}{p}t^{p}φ_{1}^{p} − F (x, tφ_{1}(x))]dx

is uniformly bounded in ε and t for ε ≤ ε_{0}. Let

M_{ε}(t) = ∫_{A(t)} (\frac{1}{p}tφ_{1}(x)f(x, tφ_{1}(x)) − F (x, tφ_{1}(x)))

+ ∫_{B(t)} [\frac{ε}{p}t^{p}φ_{1}^{p} − F (x, tφ_{1}(x))] dx

Then, from (H1) and Fatou lemma, M_{ε}(t) < −2m for t big enough, and ε ≤ ε_{0}. By (H2) there exists 0 < ε_{t} ≤ ε_{0} such that

ε_{t}u^{p−1}g(x) < f(x, u) in \overline{B_{ρ}} × [R, t]

Then if φ_{1}(x) > R/t and |x| ≤ ρ we have:

ε_{t}t^{p−1}φ_{1}(x)^{p−1}g(x) < f(x, tφ_{1})

and

J_{ε}^{1}(tφ_{1}) ≤ M_{ε}(t) < −2m .

From (H2), since F (x, tφ_{1}) ≥ 0, if we choose ε_{t} satisfying ε_{t} < \frac{1}{t^{p}} then,

J_{ε_{t}}^{2}(tφ_{1}) ≤ \frac{1}{p} ∫_{|x|>ρ} ε_{t}t^{p}φ_{1}^{p}dx < \frac{m}{2}

and we conclude that J_{ε_{t}}(tφ_{1}) < −m for any ε_{t} ≤ ε_{0}. In a similar way, choosing first t big enough, and then ε_{t} small, we can prove that J_{ε_{t}}(−tφ_{1}) < −m ♢

Proof of theorem 1.4

For ε > 0 small enough, from lemmas 2.7 and 2.8 we have that

−∞ < inf_{O^{±}} J_{ε} < −m

and since (P S)_{c,O^{±}} holds for all c < −m, it follows from the deformation lemma that the above infima are attained, say at u^{−} ∈ O^{−} and u^{+} ∈ O^{+}. Since O^{±} are both open in D^{1,p} we have found two critical points of J_{ε}. Let

c = inf_{γ∈Γ} max_{t∈[0,1]} J_{ε}(γ(t))

with

Γ = {γ ∈ C([0, 1], D^{1,p}(ℝ^{n}) : γ(0) = u^{−}, γ(1) = u^{+}}

We observe that γ([0, 1]) ∩ W ≠ 0 for any γ ∈ Γ, so we conclude that

c = inf_{W} J_{ε} ≥ −m

J_{ε} verifies (P S), and from Ambrossetti-Rabinowitz’s Mountain Pass Theorem [1] we conclude that c is a third critical value of J_{ε}, and since J_{ε}(u^{±}) < −m, the corresponding critical point is different from u^{+}, u^{−}.

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Pablo L. De Nápoli (e-mail: pdenapo@dm.uba.ar) M. Cristina Mariani (e-mail: mcmarian@dm.uba.ar) Universidad de Buenos Aires FCEyN - Departamento de Matemática Ciudad Universitaria, Pabellón I Buenos Aires, Argentina