A non-smooth Brezis-Oswald uniqueness result

Theorem 1.1

Let Ω ⊆ R N be open, connected and of finite measure. For some p > 1 , suppose that

1. H : R N → [ 0 , ∞ [ is strictly convex and positively p-homogeneous.

2. f : Ω × R → R is Caratheodory and fulfils

   sup x ∈ Ω , ∣ s ∣ ≤ t ∣ f ( x , s ) ∣ < ∞ ∀ t > 0 .

3. The map t ↦ f ( x , t ) / t p − 1 is non-increasing on ] 0 , ∞ [ .

let C + ( J ) be the set of its non-negative critical points v ∈ W 0 1 , p ( Ω ) \ { 0 } . Then,

1. All elements in C + ( J ) are multiple of each other, and each of them minimise J among non-negative functions in W 0 1 , p ( Ω ) .

2. If t ↦ f ( x , t ) / t p − 1 is strictly decreasing on t > 0 , C + ( J ) contains at most one element.

3. If f does not depend on x and C + ( J ) contains more than one element, each function in C + ( J ) minimises the Rayleigh quotient

   (1.4) ∫ Ω H ( D v ) d x ∫ Ω v p d x ,

   among all v ∈ W 0 1 , p ( Ω ) \ { 0 } , v ≥ 0 .

Remark 1.2

Let us make some comments on the features of the previous theorem.

1. No regularity assumptions are made on the boundary of the open set Ω . The finite measure assumption could be further weakened requiring additional conditions on the behaviour of f ( x , t ) near 0, but we chose not to analyse this case in order to have a cleaner statement. Note that in this setting, there is no hope to have a Hopf maximum principle available.

2. The stated assumption on H enforces the two-sided bound

   c ∣ z ∣ p ≤ H ( z ) ≤ d ∣ z ∣ p , ∀ z ∈ R N

   for some 0 < c ≤ d < ∞ . However, since H is not supposed to be C 1 , the first term of J in (1.3) is only locally Lipschitz, with multi valued differential. Note also that H is not supposed to be even. The strict convexity of H is also not needed in Assertion (2), as well as in the second statement of Assertion (1).

3. Conditions (ii) and (iii) on the reaction f are equivalent to

   f ( ⋅ , t ) ∈ L ∞ ( Ω ) , f ( x , t ) ≤ C ( 1 + t p − 1 ) , t ↦ f ( x , t ) t p − 1 non-increasing

   for t > 0 , which are more commonly found in the Brezis-Oswald type results.

4. Under the sole Assumptions (ii) and (iii), even for smooth H , J is only l.s.c., and the notion of critical point has to be dealt with care. We use here the notion of energy critical point from [6], meaning those u ∈ W 0 1 , p ( Ω ) , which, roughly speaking, minimise the corresponding semi-linearised convex functional

   v ↦ ∫ Ω H ( D v ) − f ( x , u ) v d x .

   For the sake of this introductory discussion, let us note that the usually encountered notions of critical point, when available, yield an energy critical point. For more details, we refer to Definition 2.4 and Remark 2.5.

5. Note that we do not give any condition ensuring that C + ( J ) is actually not empty. One of the main features of the original Brezis-Oswald result (and of the related literature on the subject) was actually to provide necessary and sufficient condition for J to have non-negative, non-trivial critical points, under a strict monotonicity assumption on t ↦ f ( x , t ) / t p − 1 . We will not investigate this theme here.

6. Under Assumption (iii) alone, the structure of C + ( J ) is described in an optimal way by Conclusion (1) of the theorem. Indeed, when H ( z ) = ∣ z ∣ 2 , given a first positive eigenfunction u of the Dirichlet Laplacian in the unit ball B 1 and an interval [ a , b ] ⊆ [ 0 , + ∞ [ , any reaction f obeying f ( x , t ) = λ 1 ( B 1 ) t if a u ( x ) ≤ t ≤ b u ( x ) and f ( x , t ) / t strictly decreasing otherwise has C + ( J ) = { t u : t ∈ [ a , b ] } .

7. A similar statement holds true for non-positive critical points of J , with obvious modifications to (ii) and (iii). Note, however, that since we are not assuming that H is even, the infimum of the ratios (1.4) may change under the corresponding constraint v ≤ 0 . This feature of possibly non-even, positively p -homogeneous energies H and their corresponding first Dirichlet eigenvalues is discussed in Section 4.

Theorem 1.3

Let Ω ⊆ R N have finite measure and, for given p ≥ 1 , suppose that

1. H : R N → [ 0 , ∞ [ is convex and fulfils H ( z ) ≥ c ∣ z ∣ p for some c > 0 and all z ∈ R N .

2. f : Ω × R → R is Caratheodory and fulfils

   sup x ∈ Ω , ∣ s ∣ ≤ t ∣ f ( x , s ) ∣ < ∞ ∀ t > 0 .

3. There exist μ , θ ≥ 0 and 1 ≤ q ≤ p ∗ such that

   (1.5) sign ( t ) f ( x , t ) ≤ μ ( ∣ t ∣ + θ ) q f o r a . e . x ∈ Ω a n d a l l t ∈ R .

holds true for some C = C ( N , p , q ) which, for any N, is positive and continuous in its other arguments.

Remark 1.4

Regarding this second result, the following should be noted.

1. Not only H is only assumed to be locally Lipschitz, but no growth control from above is required in order to obtain boundedness of the critical points of J .

2. Assumption (iii) is understood to be empty in the case q = ∞ , which forces p > N thanks to q ≤ p ∗ . Clearly, in this case, boundedness is trivial thanks to Morrey embedding. Anyway, (1.6) only holds for q < p ∗ and therefore for finite q ’s.

3. Assumptions (ii) and (iii) are required in order for the functional J in (1.3) to be well defined (possibly assuming the value + ∞ ) on W 0 1 , p ( Ω ) . In any case, under these assumptions, J turns out to be only l.s.c. on W 0 1 , p ( Ω ) with the strong topology, and again, we understand by critical point an energy one (see Section 2.4).

4. Form (1.6) of the sup bound is optimal, as discussed in Remark 3.4. Moreover, it is invariant by scaling and scalar multiples, meaning the following. A function u is a critical point for J on W 0 1 , p ( Ω ) if and only if for any λ , r > 0 u λ , r ( x ) = λ u ( x / r ) is a critical point for a suitable functional J λ , r on W 0 1 , p ( r Ω ) (where r Ω = { r x : x ∈ Ω } ). More precisely, the functional J λ , r is of form (1.3) for a reaction f ˆ obeying (1.5) with θ ˆ = λ θ , μ ˆ = μ / λ q and H ˆ obeying H ˆ ( z ) ≥ c ˆ ∣ z ∣ p with c ˆ = c λ / r . The corresponding estimates (1.6) turn out to be independent of λ and r .

5. Regarding the continuity statement for the constant C in the last part of the theorem, it is to be understood in its domain p ≥ 1 , q < p ∗ . As is to be expected, the constant C blows up as q ↑ p ∗ .

Proposition 2.1

For 1 ≤ p , q < ∞ , let X p , q ( R N ) be the vector space of functions in L q ( R N ) with distributional derivative in L p ( R N ) , equipped with the norm ‖ u ‖ = ‖ D u ‖ p + ‖ u ‖ q . Then, C c ∞ ( R N ) is dense in X p , q ( R N ) .

Proof

The set C ∞ ( R N ) ∩ X p , q is dense in X p , q by known results (see [19, Theorem 1, Section 1.1.5]), so it suffices to show that any u ∈ C ∞ ( R N ) ∩ X p , q can be approximated by functions in C c ∞ ( R N ) . Fix a cutoff η ∈ C c ∞ ( B 2 ; [ 0 , 1 ] ) such that η ≡ 1 in B 1 and, for any u ∈ C ∞ ( R N ) ∩ X p , q , define for k ∈ N

so that u k ∈ C c ∞ ( R N ) , supp ( D η k ) ⊆ A k and ‖ D η k ‖ ∞ = ‖ D η ‖ ∞ / k . Clearly ‖ u − u k ‖ q ≤ ‖ u ‖ L q ( R N \ B k ) → 0 as k → ∞ , so we only have to estimate ‖ D ( u − u k ) ‖ p . It holds

and the first term on the right vanishes for k → ∞ since ∣ D u ∣ ∈ L p ( R N ) , so we are left to estimate the second term.

The case p < N is standard: by Chebyshev inequality, any u ∈ L q ( R N ) vanishes at infinity, hence, X p , q ⊆ D 1 , p ( R N ) ⊆ L p ∗ ( R N ) . Therefore, by Hölder inequality

and the last factor on the right vanishes as k → ∞ due to u ∈ L p ∗ ( R N ) ; hence, also the last term in (2.3) does so. If p ≥ N , we use the inequality

which is trivial by Hölder when q ≥ p and holds true thanks to the Gagliardo-Nirenberg inequality when q < p . Applying it to v ( x ) = u ( k x ) and changing variables, we obtain

As k → ∞ , the first term on the right vanishes again, as well as the second, since

Proposition 2.2

(Gagliardo-Nirenberg) Let N ≥ 2 and 1 ≤ p , q < ∞ . Then, for any u ∈ X p , q , it holds

where

Proof

By the previous proposition we can suppose that u ∈ C c ∞ ( R N ) . For p = 1 (2.4) reduces to (2.2), so we can suppose that p > 1 . Let

and apply (2.2) to v = ∣ u ∣ r and then Hölder inequality, to obtain

Proposition 2.3

Let H : R N → [ 0 , ∞ [ be convex and positively p-homogeneous for some p ≥ 1 . Then, for any w ∈ R N and ξ ∈ ∂ H ( w ) , it holds

Proof

Let ξ ∈ ∂ H ( z ) for some z ∈ R N . By p -homogeneity and convexity

for all t ≥ 1 , while the left-hand side coincides with the right-hand side when t = 1 . Therefore,

which gives (2.5).□