Remarks on nonlinear equations with measures

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Gw$.

In addition we define a notion of solution in the case that µ is a positive Borel measure which may explode on a compact subset of the domain and discuss the question of existence and uniqueness in this case. We always assume that g ∈ C(R) is a monotone increasing function such that g(0) = 0.
To simplify the presentation we also assume that g is odd.
A function u ∈ L 1 (Ω) is a weak solution of the boundary value problem (1.1), µ ∈ M ρ , if u ∈ L g ρ (Ω), i.e. if u and g • u are in L 1 loc (Ω) and (1.2) holds for every φ ∈ C 2 c (Ω). Brezis and Strauss [6] proved that, if µ is an L 1 function the problem possesses a unique solution. This result does not extend to arbitrary measures in M ρ (Ω).
Denote by M g ρ the set of measures µ ∈ M ρ for which (1.1) is solvable. A measure in M g ρ is called a g-good measure. It is known that, if a solution exists then it is unique.
We say that g is subcritical if M g ρ = M ρ . Benilan and Brezis, [5] and [4] proved that the following condition is sufficient for g to be subcritical: In the case that g is a power non-linearity, i.e., g = g q where g q (t) = |t| q sign t in R, q > 1, this condition means that q < q c := N/(N − 2). Benilan and Brezis also proved that, if g = g q and q ≥ q c , problem (1.1) has no solution when µ is a Dirac measure.
Later Baras and Pierre [3] gave a complete characterization of M g ρ in the case that g = g q with q ≥ q c . They proved that a finite measure µ is g q -good if and only if |µ| does not charge sets ofC 2,q ′ capacity zero, q ′ = q/(q − 1). HereC α,p denotes Bessel capacity with the indicated indices.
In the present paper we extend the result of Baras and Pierre to a large class of non-linearities and also discuss the notion of solution in the case that µ is a positive measure which explodes on a compact subset of Ω.

Statement of results
Denote by H the set of even functions h such that For h ∈ H denote by L h (Ω) the corresponding Orlicz space in a domain Ω ⊂ R N : Further denote by h * the conjugate of h. Since, by assumption, h is strictly convex, h ′ is strictly increasing so that, Let G be the Green kernel for −∆ in Ω and denote For every h ∈ H, the capacity C 2,h in Ω is defined as follows. For every compact set E ⊂ Ω put: For an arbitrary set A ⊂ Ω put This definition is compatible with (2.2) : when E is compact the value of C 2,h (E) given by the above formula coincides with the value given by (2.2), (see [2]).
We say that h satisfies the ∆ 2 condition if there exists C > 0 such that If h ∈ H satisfies this condition then, L h is separable (see [8]) and the capacity C 2,h has the following additional properties (see [2]).
Let Ω be a bounded domain in R N . For every A ⊂ Ω, and for every increasing sequence of sets {A n } Furthermore, for every A ⊂ Ω (2.5) If h ∈ H and both h and h * satisfy the ∆ 2 condition then L h is reflexive [8].
Finally we denote by G the space of odd functions in C(R) such that h := |g| ∈ H and by G 2 the set of functions g ∈ G such that h and h * satisfy the ∆ 2 condition. For g ∈ G put L g := L |g| , C 2,g := C 2,|h| , g * (t) = |g| * (t)sign t ∀t ∈ R.
In the sequel we assume that Ω is a bounded domain of class C 2 . The first theorem provides a necessary and sufficient condition for the existence of a solution of (1.1) in the spirit of [3].
Theorem 2.1. Let g ∈ G 2 and let µ be a measure in M ρ (Ω). Then problem (1.1) possesses a solution if and only if µ vanishes on every compact set E ⊂ Ω such that C 2,g * (E) = 0. This condition will be indicated by the notation µ ≺ C 2,g * .
Next we consider problem (1.1) when µ is a positive Borel measure which may explode on a compact set F ⊂ Ω. In this part of the paper we assume that g ∈ G 2 and that g satisfies the Keller -Osserman condition [9] and [12]. This condition ensures that the set of solutions of in Ω is uniformly bounded in compact subsets of Ω. Therefore, if E ⊂ Ω and E is compact then there exists a maximal solution of This solution will be denoted by U E .
Notation. Consider the family of positive Borel measures µ in Ω such that: (2) µ(A) = ∞ for every non-empty Borel set A ⊂ F . The set F will be called the singular set of µ. The family of measures µ of this type will be denoted by B ∞ (Ω).

Definition 2.2. Assume that g ∈ G and that g satisfies the Keller -Os
Assume that g ∈ G 2 and that g satisfies the Keller -Osserman condition. Let µ ∈ B ∞ with singular set F . Then:  .7)) then the generalized solution is unique.

) has a generalized solution if and only if µ vanishes on every compact set
(ii) If g satisfies the subcriticality condition (1.4) then problem (1.1) possesses a unique generalized solution for every µ ∈ B ∞ .

Proof of Theorem 2.1
The proof is based on several lemmas. We assume throughout that the conditions of the theorem are satisfied.
Denote by L 1 ρ (Ω) the Lebesgue space with weight ρ and by L g ρ (Ω) the Orlicz space with weight ρ.
Further denote by W k L g (Ω), k ∈ N, the Orlicz-Sobolev space consisting of functions v ∈ L g (Ω) such that D α v ∈ L g (Ω) for |α| ≤ k.
Under our assumptions the set of bounded functions in L g is dense in this space (see [8]). Consequently, by [7], C ∞ (Ω) is dense in W k L g (Ω). As a consequence of the reflexivity of L g the space and the norm in W −k L g * is defined as the norm of the dual space of W k 0 L g . The spaces W k L g ρ and W −k L g * ρ are defined in the same way. Lemma 3.1. If µ ∈ M ρ (Ω) is a g-good measure then (1.1) has a unique solution, which we denote by v µ . The solution satisfies the inequality where C is a constant depending only on g and Ω.
Proof. Let {Ω n } be a C 2 uniform exhaustion of Ω. Then G µ ∈ L g (Ω n ) is a positive supersolution of problem (1.1) in Ω n . Therefore -as the zero function is a subsolution -there exists a solution, say u n , of (1.1) in Ω n and, by Lemma 3.1, where ρ n (x) = dist (x, ∂Ω n ) and C is a constant depending only on g and the C 2 character of Ω n . Since Ω n } is uniformly C 2 , the constant may be chosen to be independent of n. Moreover {u n } is increasing. Therefore u = lim u n ∈ L 1 (Ω) ∩ L g ρ (Ω) is the solution of (1.1).
Proof. (a) Assuming that |µ| is g -good, let v be the solution of (1.1) with µ replaced by |µ|. Then v is a supersolution and −v is a subsolution of (1.1). Therefore (1.1) has a solution.
(b) If T = ∆h then, for every φ ∈ C ∞ c (Ω), T defines a continuous linear functional on this space; consequently T ∈ W −2 L g (Ω).
On the other hand if T ∈ W −2 L g (Ω), put S(∆φ) := T (φ) ∀φ ∈ W 2 0 L g * . Note that for φ in this space we have φ = G −∆φ . Therefore S is well defined on the subspace of L g * given by {∆φ : φ ∈ W 2 0 L g * }. Therefore there exists h ∈ L g (Ω) such that It follows that T = ∆h.
Proof. Assume that µ is g-good and let u be the solution of (1.1). Then is continuous on C 2 0 (Ω) with respect to the norm of W 2 L g * ρ (Ω). Therefore, the functional can be extended to a continuous linear functional on W 2 L g * (Ω ′ ) for every Ω ′ ⋐ Ω. Thus µ 0 ∈ W −2 L g loc (Ω) ∩ M ρ (Ω). (ii) In view of (2.3) it is sufficient to prove that µ vanishes on compact sets E such that C 2,g * (E) = 0. Assertion. If ν ∈ W −2 L g loc (Ω) then ν(E) = 0 for every compact set E such that C 2,g * (E) = 0. This assertion and part (i) imply part (ii). Suppose that there exists a set E such that C 2,g * (E) = 0 and ν(E) = 0. Then there exists a compact subset of E on which ν has constant sign. Therefore we may assume that E is compact and that ν is positive on E. We may assume that ν ∈ W −2 L g (Ω); otherwise we replace Ω by a C 2 domain Ω ′ ⋐ Ω.
Let {V n } be a sequence of open neighborhoods of E such thatV n+1 ⊂ V n and V n ↓ E. Then there exists a sequence {ϕ n } in C ∞ c (Ω) such that 0 ≤ ϕ n ≤ 1, ϕ n = 1 in V n+1 , supp ϕ n ⊂ V n and ϕ n g * → 0. This is proved in the same way as in the case of Bessel capacities. We use (2.5) and the fact that C ∞ (Ω) is dense in W 2 L g ρ (Ω) [7]). Furthermore we use an extension of the lemma on smooth truncation [1, Theorem 3.3.3] to Sobolev-Orlicz spaces with an integral number of derivatives. The extension is straightforward. Hence, On the other hand, This contradiction proves the assertion. Proof. Since µ is the limit of an increasing sequence of measures in M(Ω) it is sufficient to prove the lemma for µ ∈ M(Ω). Let ϕ ∈ W 2 0 L g * (Ω) and denoteφ = G ∆ϕ . Thenφ is equivalent to ϕ.
Therefore the functional p : is well defined. The functional is sublinear, convex and l.s.c.: if ϕ n → ϕ in Furthermore, p(aϕ) = ap(ϕ) ∀a > 0. Therefore the result follows by an application of the Hahn-Banach theorem, in the same way as in [3,Lemma 4.2].
Proof of Theorem 2.1. By Lemma 3.4 the condition µ ≺ C 2,g * is necessary for the existence of a solution. We show that the condition is sufficient.
If µ ≺ C 2,g * then |µ| ≺ C 2,g * . By Lemma 3.3 if |µ| is g-good then µ is g-good. Therefore it remains to prove the sufficiency of the condition for positive µ. In this case, by Lemma 3.5, there exists an increasing sequence of positive measures {µ n } ⊂ W −2 L g (Ω) such that µ n ↑ µ. By Lemma 3.3 the measures µ n are g-good. Denote by u n the solution of (1.1) with µ replaced by µ n . By Lemma 3.1, u n ≥ 0, {u n } increases and {u n } is bounded in L 1 (Ω) ∩ L g ρ (Ω). Therefore u = lim u n ∈ L 1 (Ω) ∩ L g ρ (Ω) and u n → u in this space. Consequently u is the solution of (1.1). − ∆u + g • u = µ in Ω n := Ω \Ō n such that u = 0 on the boundary. By a standard argument, it follows that, under this condition: for every f ∈ L 1 (∂Ω ∪ ∂O n ), (4.1) has a solution such that u = f on the boundary. As g satisfies the Keller -Osserman condition, it also follows that (4.1) has a solution u n such that u n = 0 on ∂Ω and u n = ∞ on ∂O n . Denote by v n the solution of (4.1) vanishing on ∂Ω ∪ ∂O n and put v 0,µ = lim v n ,ū µ = lim u n . Then v 0,µ is the smallest positive solution of (4.1) vanishing on ∂Ω whileū µ is the largest such solution. In particularū µ ≥ v ν for every ν ∈ M g ρ such that supp ν ⊂ F . Thusū µ is the largest generalized solution of (1.1).

Proof of
Next we construct the minimal generalized solution of (1.1). The function u 0,µ + V F is a supersolution and max(u 0,µ , V F ) is a subsolution of (4.1), both vanishing on the boundary. Let w n denote the solution of (4.1) such that w n = 0 on ∂Ω and w n = max(u 0,µ , V F ) on ∂O n . Then w n+1 ≤ w n ≤ u 0,µ + V F and consequently, w = lim w n is the smallest solution of (4.1) such that It follows that w is a generalized solution of (1.1). Since any such solution dominates max(u 0,µ , V F ) it follows that w is the smallest generalized solution of the problem. It is easy to see that w = u µ as given by (2.7).
Then {u n } increases and u = lim u n . Similarly, ifū n is the solution of the problem then {ū n } increases and, in view of (4.2),ū = limū n . Therefore, if V F = U F thenū µ = u µ .
(ii) We assume that in addition to the other conditions of the theorem, g satisfies the subcriticality condition. In this case, for every point z ∈ Ω and k ∈ R, there exists a solution u k,z of the problem (4.3) − ∆u + g • u = kδ z in Ω, u = 0 on ∂Ω.
Put w z = lim k→∞ u k,z . By definition w z = V {z} . We also have w z = U {z} . This follows from the fact that g satisfies the Keller -Osserman condition. This condition implies that there exists a decreasing function ψ ∈ C(0, ∞) such that ψ(t) → ∞ as t → 0 and every positive solution u of (4.3) satisfies .The constant C 1 depends only on g, N . Because of the boundary condition the constant C 2 depends on z. However for z in a compact subset of Ω one can choose C 2 to be independent of z.
Let x ∈ F ′ \ F and let z be a point in F such that |x − z| = dist (x, F ). Then there exists a positive constant C(F ) such that It follows that there exists a constant c such that for every x ∈ F ′ . Since U F and V F vanish on ∂Ω it follows that (4.4) (with possibly a larger constant) remains valid in Ω \ F ′ . This is verified by a standard argument using Harnack's inequality and the fact that g satisfies the Keller -Osserman condition. Thus (4.4) is valid in Ω \ F . By an argument similar to the one introduced in [10, Theorem 5.4], this inequality implies that U F = V F . (iii) For the case considered here, it was proved in [11] that U F = V F . Therefore uniqueness follows from part (i).