The Brezis–Oswald Result for Quasilinear Robin Problems

Remark 2.1

Hypotheses (Ha) (i)–(iii) are dictated by the nonlinear regularity theory of Lieberman [5] and by the nonlinear maximum principle of Pucci and Serrin [24]. Hypothesis (Ha) (iv) is used to prove the uniqueness of the positive solution.

Note that the primitive G0⁢(t), t≥0, is strictly convex and strictly increasing. By setting G⁢(y)=G0⁢(|y|) for all y∈ℝN, we have that G⁢(⋅) is convex and continuously differentiable. Moreover,

∇ ⁡ G ⁢ ( 0 ) = 0   and   ∇ ⁡ G ⁢ ( y ) = G 0 ′ ⁢ ( | y | ) ⁢ y | y | = a 0 ⁢ ( | y | ) ⁢ y = a ⁢ ( y )

for all y∈ℝN∖{0}. Therefore, G⁢(⋅) is the primitive of a⁢(⋅) vanishing at y=0. Finally, the convexity of G⁢(⋅) implies that

Lemma 2.2

If hypotheses (Ha)  (i)–(iii) hold, then

1. y ↦ a ⁢ ( y ) is continuous and strictly monotone, hence maximal monotone too,

2. |a⁢(y)|≤c4⁢(1+|y|p-1) for all y∈ℝN and some c4>0,

3. (a⁢(y),y)ℝN≥c1p-1⁢|y|p for all y∈ℝN.

Corollary 2.3

If hypotheses (Ha)  (i)–(iii) hold, then

c 1 p ⁢ ( p - 1 ) ⁢ | y | p ≤ G ⁢ ( y ) ≤ c 5 ⁢ ( 1 + | y | p )

for all y∈ℝN and some c5>0.

Examples

The following maps satisfy hypotheses (Ha):

1. The map

   a ⁢ ( y ) = | y | p - 2 ⁢ y   with ⁢ 1 < p < ∞ ,

   which corresponds to the p-Laplace differential operator

   Δ p ⁢ u = div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u )   for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

2. The map

   a ⁢ ( y ) = | y | p - 2 ⁢ y + | y | q - 2 ⁢ y   with ⁢ 1 < q < p < ∞ ,

   which corresponds to the (p,q)-differential operator defined by

   Δ p ⁢ u + Δ q ⁢ u   for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

   Such operators arise in problems of mathematical physics; see [4] and [6] for examples in quantum and plasma physics, respectively. Recently there have been some existence and multiplicity results for equations driven by such operators, see [7, 16, 20, 22, 23, 26].

3. The map

   a ⁢ ( y ) = ( 1 + | y | 2 ) p - 2 2 ⁢ y   with ⁢ 1 < p < ∞ ,

   which corresponds to the generalized p-mean curvature differential operator defined by

   div ⁡ ( ( 1 + | D ⁢ u | 2 ) p - 2 2 ⁢ D ⁢ u )   for all u ∈ W 1 , p ⁢ ( Ω ) .

4. The map

   a ⁢ ( y ) = | y | p - 2 ⁢ y ⁢ ( 1 + 1 1 + | y | p )   with ⁢ 1 < p < ∞ .

Proposition 2.4

If ξ∈L∞⁢(Ω) and hypothesis (Hβ) holds, then problem (2.3) admits a smallest eigenvalue λ^1⁢(ξ,β,p)∈ℝ, which is simple and isolated, and there exists a corresponding Lp-normalized eigenfunction u^1 (that is, ∥u^1∥p=1) which satisfies

u ^ 1 ∈ C 1 , η ⁢ ( Ω ¯ )   for some ⁢ η ∈ ( 0 , 1 ) , u ^ 1 ∈ int ⁡ C + .

Moreover, every eigenfunction corresponding to an eigenvalue λ^>λ^1 is nodal (that is, sign changing).

Proof.

Let γ:W1,p⁢(Ω)→ℝ be the C1-functional defined by

γ ⁢ ( u ) = ∥ D ⁢ u ∥ p p + ∫ Ω ξ ⁢ ( z ) ⁢ | u | p ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ | u | p ⁢ 𝑑 σ   for all u ∈ W 1 , p ⁢ ( Ω ) .

Here σ⁢(⋅) denotes the (N-1)-dimensional Hausdorff measure on ∂⁡Ω. Let M⊆W1,p⁢(Ω) the C1-Banach manifold defined by

M := { u ∈ W 1 , p ⁢ ( Ω ) : ∥ u ∥ p = 1 } ,

and set

Evidently,

λ ^ 1 ⁢ ( ξ , β , p ) ≥ - ∥ ξ ∥ ∞ ,

see hypothesis (Hβ) and recall that ξ∈L∞⁢(Ω).

Let {un}n≥1⊆M be a minimizing sequence for (2.4). So, γ⁢(un)↓λ^1⁢(ξ,β,p). Evidently, {un}n≥1⊆W1,p⁢(Ω) is bounded, and so we may assume that

Here we have used the Sobolev embedding theorem and the compactness of the trace map.

Because of (2.5), we have

Using (2.6)–(2.7), we obtain

Note that ∥u^1∥p=1, see (2.5). Hence, u^1∈M, and so (2.8) becomes

γ ⁢ ( u ^ 1 ) = λ ^ 1 ⁢ ( ξ , β , p ) .

The Lagrange multiplier rule (see, for example, [11, p. 701]), implies that λ^1⁢(ξ,β,p) is an eigenvalue of problem (2.3), more precisely the smallest eigenvalue, with u^1∈W1,p⁢(Ω) a corresponding eigenfunction. From [23], we know that u^1∈L∞⁢(Ω). Then the regularity result of Lieberman [15] implies that u^1∈C1,η⁢(Ω¯) for some η∈(0,1). Note that

γ ⁢ ( | u | ) = γ ⁢ ( u )   for all ⁢ u ∈ M .

So, we infer that u^1 does not change sign and we may assume that u^1≥0. The nonlinear maximum principle (see, for example, [11, p. 738] and [24, pp. 111, 120]) implies that u^1⁢(z)>0 for all z∈Ω¯, hence u^1∈int⁡C+.

Next we establish the simplicity of λ^1⁢(ξ,β,p). To this end, let v^1∈W1,p⁢(Ω) be another eigenfunction corresponding to λ^1⁢(ξ,β,p). As we did for u^1, we can show that v^1∈int⁡C+. Consider the function

R ⁢ ( u ^ 1 , v ^ 1 ) ⁢ ( z ) = | D ⁢ u ^ 1 ⁢ ( z ) | p - | D ⁢ v ^ 1 ⁢ ( z ) | p - 2 ⁢ ( D ⁢ v ^ 1 ⁢ ( z ) , D ⁢ ( u ^ 1 p v ^ 1 p - 1 ) ⁢ ( z ) ) ℝ N .

Invoking the nonlinear Picone’s identity (see [2] and [17, p. 255]) and the nonlinear Green’s identity (see [11, p. 211]), we have

Recalling that R⁢(u^1,v^1)⁢(z)≥0, we get R⁢(u^1,v^1)⁢(z)=0 for a.a. z∈Ω, and so, by the nonlinear Picone’s identity, u^1=μ⁢v^1 for some μ>0.

Now, suppose that λ^>λ^1⁢(ξ,β,p) is another eigenvalue of (2.3) with u∈W1,p⁢(Ω) a corresponding Lp-normalized eigenfunction. Assume that u^ has constant sign and to fix things suppose that u^1≥0. As before, the nonlinear regularity theory (see [15]) and the nonlinear maximum principle (see [11, 24]), imply that u^∈int⁡C+. Using once again the nonlinear Picone’s identity and the nonlinear Green’s identity, we have

a contradiction. So, u^∈C1⁢(Ω¯) must be nodal.

Finally, we show that λ^1⁢(ξ,β,p) is isolated in the spectrum of (2.3). Again we argue by contradiction. So, suppose that λ^1⁢(ξ,β,p) is not isolated. Then we can find eigenvalues {λn}n≥1 such that λn↓λ^1⁢(ξ,β,p) as n→∞. Let {un}n≥1⊆W1,p⁢(Ω) be a sequence of corresponding Lp-normalized eigenfunctions. Evidently, {un}n≥1⊆W1,p⁢(Ω) is bounded. Then, from [23], we know that there exists M1>0 such that

∥ u n ∥ ∞ ≤ M 1   for all ⁢ n ∈ ℕ .

Therefore, from [15], it follows that there exist η∈(0,1) and M2>0 such that

u n ∈ C 1 , η ⁢ ( Ω ¯ )   and   ∥ u n ∥ C 1 , η ⁢ ( Ω ¯ ) ≤ M 2   for all ⁢ n ∈ ℕ .

Because of the compact embedding of C1,η⁢(Ω¯) into C1⁢(Ω¯), we may assume (at least for a subsequence) that

Let A:W1,p⁢(Ω)→W1,p⁢(Ω)* be the nonlinear map defined by

We have

〈 A ( u n ) , h 〉 + ∫ Ω ξ ( z ) | u n | p - 2 u n h d z + ∫ ∂ ⁡ Ω β ( z ) | u n | p - 2 u n h d σ = λ n ∫ Ω | u n | p - 2 u n h d z

for all h∈W1,p⁢(Ω) and all n∈ℕ. Hence, passing to the limit as n→∞ and using (2.9), we find

〈 A ( u ~ 1 ) , h 〉 + ∫ Ω ξ ( z ) | u ~ 1 | p - 2 u ~ 1 h d z + ∫ ∂ ⁡ Ω β ( z ) | u ~ 1 | p - 2 u ~ 1 h d σ = λ ^ 1 ( ξ , β , p ) ∫ Ω | u ~ 1 | p - 2 u ~ 1 h d z

for all h∈W1,p⁢(Ω), that is,

{ - Δ p ⁢ u ~ 1 ⁢ ( z ) + ξ ⁢ ( z ) ⁢ | u ~ 1 ⁢ ( z ) | p - 2 ⁢ u ~ 1 ⁢ ( z ) = λ ^ 1 ⁢ ( ξ , β , p ) ⁢ | u ~ 1 ⁢ ( z ) | p - 2 ⁢ u ~ 1 ⁢ ( z ) for a.a. ⁢ z ∈ Ω , ∂ ⁡ u ~ 1 ∂ ⁡ n p + β ⁢ ( z ) ⁢ | u ~ 1 | p - 2 ⁢ u ~ 1 = 0 on ⁢ ∂ ⁡ Ω ,

see [21]. This implies u~1=±u^1, see (2.9) and recall that λ^1⁢(ξ,β,p) is simple. Then we can suppose that u~1∈int⁡C+, and from (2.9) we have that un∈int⁡C+ for all n≥n0, which contradicts the fact established earlier, that every non principal eigenvalue has nodal eigenfunctions. This proves that λ^1⁢(ξ,β,p) is isolated in the spectrum of (2.3). ∎

Remark 3.1

1. Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis ℝ+=[0,+∞), we may assume that

   f ⁢ ( z , x ) = f ⁢ ( z , 0 )   for a.a. z ∈ Ω and all ⁢ x ≤ 0 .

2. Hypothesis (Hf) (i) is a unilateral growth condition on f⁢(z,⋅). Clearly, both functions μ⁢(⋅) and μ0⁢(⋅) in (Hf) (iii) and (iv), respectively, are measurable functions.

3. Hypothesis (Hf) (iii) covers the case μ=-∞ in a set of positive measure.

4. Hypothesis (Hf) (ii) implies that the function x↦f⁢(z,x)/xp-1 is strictly decreasing on (0,+∞) for a.a. z∈Ω. Then, using hypothesis (Hf) (i), we have

   f ⁢ ( z , x ) x p - 1 ≤ f ⁢ ( z , 1 ) ≤ ∥ f ⁢ ( ⋅ , 1 ) ∥ ∞ = c 7   for a.a. ⁢ z ∈ Ω ⁢ and all ⁢ x ≥ 1 .

   This implies μ⁢(z)≤c7 for a.a. z∈Ω (see hypothesis (Hf) (iii)), hence

   λ ^ 1 ⁢ ( - μ ^ , β ^ , p ) ∈ ( - ∞ , + ∞ ] .

   On the other hand, again from hypotheses (Hf) (i) and (ii), we have

   f ⁢ ( z , x ) x q - 1 ≥ f ⁢ ( z , 1 ) ≥ - ∥ f ⁢ ( ⋅ , 1 ) ∥ ∞ = - c 7   for a.a. ⁢ z ∈ Ω ⁢ and all ⁢ x ∈ ( 0 , 1 ] .

   Thus, μ0⁢(z)≥-c7 for a.a. z∈Ω and

   λ ^ 1 ⁢ ( - μ ~ 0 , β ~ , q ) ∈ [ - ∞ , + ∞ ) .

   Of course, if μ,μ0∈L∞⁢(Ω), then λ^1⁢(-μ^,β^,p) and λ^1⁢(-μ~0,β~,q)∈ℝ are principal eigenvalues of nonlinear eigenvalue problems similar to (2.3).

5. If f⁢(z,x)=f⁢(x) (autonomous case), then hypotheses (Hf) (iii) and (iv) are equivalent to writing

   μ < 0 < μ 0 .

Example

Consider the function

f ⁢ ( x ) = λ ⁢ ( x q - 1 - x r - 1 )   for all ⁢ x ≥ 0

with 1<q≤p<r and λ>0. This function satisfies hypotheses (Hf) with μ0=λ>0 and μ=-∞. Note that f⁢(x) is the reaction term in the nonlinear logistic equation of sub-diffusive type, see [19].

Proposition 3.2

If hypotheses (Ha)  (i)–(iii), (Hβ), (Hf) hold, then the functional φ is coercive.

Proof.

We argue indirectly. So, suppose that φ is not coercive. Then, we can find {un}n≥1⊆W1,p⁢(Ω) such that

From (3.1) and hypothesis (Hf) (i), we have

Using Corollary 2.3 and (3.3), we have

for some c9>0 and all n∈ℕ. From (3.2), (3.4) and hypothesis (Hβ), it follows that

Let yn=un/∥un∥p, n∈ℕ. Then

and from (3.4) we have

c 1 p ⁢ ( p - 1 ) ⁢ ( ∥ D ⁢ y n ∥ p p + 1 ) ≤ c 9 ⁢ ( 1 ∥ u n ∥ p p + 1 )   for all ⁢ n ∈ ℕ ,

which implies that {yn}n≥1⊆W1,p⁢(Ω) is bounded by (3.5) and (3.6). So, we may assume that

From (3.2) and Corollary 2.3, we have

for all n∈ℕ (see (3.1)), where F⁢(z,s)=∫0sf⁢(z,t)⁢𝑑t.

Hypothesis (Hf) (ii) implies that

f ⁢ ( z , x ) x p - 1 ≥ f ⁢ ( z , 1 )   for a.a. ⁢ z ∈ Ω ⁢ and all ⁢ x ∈ ( 0 , 1 ] ,

hence, by hypothesis (Hf) (i),

f ⁢ ( z , x ) ≥ - ∥ f ⁢ ( ⋅ , 1 ) ∥ ∞ ⁢ x p - 1   for a.a. ⁢ z ∈ Ω ⁢ and all ⁢ x ∈ ( 0 , 1 ] .

Therefore, f⁢(z,0)≥0 for a.a. z∈Ω, and so

Using (3.9) in (3), we obtain

If {un+}n≥1⊆Lp⁢(Ω) is bounded, then since yn+=un+/∥un∥p for all n∈ℕ, from (3.5) we infer that yn+→0 in Lp⁢(Ω), hence y≤0. Hypothesis (Hf) (i) implies that

hence

∫ Ω F ⁢ ( z , u n + ) ∥ u n ∥ p p ⁢ 𝑑 z ≤ c 10 ∥ u n ∥ p p + c 10 ⁢ ∥ y n + ∥ p p   for all ⁢ n ∈ ℕ ,

and by (3.5),

So, if in (3.10) we pass to the limit as n→∞ and use (3.7) and (3.12), then

c 1 p ⁢ ( p - 1 ) ⁢ ( ∥ D ⁢ y ∥ p p + ∥ y ∥ p p ) ≤ 0 ,

and so y=0, in contradiction with (3.7).

If {un+}n≥1⊆Lp⁢(Ω) is unbounded, then we may assume that ∥un+∥p→+∞, and from (3.2) and (3.9) we have

Note that

From (3.7) we have that yn+→y+ in Lp⁢(Ω), hence

y n + ⁢ ( z ) → y + ⁢ ( z )   for a.a. ⁢ z ∈ Ω ,

at least for a subsequence. Then

and

Let z∈{μ>-∞}∖E with |E|N=0 (by |⋅|N we denote the Lebesgue measure on ℝN) be such that (see hypothesis (Hf) (iii))

f ⁢ ( z , x ) x p - 1 → μ ⁢ ( z )   as ⁢ x → + ∞ .

Given ϵ>0, we can find M2=M2⁢(z)>0 such that

f ⁢ ( z , x ) ≤ ( μ ⁢ ( z ) + ϵ ) ⁢ x p - 1   for all ⁢ x ≥ M 2 ,

hence

F ⁢ ( z , x ) ≤ 1 p ⁢ ( μ ⁢ ( z ) + ϵ ) ⁢ x p   for all ⁢ x ≥ M 2 ,

so that

lim sup x → + ∞ ⁡ F ⁢ ( z , x ) x p ≤ 1 p ⁢ ( μ ⁢ ( z ) + ϵ ) .

Since ϵ>0 is arbitrary, we let ϵ↓0 to conclude that

lim sup x → + ∞ ⁡ F ⁢ ( z , x ) x p ≤ 1 p ⁢ μ ⁢ ( z ) .

Next suppose z∈{μ=-∞}∖E (recall that |E|N=0). We have

f ⁢ ( z , x ) x p - 1 → - ∞   as ⁢ x → + ∞ .

Then, given S>0, we can find M3=M3⁢(S)>0 such that

f ⁢ ( z , x ) ≤ - S ⁢ x p - 1   for all ⁢ x ≥ M 3 ,

thus

F ⁢ ( z , x ) x p ≤ - S p   for all ⁢ x ≥ M 3 ,

and so

lim sup x → + ∞ ⁡ F ⁢ ( z , x ) x p ≤ - S p .

Since S>0 is arbitrary, we let S→+∞ to conclude that

lim x → + ∞ ⁡ F ⁢ ( z , x ) x p = - ∞ = μ ⁢ ( z ) .

From the discussion above, we infer that

Using (3.15), (3.16), (3.17) and Fatou’s lemma, we have

Also, we have

| ∫ { y + = 0 } F ⁢ ( z , u n + ) ∥ u n + ∥ p p d z | ≤ c 10 ∫ { y + = 0 } ( 1 ∥ u n + ∥ p p + ( y n + ) p ) d z   for all n ∈ ℕ

(see (3.11)), thus

We return to (3.14), pass to the limit as n→∞ and use (3.18) and (3.19). Then

In (3.13), if we pass to the limit as n→∞ and use (3.7) and (3.20), then we obtain

c 1 p ⁢ ( p - 1 ) [ ∥ D y + ∥ p p + ∫ ∂ ⁡ Ω β ^ ( z ) ( y + ) p d σ ] ≤ 1 p ∫ { y + ≠ 0 } μ ( y + ) p d z .

Hence,

If y+=0, then from (3.7), (3.10) and (3.19), we have ∥D⁢y∥pp+∥y∥pp≤0, and thus y=0, in contradiction with (3.7). Therefore, y+≢0, and so from (3.21) we have (see the proof of Proposition 2.4) λ^1⁢(-μ^,β^,p)≤0, which contradicts hypothesis (Hf) (iii). This proves the coercivity of φ. ∎