2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 183–202. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Nonlinear equations with natural growth terms and measure data ^{∗}

Alessio Porretta

Abstract We consider a class of nonlinear elliptic equations containing a p-Laplacian type operator, lower order terms having natural growth with respect to the gradient, and bounded measures as data. The model example is the equation

−Δ_{p}(u) + g(u)|∇u|^{p} = μ

in a bounded set Ω ⊂ ℝ^{N}, coupled with a Dirichlet boundary condition. We provide a review of the results recently obtained in the absorption case (when g(s)s ≥ 0) and prove a new existence result without any sign condition on g, assuming only that g ∈ L^{1}(R). This latter assumption is proved to be optimal for existence of solutions for any measure μ.

1 Introduction

In this work we focus our attention on nonlinear Dirichlet problems whose model is

\begin{matrix}−Δ_{p}(u) + g(u)|∇u|^{p} = μ in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (1.1)

where p > 1, g : ℝ → ℝ is a continuous function, and μ is a bounded Radon measure on Ω which is a bounded subset of ℝ^{N}.

Recently, many researchers have investigated the possibility to find solutions of (1.1) under the assumption that g(s)s ≥ 0, in which case the term g(u)|∇u|^{p} is said to be an absorption term. In this case a detailed picture of what happens is now available, according to the growth at infinity of g(s) and to whether the measure μ charges or not sets of zero p-capacity (the capacity defined in W_{0}^{1,p}(Ω)). In the next section, we try to give a quick review of these results and explain the main features of the problem in the absorption case, both for elliptic and for parabolic equations.

No results for general measures μ are known to our knowledge if the sign condition is not assumed to hold, possibly including the reaction case in which g(s)s ≤ 0. It is the purpose of the third section of this paper to give new results in this situation. Eventually, these new results seem to fit perfectly those proved in the absorption case, and we will prove (stated in more generality in Section 3) the following theorem, which extends that proved in [44] (under the same assumptions) for data μ ∈ L^{1}(Ω).

∗Mathematics Subject Classifications: 35J60, 35J65, 35R05.

Key words: Nonlinear elliptic equations, natural growth terms, measure data.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

Theorem 1.1 Let μ be a nonnegative bounded Radon measure on Ω. Assume that g ∈ L^{1}(ℝ). Then there exists a distributional solution u of (1.1).

Next we will give an example which somehow expresses that the assumption g ∈ L^{1}(ℝ) in Theorem 1.1 is optimal; if μ is the Dirac mass, we prove that no solution can be obtained by approximation. In particular, in the reaction case (g(s)s ≤ 0), if μ is approximated by a sequence of smooth functions, the sequence of approximating solutions converges to a solution of (1.1) if g ∈ L^{1}(ℝ), while it blows up everywhere in Ω if g ∉ L^{1}(ℝ). We recall that in [34] the absorption case g(s)s ≥ 0 had already been studied; in that situation if the Dirac mass is approximated by smooth functions, the approximated solutions still converge to a solution of the problem if g ∈ L^{1}(ℝ), while they converge to zero if g ∉ L^{1}(ℝ). Thus, even if for different reasons, in both cases the assumption g ∈ L^{1}(ℝ) turns out to be optimal.

2 The absorption case: a quick review

A wide literature has dealt with elliptic and parabolic equations with measure data in the last decades. In particular, the techniques of a priori estimates and compactness of approximating solutions, firstly introduced in [14], have been proved to work well enough for pseudomonotone operators of Leray-Lions type ([32]), providing several existence results in case of L^{1} data. The presence of absorbing lower order terms (i.e. satisfying a sign condition) often brings in this kind of problems new features; for instance, as in [18], [15], lower order terms may have a regularizing effect on solutions of problems with L^{1} data. The two main examples are the following problems:

\begin{matrix}−Δ_{p}u + |u|^{r−1}u = μ in Ω, \\ u = 0 on ∂Ω , \end{matrix}  (2.1)

and

\begin{matrix}−Δ_{p}u + u|∇u|^{p} = μ in Ω, \\ u = 0 on ∂Ω . \end{matrix}  (2.2)

If μ ∈ L^{1}(Ω), problem (2.1) has a solution in W_{0}^{1,q}(Ω) for any q < \frac{pr}{r+1}, while problem (2.2) has a finite energy solution u, which belongs to W_{0}^{1,p}(Ω). In general, if the lower order term is absorbing, one can prove the existence of a solution with L^{1}(Ω) data; for instance, the problem:

\begin{matrix}−Δ_{p}(u) + H(x, u, ∇u) = f in Ω, \\ u = 0 on ∂Ω , \end{matrix}  (2.3)

with ξ ↦ H(x, s, ξ) growing at most like |ξ|^{p} (the so-called natural growth), always admits a solution if f ∈ L^{1}(Ω) (see [42]). In fact, dealing with the limit growth for H(x, s, ξ) is not that easy and requires the strong compactness of truncations in the energy space; on the other hand, these truncation methods can be adapted to several different contexts if still dealing with L^{1}(Ω) data, as obstacle problems or more general operators (see [9], [10], [28]).

When trying to extend the previous results to measure data, it turns out that precisely the regularizing effect mentioned above may be responsible for nonexistence of solutions. Actually, this fact was first observed in the pioneering works of H. Brezis ([20, 21]) and in a whole series of papers (see [4, 25, 30, 47, 45, 46] and the references therein) concerning problem (2.1) in the linear case p = 2. More recently, the nonlinear case p ≠ 2 has been dealt with in [37, 38, 11]. Summing up these results, it is proved that problem (2.1) has a solution for every given bounded measure μ only if r < \frac{N(p−1)}{N−p}, while if r ≥ \frac{N(p−1)}{N−p} then no

solution exists if μ charges sets of zero q-capacity with \frac{q(p−1)}{q−p} < r (a necessary and sufficient condition in the linear case p = 2 is given in [30]). As an example, if μ is the Dirac mass, then a solution of (2.1) exists if and only if r < \frac{N(p−1)}{N−p}.

However, the statement of nonexistence of solutions needs to be suitably precised; how shall we express such a failure of existence? Three different ways have been suggested so far in previous works: firstly, nonexistence of solutions for a general problem as (2.3) may be deduced from removable singularity type results. This is a classical approach, and mostly used for linear operators; a set K is removable if any solution of

−Δ_{p}(u) + H(x, u, ∇u) = f in Ω \ K,
u ∈ W^{1,p}(Ω \ K) ,

can be proved to be a solution in the whole of Ω. If K is removable, then we cannot have a solution of the equation with data concentrated on K.

Alternatively, one studies the limit of approximating equations:

\begin{matrix}−Δ_{p}(u_{n}) + H(x, u_{n}, ∇u_{n}) = f_{n} in Ω , \\ u_{n} = 0 on ∂Ω , \end{matrix}  (2.4)

if f_{n} converges to a measure μ in the so-called narrow topology, which means

∫_{Ω} f_{n}φdx →^{n→+∞} ∫_{Ω} φdμ ,  (2.5)

for any function φ bounded and continuous on Ω. This is the most natural way to approximate a bounded Radon measure, so that, if a solution exists, we expect that we can prove the convergence of u_{n} towards a solution u of

\begin{matrix}−Δ_{p}(u) + H(x, u, ∇u) = μ in Ω \\ u = 0 on ∂Ω , \end{matrix}  (2.6)

like it happens if f_{n} strongly converges in L^{1}(Ω). Thus studying the limit of u_{n} is a constructive way to see whether and why existence may fail; thanks to (2.5) f_{n} is bounded in L^{1}(Ω) so that “a priori” estimates are available, and usually compactness of u_{n} can also be proved. The main task is to understand which is the limit of u_{n} and what equation it satisfies.

Finally, a third approach is in some sense the combination of the previous two. One studies (2.4) assuming only that f_{n} converges in L_{loc}^{1}(Ω \ K) towards a function f, where K is a compact subset of Ω. Here no assumption is made on the behaviour of f_{n} on K, so that no estimates on K are obtained for u_{n}. If one proves that u_{n} still converges to a solution u of (2.3), this means that perturbations of f whatever singular, but localized on K, are not seen by the equation. As in the viewpoint of removable singularity, no solution can be expected for data concentrated on K. These three possible approaches were all investigated as far as problem (2.1) is concerned in some of the papers mentioned above.

In a series of recent works, these questions have been studied for problems with gradient dependent lower order terms. A particular case is given when the lower order term has natural growth. When trying to find solutions for the model equation

\begin{matrix}−Δ_{p}u + g(u)|∇u|^{p} = μ \\ u = 0, \end{matrix}  (2.7)

the growth at infinity of g(s) and the regular or singular nature of μ play a crucial role. Removable singularity results were proved by H. Brezis and L. Nirenberg in [24] for p = 2, showing that if sg(s) ≥ \frac{γ}{s^{2}} with γ > 1, then any compact set of zero capacity (the standard Newtonnian capacity) is removable. In [17], [40], [34], the behaviour of sequences of approximating solutions was studied if μ is approximated in the narrow topology, say by a standard convolution. It is proved that if g ∈ L^{1}(ℝ), then there exists a solution u of (2.7) for any measure μ, while if g ∉ L^{1}(ℝ) approximating solutions converge to a solution u of the same problem but with datum μ_{0}, the absolutely continuous part of μ with respect to p-capacity. Here p-capacity denotes the capacity defined in W_{0}^{1,p}(Ω) and we recall (see [29]) that any measure μ admits a unique decomposition as μ = μ_{0} + λ, where λ is concentrated on a set of zero p-capacity and μ_{0}(E) = 0 for any set E of zero p-capacity. In other words, in the approximation method, one looses the singular part of the measure which is concentrated on sets of zero p-capacity; if μ does not charge sets of zero p-capacity then existence is proved for any function g(s).

Removability properties in the stability approach are investigated in [36], where the approximating equations of (2.7) are considered with data f_{n} only converging to a function f in L_{loc}^{1}(Ω \ K), where K has p-capacity zero. It is proved that, setting G(s) = ∫_{0}^{s} g(t)dt, if, roughly speaking, exp(−G(s)/(p − 1)) ∈ L^{1}(ℝ) then u_{n} still converges to a solution with datum f; thus, whatever singular perturbations, provided they are localized on sets of zero p-capacity, are not seen by the equation. This result somehow includes the removable singularity point of view, and extends the result in [24] since the assumption that exp(−G(s)/(p − 1)) ∈ L^{1}(ℝ), in the case p = 2, is weaker than assuming that sg(s) ≥ \frac{γ}{s^{2}} with γ > 1.

This kind of phenomena due to absorption terms has been investigated for parabolic equations as well. As it happens for the stationary case, the semilinear evolution problem

u_{t} − Δu + |u|^{r−1}u = μ in Q := Ω × (0, T)
u = 0 on Σ := ∂Ω × (0, T)
u(0) = u_{0} in Ω,

(2.8)

does not always have a solution for any measure μ on Q and any measure initial datum u_{0}. In [22], the authors study the problem with μ = 0 concentrating the attention on the initial measure u_{0}. They point out that, ir r is large enough, nonexistence phenomena may occurr, and can appear as initial layer phenomena. In fact, a singular measure as initial condition may be lost while approximating the problem with smooth approximating problems. Subsequently, in [5], necessary and sufficient conditions are given on the measures μ and u_{0} in order to have a solution of (2.8); as expected, these conditions involve some notions of space-time dependent capacity. Further results on nonlinear analogue of (2.8) are proved in [3], [33], [12] (see also the references in these papers).

In view of the results mentioned above for elliptic equations, recent study has been devoted to evolution problems as the following:

u_{t} − Δ_{p}u + g(u)|∇u|^{p} = 0 in Q := Ω × (0, T)
u = 0 on Σ := ∂Ω × (0, T)
u(0) = u_{0} in Ω,

(2.9)

in case u_{0} is a bounded measure. The existence of a solution in case u_{0} ∈ L^{1}(Ω) is proved in [41]. The possibility to extend this result to a general measure initial datum is studied in [13]. Again, under the assumption that g ∈ L^{1}(ℝ), it is proved the existence of a solution for any measure u_{0}. On the other hand, if g ∉ L^{1}(ℝ), then initial layer phenomena occur; in particular, if u_{0n} is a convolution approximation of the measure u_{0}, the sequence of approximating solutions u_{n} of the same problem, with initial datum u_{0n}, converges to a solution u of the problem having, as initial value, the absolutely continuous part of u_{0} with respect to Lebesgue measure. Sharp removable singularity type results, which in a stronger way express the nonexistence of solution, still depend on the growth at infinity of g(s) and are obtained in [43].

Eventually, one obtains for the evolution problem (2.9) the same type of results obtained for the elliptic problem (2.7) replacing the role of μ with u_{0} and the p-capacity (capacity in W_{0}^{1,p}(Ω)) with the Lebesgue measure in Ω. Are then these results consistent? The answer has to be found in the study of the notion of capacity for parabolic equations. A functional type presentation and construction of the parabolic p-capacity (capacity defined in the space W = {u ∈ L^{p}(0, T; W_{0}^{1,p}(Ω)) , u_{t} ∈ L^{p′}(0, T; W^{−1,p′}(Ω))}) is given in [39] for p = 2 and in [27] for p ≠ 2. In this last paper, it is proved that given B ⊂ Ω, the set {t = 0} × B has zero parabolic capacity in (0, T) × Ω if and only if B has zero Lebesgue measure. Thus, if one looks at singularities at initial time as singularities on Q concentrated at t = 0, the results obtained on (2.9) reflect perfectly those on (2.7). Moreover, it becomes clear that in order to deal with problem (2.9) with interior space-time dependent measures as data, one has to follow the outlines of the stationary case and use a decomposition theorem for measures with respect to parabolic p-capacity. This latter result, which extends the stationary one given in [16], is proved in [27] and states that any measure μ on (0, T) × Ω which does not charge sets of zero parabolic p-capacity admits the decomposition (as a distribution)

μ = f + g_{1} − (g_{2})_{t}

with f ∈ L^{1}(Q), g_{1} ∈ L^{p′}(0, T; W^{−1,p′}(Ω)) and g_{2} ∈ L^{p}(0, T; W_{0}^{1,p}(Ω)).

Finally, let us mention that, in the linear case (p = 2), other existence and nonexistence results with gradient dependent lower order terms (absorbing or repulsive) and measure data are obtained in [1, 2, 6, 7] (see also the references cited therein). We point out that the techniques used in these papers are mainly based on a linear operator and on the concept of distributional solution (with two integration by parts), or on semigroup theory and the concept of integral solution. These approaches allow to have sharper nonexistence results especially for the case of subcritical growth, on the other hand their study is mostly restricted to the case g(u) ≡ 1.

2.1 Natural growth reaction terms and measure data

As explained in the previous section, if the term H(x, u, ∇u) is an absorption term and has natural growth, the borderline case which allows to have solutions of (2.6) for all measures μ is the case in which

|H(x, u, ∇u)| ≤ g(u)|∇u|^{p} , with g ∈ L^{1}(ℝ).

Our aim is now to show that, somehow surprisingly, the same assumption is necessary and sufficient to have solutions for any measure even in the reaction case, that is without assuming any sign condition on H(x, s, ξ). In particular, if we aim to have solutions of (2.7) for any given measure data, there is no difference between the reaction and the absorption case.

Heuristically, this feature can be easily explained. In fact, the model equation

−Δu = g(u)|∇u|^{2} + μ ,  (2.10)

can be transformed, through a change of unknown, into the equation

−Δv = exp(G(u))μ ,  (2.11)

with v = ∫_{0}^{u} exp(G(s))ds and G(s) = ∫_{0}^{s} g(r)dr.

In [34] we proved that equation (2.11) has a solution if exp(G(u)) has a finite limit at infinity, which is the case whenever g ∈ L^{1}(ℝ), so that in this case (2.10) is also expected to have a solution. On the other hand, if g ∉ L^{1}(ℝ), then the right hand side of (2.11) can be hardly handled since exp(G(u)) is not bounded.

We are going to provide an example where μ is the Dirac mass and no solution of (2.11) can be found by approximation, precisely proving that approximated solutions of (2.10) in this case blow up completely (i.e. at every point of Ω).

We will prove our result in a more general situation. Assume that a(x, s, ξ) and H(x, s, ξ) are Carathéodory functions satisfying, for almost every x ∈ Ω, for every s ∈ ℝ, ξ, η ∈ ℝ^{N} (ξ ≠ η):

a(x, s, ξ) · ξ ≥ α|ξ|^{p} , α > 0 , p > 1 ,  (2.12)
|a(x, s, ξ)| ≤ β(k(x) + |s|^{p−1} + |ξ|^{p−1}) k(x) ∈ L^{p′}(Ω), β > 0 ,  (2.13)
(a(x, s, ξ) − a(x, s, η)) · (ξ − η) > 0 ,  (2.14)

and

|H(x, s, ξ)| ≤ γ(x) + g(s)|ξ|^{p} , γ(x) ∈ L^{1}(Ω)
and g : R → R^{+} continuous , g ≥ 0 , g ∈ L^{1}(ℝ) .

(2.15)

In the following we denote by cap_{p}(B) the p-capacity of a borelian set B ⊂ Ω, where the p-capacity is the standard notion of capacity defined in the Sobolev space W_{0}^{1,p}(Ω). Let us recall (see [29]) that any bounded Radon measure μ has a unique decomposition as

μ = μ_{0} + λ ,  (2.16)

where μ_{0}, λ are bounded measures such that μ_{0} does not charge sets of zero p-capacity (i.e. μ_{0}(B) = 0 for every B with cap_{p}(B) = 0) and λ is concentrated on a set E ⊂ Ω such that cap_{p}(E) = 0. Moreover, if μ is nonnegative, then both μ_{0} and λ are nonnegative. For a presentation of the basic notions concerning measures and capacity the reader may refer to [31], [26]. We also have, from [16], that μ_{0} furtherly admits a decomposition (in distributional sense) as

μ_{0} = f − div(F) , f ∈ L^{1}(Ω), F ∈ L^{p′}(Ω)^{N}.  (2.17)

Hereafter, let μ be a bounded nonnegative Radon measure on Ω. Referring to the previous decomposition of μ and μ_{0} in (2.16), (2.17), there exists a sequence μ_{n} of bounded functions such that

μ_{n} = μ_{0n} + λ_{n} , μ_{0n} ≥ 0, λ_{n} ≥ 0,
μ_{0n} = f_{n} − div(F_{n}) , f_{n} ∈ L^{∞}(Ω) , F_{n} ∈ L^{∞}(Ω)^{N} ,
f_{n} → f strongly in L^{1}(Ω),
F_{n} → F strongly in L^{p′}(Ω)^{N},
∫_{Ω} φλ_{n}dx → ∫_{Ω} φdλ ∀φ ∈ C_{b}(Ω) ,

(2.18)

where C_{b}(Ω) denotes the space of bounded continuous functions in Ω. Such a sequence μ_{n} can be constructed using convolution and a suitable compactly supported approximation of μ.

For fixed n ∈ ℕ, since μ_{n} ∈ L^{∞}(Ω), under the previous assumptions it is proved in [19] that there exists a weak solution u_{n} ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) of the problem:

− div(a(x, u_{n}, ∇u_{n})) = H(x, u_{n}, ∇u_{n}) + μ_{n} in Ω ,

u_{n} = 0 on ∂Ω . (2.19)

Our main result is the following.

Theorem 2.1 Let a(x, s, ξ) and H(x, s, ξ) satisfy assumptions (2.12)–(2.15). Let μ be a nonnegative bounded Radon measure on Ω. Then there exists a solution u of the problem

− div(a(x, u, ∇u)) = H(x, u, ∇u) + μ in Ω ,

u = 0 on ∂Ω . (2.20)

Proof. We essentially follow the method used in [44], which consists in multiplying the equation (2.19) by exp(G(u_{n})) or by exp(−G(u_{n})), where G(s) = ∫_{0}^{s} g(t)/αdt (the function g appears in (2.15)). In other words this replaces the idea of the change of unknown which transforms the model problem (2.10) into (2.11). After this multiplication, we will apply the techniques fully developed in [40], [34] to obtain the strong convergence of truncations.

In the following, we omit for shortness the dependence on x in the integrals, and we denote by c any positive constant independent on n. Let φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω); choosing exp(G(u_{n}))φ as test function in (2.19) we have

∫Ω exp(G(u_{n}))a(u_{n}, ∇u_{n})∇φ + ∫_{Ω} \frac{g(u_{n})}{α} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇u_{n}φ
= ∫_{Ω} H(u_{n}, ∇u_{n}) exp(G(u_{n}))φ + ∫_{Ω} φ exp(G(u_{n}))μ_{n} .

For any φ ≥ 0, thanks to (2.12) and (2.15) we obtain

∫_{Ω} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇φ ≤ ∫_{Ω} γ(x)φ exp(G(u_{n})) + ∫_{Ω} φ exp(G(u_{n}))μ_{n}

∀φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) , φ ≥ 0 .

(2.21)

Similarly, taking exp(−G(u_{n}))φ as test function in (2.19) we obtain

∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇φ + ∫_{Ω} γ(x)φ exp(−G(u_{n}))

≥ ∫_{Ω} φ exp(−G(u_{n}))μ_{n} ∀φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) , φ ≥ 0 . (2.22)

Let φ = T_{k}(u_{n})^{+} in (2.21) and φ = T_{k}(u_{n})^{−} in (2.22). Also let G(±∞) = \frac{1}{α} ∫_{0}^{±∞} g(s)ds which are well defined since g ∈ L^{1}(ℝ). Since exp(G(−∞)) ≤ exp(G(s)) ≤ exp(G(+∞)) and exp(|G(±∞)|) ≤ exp(∥g∥_{L^{1}(ℝ)}/α), using (2.12), we obtain

∥T_{k}(u_{n})∥_{W_{0}^{1,p}(Ω)}^{p} ≤ \frac{1}{α} exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} )k(∥γ∥_{L^{1}(Ω)} + ∥μ_{n}∥_{L^{1}(Ω)}) ≤ ck . (2.23)

Standard estimates (see [8]) imply that u_{n} is bounded in the Marcinkiewicz space M^{\frac{N(p−1)}{N−p}} (Ω) and |∇u_{n}| is bounded in the Marcinkiewicz space M^{\frac{N(p−1)}{N−1}} (Ω). In particular we have from (2.13) that a(x, u_{n}, ∇u_{n}) is bounded in L^{q}(Ω)^{N} for any q < \frac{N}{N−1}. Furthermore, there exist a function u and a subsequence such that

u_{n} → u a.e. in Ω,
T_{k}(u_{n}) → T_{k}(u) weakly in W_{0}^{1,p}(Ω) and a.e. in Ω for any k > 0.

Let us take φ = T_{1}(u_{n} − T_{j}(u_{n}))^{−} in (2.22); we obtain

∫_{{−(j+1)≤u_{n}≤−j}} a(u_{n}, ∇u_{n})∇u_{n} + ∫_{Ω} exp(−G(u_{n}))T_{1}(u_{n} − T_{j}(u_{n}))^{−}μ_{n}

≤ γ ∫_{Ω} exp(−G(u_{n}))T_{1}(u_{n} − T_{j}(u_{n}))^{−} . (2.24)

The term with μ_{n} can be neglected since it is nonnegative. In the right hand side we can pass to the limit in n and in j by Lebesgue’s theorem, using that G is bounded; indeed, since

exp(−G(u))T_{1}(u − T_{j}(u))^{−} ≤ exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} )χ_{\{u<−j\}}

we have

∫_{Ω} exp(−G(u_{n}))T_{1}(u_{n} − T_{j}(u_{n}))^{−} →^{n→∞} ∫_{Ω} exp(−G(u))T_{1}(u − T_{j}(u))^{−} →^{j→∞} 0 ,

so that we deduce from (2.24)

lim_{j→∞} lim sup_{n→∞} ∫_{{−(j+1)≤u_{n}≤−j}} a(u_{n}, ∇u_{n})∇u_{n} = 0 .  (2.25)

We are going now to prove that the truncations strongly converge in W_{0}^{1,p}(Ω). Following the idea introduced in [26], this is done by using a suitable sequence of cut-off functions. Indeed, let δ > 0; since λ is a regular measure concentrated on E and since E has zero p-capacity, there exist a compact set K_{δ} ⊂ E and a sequence {ψ_{δ}} of functions in C_{c}^{∞}(Ω) with the properties that

λ(E \ K_{δ}) < δ , 0 ≤ ψ_{δ} ≤ 1 ,
ψ_{δ} ≡ 1 on an open neighbourhood A_{δ} of K_{δ}
ψ_{δ} →^{δ→0} 0 strongly in W_{0}^{1,p}(Ω).

(2.26)

Take now φ = (k − T_{k}(u_{n}))(1 − |T_{1}(u_{n} − T_{j}(u_{n})|)ψ_{δ} in (2.22), with j > k. Observe that φ = (k − u_{n})ψ_{δ} if |u_{n}| < k and φ = 0 if u_{n} > k. Thus we get, using also that exp(−G(u_{n})) ≤ exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) and ψ_{δ} ≤ 1,

∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}(k − T_{k}(u_{n}))(1 − |T_{1}(u_{n} − T_{j}(u_{n})|)

+ 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{\{−(j+1)≤u_{n}≤−j\}} a(u_{n}, ∇u_{n})∇u_{n} + 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω} γψ_{δ}

≥ exp ( − \frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})ψ_{δ}  (2.27)

+ ∫_{Ω}(k − T_{k}(u_{n})) exp(−G(u_{n}))(1 − |T_{1}(u_{n} − T_{h}(u_{n})|)ψ_{δ}μ_{n} .

Since

|a(u_{n}, ∇u_{n})(1 − |T_{1}(u_{n} − T_{j}(u_{n})|)| ≤ |a(T_{j+1}(u_{n}), ∇T_{j+1}(u_{n}))| ,

and since last term is bounded in L^{p′}(Ω) and G is bounded, we have that there exists Λ_{j} ∈ L^{p′}(Ω)^{N} such that

exp(−G(u_{n}))a(u_{n}, ∇u_{n})(k − T_{k}(u_{n}))(1 − |T_{1}(u_{n} − T_{j}(u_{n})|) → Λ_{j}

weakly in L^{p′}(Ω)^{N}. Thus we get

lim_{n→∞} ∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}(k − T_{k}(u_{n}))(1 − |T_{1}(u_{n} − T_{j}(u_{n})|)

= ∫_{Ω} Λ_{j}∇ψ_{δ} ,

and then, as δ tends to zero, thanks to (2.26) we have

lim_{δ→0} lim_{n→∞} ∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}(k−T_{k}(u_{n}))(1−|T_{1}(u_{n}−T_{j}(u_{n})|) = 0 .

The third integral in (2.27) easily goes to zero since ψ_{δ} converges to zero and γ ∈ L^{1}(Ω). Furthermore, the term with μ_{n} can again be neglected since it is nonnegative. Therefore, passing to the limit first in n, then in δ we obtain from (2.27)

lim_{δ→0} lim sup_{n→∞} ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})ψ_{δ}

≤ lim sup_{n→∞} 2k exp (\frac{2∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{\{−(j+1)≤u_{n}≤−j\}} a(u_{n}, ∇u_{n})∇u_{n} .

Then, as j goes to infinity, using (2.25) and since a(x, s, ξ) · ξ ≥ 0, we get

lim_{δ→0} lim sup_{n→∞} ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})ψ_{δ} = 0 .  (2.28)

Let now w_{n} = T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)), we take φ = w_{n}^{+}(1 − ψ_{δ}) in (2.21) and φ = w_{n}^{−}(1 − ψ_{δ}) in (2.22) to obtain

∫_{{w_{n}≥0}} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇w_{n}(1 − ψ_{δ})

≤ ∫_{Ω} γw_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ}) + ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})μ_{n}

+ ∫_{Ω} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}w_{n}^{+}

and

∫_{{w_{n}≤0}} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇w_{n}(1 − ψ_{δ})
≤ ∫_{Ω} γw_{n}^{−} exp(−G(u_{n}))(1 − ψ_{δ}) − ∫_{Ω} w_{n}^{−} exp(−G(u_{n}))(1 − ψ_{δ})μ_{n}
− ∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}w_{n}^{−} .

Setting M = h + 4k and using a(x, s, ξ) · ξ ≥ 0, we have

a(u_{n}, ∇u_{n})∇w_{n} ≥a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇(T_{k}(u_{n}) − T_{k}(u))
− |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{{|u_{n}|>k}} .

Then

∫_{{w_{n}≥0}} exp(G(u_{n}))a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ})
≤ ∫_{Ω} γw_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ}) + ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})μ_{n}
+ ∫_{Ω} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}w_{n}^{+}
+ ∫_{Ω} exp(G(u_{n}))|a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{{|u_{n}|>k}}(1 − ψ_{δ})

(2.29)

and

∫_{{w_{n}≤0}} exp(−G(u_{n}))a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ})

≤ ∫_{Ω} γw_{n}^{−} exp(−G(u_{n}))(1 − ψ_{δ}) − ∫_{Ω} w_{n}^{−} exp(−G(u_{n}))(1 − ψ_{δ})μ_{n}

− ∫_{Ω} exp(−G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}w_{n}^{−}

+ ∫_{Ω} exp(−G(u_{n}))|a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{{|u_{n}|>k}}(1 − ψ_{δ}) .

(2.30)

Since a(u_{n}, ∇u_{n}) is bounded in L^{q}(Ω)^{N} for any q < \frac{N}{N−1}, there exists ν ∈ L^{q}(Ω)^{N} such that a(u_{n}, ∇u_{n}) weakly converges to ν in L^{q}(Ω)^{N}. Since ψ_{δ} ∈ W_{0}^{1,∞}(Ω), and G is bounded, we get

∫_{Ω} exp(G(u_{n}))a(u_{n}, ∇u_{n})∇ψ_{δ}w_{n}^{+}

→^{n→∞} ∫_{Ω} exp(G(u))ν∇ψ_{δ}T_{2k}(u − T_{h}(u)) →^{h→∞} 0 .

(2.31)

Using that |∇T_{k}(u)|χ_{{|u_{n}|>k}} strongly converges to zero in L^{p}(Ω) and that ∇T_{M}(u_{n}) is bounded in L^{p′}(Ω)^{N} we also have that

∫_{Ω} exp(G(u_{n}))|a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{{|u_{n}|>k}}(1 − ψ_{δ}) →^{n→∞} 0 .

(2.32)

Similarly, using the weak convergence of T_{k}(u_{n}) to T_{k}(u) in W_{0}^{1,p}(Ω), we have

∫_{Ω} exp(−G(u_{n}))a(T_{k}(u_{n}), ∇T_{k}(u))∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ}) →^{n→∞} 0 , (2.33)

and, since γ ∈ L^{1}(Ω),

∫_{Ω} γw_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})

→^{n→∞} ∫_{Ω} γ exp(G(u))(1 − ψ_{δ})T_{2k}(u − T_{h}(u))^{+} →^{h→∞} 0 .

(2.34)

Moreover, we have, using the decomposition of μ_{n} in (2.18),

∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})μ_{n}

= ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})dμ_{0n} + ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})λ_{n}

≤ exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω} w_{n}^{+}(1 − ψ_{δ})dμ_{0n} + 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω}(1 − ψ_{δ})λ_{n} .

Since w_{n}^{+} converges to T_{2k}(u − T_{h}(u))^{+} weakly-∗ in L^{∞}(Ω) and weakly in W_{0}^{1,p}(Ω), using the convergence of μ_{0n} (which is strong in L^{1}(Ω) + W^{−1,p′}(Ω)) and λ_{n} we obtain

lim sup_{n→∞} ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})μ_{n}

≤ exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω} T_{2k}(u − T_{h}(u))^{+}(1 − ψ_{δ})dμ_{0}

+ 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω}(1 − ψ_{δ}) dλ .

(2.35)

Since T_{k}(u) ∈ W_{0}^{1,p}(Ω) for any k > 0 and (2.23) holds true, we have (see e.g. Remark 2.11 in [26]) that u has a cap-quasi continuous representative which is cap-quasi everywhere finite, that is there exists a function ũ such that ũ = u almost everywhere and cap{|ũ| = +∞} = 0. In particular, since μ_{0} does not charge sets of zero capacity, we have that ũ is finite μ_{0}-quasi everywhere, hence T_{2k}(ũ − T_{h}(ũ)) converges to zero μ_{0}-quasi everywhere. Letting h go to infinity we deduce that

lim_{h→∞} ∫_{Ω} T_{2k}(u − T_{h}(u))^{+}(1 − ψ_{δ}) dμ_{0} = 0 ,

so that (2.35) implies

lim_{h→∞} lim sup_{n→∞} ∫_{Ω} w_{n}^{+} exp(G(u_{n}))(1 − ψ_{δ})μ_{n} ≤ 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} ) ∫_{Ω}(1 − ψ_{δ}) dλ

(2.36)

Then, as n and then h go to infinity, using (2.31), (2.32), (2.33), (2.34), (2.36), we obtain from (2.29),

lim sup_{h→∞} lim sup_{n→∞} ∫_{{w_{n}≥0}} exp(G(u_{n}))[a(T_{k}(u_{n}), ∇T_{k}(u_{n}))

− a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ})

≤ 2k exp(\frac{∥g∥_{L^{1}(ℝ)}}{α}) ∫_{Ω}(1 − ψ_{δ})dλ ≤ 2k exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} )λ(Ω \ K_{δ}) .

By means of (2.14) and recalling (2.26) we deduce

lim sup_{δ→0} lim sup_{h→∞} lim sup_{n→∞} ∫_{{w_{n}≥0}} [a(T_{k}(u_{n}), ∇T_{k}(u_{n}))

− a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ}) ≤ 0.

In the same way we work on (2.30), obtaining

lim sup_{δ→0} lim sup_{h→∞} lim sup_{n→∞} ∫_{{w_{n}≤0}} [a(T_{k}(u_{n}), ∇T_{k}(u_{n}))

− a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ}) ≤ 0.

Adding the two inequalities we conclude

lim sup_{δ→0} lim sup_{n→∞} ∫_{Ω} [a(T_{k}(u_{n}), ∇T_{k}(u_{n}))

− a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ}) = 0 . (2.37)

Now, we have

∫_{Ω}[a(T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))

= ∫_{Ω}[a(T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u))(1 − ψ_{δ})

+ ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})ψ_{δ} − ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u)ψ_{δ}

− ∫_{Ω} a(T_{k}(u_{n}), ∇T_{k}(u))∇(T_{k}(u_{n}) − T_{k}(u))ψ_{δ} .

Using the weak convergence of T_{k}(u_{n}) to T_{k}(u) last term converges to zero as n goes to infinity. Similarly, we have that a(T_{k}(u_{n}), ∇T_{k}(u_{n})) is bounded in L^{p′}(Ω)^{N} uniformly on n while ∇T_{k}(u)ψ_{δ} converges to zero in L^{p}(Ω)^{N} as δ tends to zero. Using also (2.37) and (2.28) we finally get, letting first n go to infinity and then δ to zero,

lim_{n→∞} ∫_{Ω}[a(T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(T_{k}(u_{n}), ∇T_{k}(u))]∇(T_{k}(u_{n}) − T_{k}(u)) = 0.

Under assumptions (2.12)–(2.14), it is well known that this implies

T_{k}(u_{n}) → T_{k}(u) strongly in W_{0}^{1,p}(Ω) for any k > 0.  (2.38)

Moreover, using that meas{|u_{n}| > k} goes to zero as k goes to infinity uniformly on n, as a consequence of (2.38) we also have that, up to subsequences, ∇u_{n} almost everywhere converges to ∇u in Ω. In turns, this implies that

a(x, u_{n}, ∇u_{n}) → a(x, u, ∇u) strongly in L^{q}(Ω)^{N} for any q < \frac{N}{N−1}. (2.39)

Let φ = ∫_{0}^{u_{n}} g(s)χ_{{s>h}}ds in (2.21); since |φ| ≤ ∫_{h}^{∞} g(s)ds we have

∫_{Ω} a(u_{n}, ∇u_{n})∇u_{n}g(u_{n})χ_{{u_{n}>h}}

≤ exp (\frac{∥g∥_{L^{1}(ℝ)}}{α} )( ∫_{h}^{∞} g(s)ds)(∥γ∥_{L^{1}(Ω)} + ∥μ_{n}∥_{L^{1}(Ω)}) .

Using (2.12) and the fact that μ_{n} is bounded in L^{1}(Ω) gives

α ∫_{{u_{n}>h}} g(u_{n})|∇u_{n}|^{p} ≤ c( ∫_{h}^{∞} g(s)ds) ,

and then since g ∈ L^{1}(ℝ) we obtain

lim_{h→∞} sup_{n∈ℕ} ∫_{{u_{n}>h}} g(u_{n})|∇u_{n}|^{p} = 0 .

Similarly, taking φ = ∫_{u_{n}}^{0} g(s)χ_{{s<−h}}ds in (2.22) we obtain the corresponding result on the set {u_{n} < −h}, hence

lim_{h→∞} sup_{n∈ℕ} ∫_{{|u_{n}|>h}} g(u_{n})|∇u_{n}|^{p} = 0 .  (2.40)

A standard argument allows to conclude from (2.38) and (2.40) that g(u_{n})|∇u_{n}|^{p} strongly converges in L^{1}(Ω) to g(u)|∇u|^{p}. Then from (2.15), the almost everywhere convergence of u_{n} and ∇u_{n} and Lebesgue’s theorem we conclude that

H(x, u_{n}, ∇u_{n}) → H(x, u, ∇u) strongly in L^{1}(Ω).  (2.41)

Thanks to (2.39) and (2.41) we can pass to the limit in (2.19) and we obtain that u is a distributional solution of (2.20). □

Remark 2.2 The assumption that μ is nonnegative is not essential in Theorem 2.1. In order to deal with changing sign measures it is enough to follow the same lines of the previous proof with suitable modifications while proving the strong convergence of truncations similar to those developed in [26].

Example 2.3 Let μ = δ_{0} be the Dirac mass at the origin and let Ω = B(0, 1) be the unit ball in ℝ^{N}, with N ≥ 3. Let μ_{n} = n^{N}χ_{B(0, \frac{1}{n})}; clearly μ_{n} converges, in the narrow topology, to λδ_{0} for some constant λ > 0. Note that in particular μ_{n} satisfies (2.18) (with f_{n} = F_{n} = 0). Let u_{n} be any sequence of solutions of

\begin{matrix}−Δu_{n} = g(u_{n})|∇u_{n}|^{2} + μ_{n} in Ω , \\ u_{n} = 0 on ∂Ω. \end{matrix}  (2.42)

We claim that if the following assumption holds:

∃h ∈ C(ℝ, ℝ^{+}): g(s) ≥ h(s) for every s ∈ ℝ^{+},
h is nonincreasing, lim_{s→+∞} h(s) = 0 and h ∉ L^{1}(ℝ^{+}),

(2.43)

then the sequence u_{n} blows up completely, namely u_{n}(x) → +∞ for every x ∈ Ω.

As far as assumption (2.43) is concerned, observe that if g is nonincreasing, converges to zero at infinity and g ∉ L^{1}(ℝ), we can clearly take h = g in (2.43); this includes the main examples of g around the borderline case g ∈ L^{1}(ℝ), as g(s) = 1/(|s| + 1) or g(s) = 1/((1 + |s|) log(1 + |s|)). Anyway, assumption (2.43) is stated in this generality to include most examples of g; in particular, note that the it requires g to be larger than a nonincreasing function which is not integrable, so that g itself may also be unbounded.

In order to prove our claim, we adapt an idea used in a context of sublinear equations by L. Orsina ([35]). Let us set H(s) = ∫_{0}^{s} h(ξ)dξ, ψ(s) = ∫_{0}^{s} exp(H(ξ))dξ and define v_{n} := ψ(u_{n}) (the function h is defined in (2.43)). Observe that ψ is an increasing unbounded function, so that v_{n} goes to infinity if and only if u_{n} goes to infinity. Since g(u_{n}) ≥ h(u_{n}), v_{n} satisfies

−Δv_{n} ≥ exp(H(u_{n}))μ_{n} in Ω ,

v_{n} = 0 on ∂Ω. (2.44)

In particular, by definition of μ_{n}, we have that v_{n} is a supersolution of the problem

−Δz = exp(H(ψ^{−1}(z)))n^{N} in B(0, \frac{1}{n}),

z = 0 on ∂B(0, \frac{1}{n}) .

(2.45)

Let φ_{1,n} be the first eigenfunction of the Laplacian on B(0, \frac{1}{n}), normalized so that ∥φ_{1,n}∥_{L^{∞}(Ω)} = 1, and let λ_{1,n} be the first eigenvalue. Let us set

B(s) := \frac{exp(H(ψ^{−1}(s)))}{s}.

Since h is nonincreasing we have

\frac{d}{dr} (\frac{exp(H(r))}{ψ(r)}) =\frac{exp(H(r))}{ψ(r)^{2}}(h(r) ∫_{0}^{r} exp(H(ξ))dξ − exp(H(r)))
≤\frac{exp(H(r))}{ψ(r)^{2}}( ∫_{0}^{r} exp(H(ξ))h(ξ)dξ − exp(H(r))) < 0 ,

so that B(ψ(s)) is decreasing. Since ψ is increasing, we deduce that B is a decreasing function. Let us set T_{n} = B^{−1}(\frac{λ_{1,n}}{n^{N}}). Since B is decreasing, we deduce that

\frac{λ_{1,n}}{n^{N}} = B(T_{n}) = B(T_{n}∥φ_{1,n}∥_{L^{∞}(Ω)}) ≤ B(T_{n}φ_{1,n}(x)) ∀x ∈ B(0, \frac{1}{n}) ,

which implies, by definition of B,

λ_{1,n}T_{n}φ_{1,n}(x) ≤ exp(H(ψ^{−1}(T_{n}φ_{1,n}(x))))n^{N} ∀x ∈ B(0, \frac{1}{n}) .

Since λ_{1,n}T_{n}φ_{1,n} = −Δ(T_{n}φ_{1,n}) we conclude that T_{n}φ_{1,n} is a subsolution of (2.45). Since exp (H(ψ^{−1}(z)))/z = B(z) is decreasing, a well-known comparison principle holds for positive sub-super solutions of (2.45) (see for example [23]), so that we get v_{n} ≥ T_{n}φ_{1,n} in B(0, \frac{1}{n}). By scaling arguments we know that

φ_{1,n}(x) = φ_{1,1}(nx) , λ_{1,n} = λ_{1,1}n^{2} ,

hence we obtain

∀x ∈ B(0, \frac{1}{2n}) : v_{n}(x) ≥ B^{−1}(\frac{λ_{1,1}}{n^{N−2}} ) min_{B(0, \frac{1}{2} )} φ_{1,1} .

Since φ_{1,1} is radial, we have min_{B(0, \frac{1}{2} )} φ_{1,1} = φ_{1,1}(\frac{1}{2}), so that

min_{B(0, \frac{1}{2n})} v_{n} ≥ B^{−1}(\frac{λ_{1,1}}{n^{N−2}} )φ_{1,1}(\frac{1}{2}) .  (2.46)

Now observe that, using De L’Hospital’s theorem and the fact that h(s) goes to zero at infinity, we have lim_{s→+∞} B(s) = 0. Since λ_{1,1}/n^{N−2} converges to zero as n tends to infinity, we end up with

lim_{n→+∞} B^{−1}(\frac{λ_{1,1}}{n^{N−2}} ) = +∞ ,

and then from (2.46)

lim_{n→+∞} min_{B(0, \frac{1}{2n})} v_{n} = +∞ .  (2.47)

Let now G(x, y) be the kernel of the Laplacian with zero boundary condition; we have from (2.44)

v_{n}(x) ≥ ∫_{Ω} G(x, y) exp(H(u_{n}))(y)μ_{n}(y)dy

≥ min_{B(0, \frac{1}{2n})} (exp(H(ψ^{−1}(v_{n})))) ∫_{B(0, \frac{1}{2n})} G(x, y)n^{N} dy .

(2.48)

Since there exists a constant c > 0 such that

∫_{B(0, \frac{1}{2n})} G(x, y)n^{N}dy → c ∫_{Ω} G(x, y)dδ_{0}(y) > 0 ,

and since both ψ^{−1} and H go to infinity at infinity (because h ∉ L^{1}(ℝ^{+})), we deduce using (2.47) that the right hand side of (2.48) goes to infinity as n goes to infinity. We then conclude

lim_{n→+∞} v_{n}(x) = +∞ ∀x ∈ Ω .

Since ψ is unbounded and u_{n} = ψ^{−1}(v_{n}), we have proved that the solutions u_{n} of (2.42) blow up completely in Ω. This is in sharp contrast with what proved in Theorem 2.1 when g ∈ L^{1}(ℝ), so that this assumption is optimal in the existence result above.

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Alessio Porretta Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italia. email: porretta@mat.uniroma2.it