A Singular Semilinear Elliptic Equation with a Variable Exponent

Let f∈L2⁢N/(N+2)⁢(Ω), γ⁢(x)<1 on ∂⁡Ω or γ⁢(x)≤1 on ∂⁡Ω with ∂⁡γ⁢(x)∂⁡ne≥0, and assume that (1.2) holds. Then, there exists a solution u∈H01⁢(Ω) to problem (1.1).

Assume that for some γ*>1 and some δ>0 we have that ∥γ∥L∞⁢(Ωδ)≤γ*, and that (1.2) holds. Assume also that f∈Lm⁢(Ω) with m=N⁢(γ*+1)N+2⁢γ*. Then, there exists a solution u∈Hloc1⁢(Ω) to problem (1.1) such that u(γ*+1)/2∈H01⁢(Ω). Furthermore, if there exists Γ⊂∂⁡Ω such that γ⁢(x)≤1 in the set {x∈Ω:dist⁢(x,Γ)<ν} for some ν>0, then u∈HΓ1⁢(ω) for every open set ω⊂Ω with ω¯⊂Ω∪Γ.

1. For every s∈ℝ, we consider the positive and negative parts given by s+=max⁡{s,0} and s-=min⁡{s,0}, respectively.

2. For any k>0, we set Tk⁢(s)=min⁡(k,max⁡(s,-k)) and Gk⁢(s)=s-Tk⁢(s).

3. We define the set Ωδ:={x∈Ω:dist⁢(x,∂⁡Ω)<δ} for δ>0 fixed.

4. We denote by |E| the Lebesgue measure of a measurable set E in ℝℕ.

5. For 1≤p≤+∞, ∥u∥p denotes the usual norm of a function u∈Lp⁢(E).

6. We equip the standard Sobolev space H01⁢(E) with the usual norm ∥u∥=(∫E|∇u|2)1/2.

7. For any 1<p<N, p*=N⁢pN-p denotes the Sobolev conjugate exponent of p.

8. 𝒮 denotes the best Sobolev constant, i.e.,

   𝒮 = sup ∥ u ∥ H 0 1 ⁢ ( Ω ) = 1 ⁡ ∥ u ∥ L 2 * ⁢ ( Ω ) .

9. We recall that, for 1<p<∞, the dual space of Lp⁢(Ω) can be identified with Lp′⁢(Ω), where p′=pp-1 is the Hölder conjugated exponent of p.

We say that u∈Hloc1⁢(Ω) is a positive solution for (1.1) if u>0 almost everywhere in Ω,

f ⁢ ( x ) u γ ⁢ ( x ) ∈ L loc 1 ⁢ ( Ω )

and

for every ϕ∈C01⁢(Ω).

Problem (2.1) has a nonnegative solution un∈H01⁢(Ω)∩L∞⁢(Ω).

Proof.

For every fixed n∈ℕ we can deduce the existence of un by means of the Schauder’s fixed point theorem applied to the operator S:L2⁢(Ω)→L2⁢(Ω) defined by S⁢(v)=w∈H01⁢(Ω) for every v∈L2⁢(Ω), where w is the unique solution of (see [12])

{ - div ⁡ ( M ⁢ ( x ) ⁢ ∇ ⁡ w ) = f n ⁢ ( x ) ( | v | + 1 n ) γ ⁢ ( x ) in ⁢ Ω , w = 0 on ⁢ ∂ ⁡ Ω .

Since γ⁢(x)∈C1⁢(Ω¯) we can define γ*=∥γ⁢(x)∥L∞⁢(Ω). Taking w as test function and using (1.2), Poincaré’s and Hölder’s inequalities, we have

α λ 1 ∫ Ω w 2 ≤ α ∫ Ω | ∇ w | 2 ≤ ∫ Ω M ( x ) ∇ w ∇ w = ∫ Ω f n ⁢ ( x ) ⁢ w ( | v | + 1 n ) γ ⁢ ( x ) ≤ n γ * + 1 ∫ Ω | w | ≤ n γ * + 1 | Ω | 1 / 2 ( ∫ Ω | w | 2 ) 1 / 2 .

In particular, a ball of large enough radius remains invariant for S. Moreover, from the compact embedding of H01⁢(Ω) in L2⁢(Ω), we deduce that S is continuous and compact on L2⁢(Ω). Thus, we can use Schauder’s fixed point theorem to prove the existence of un∈H01⁢(Ω) solving the following problem:

{ - div ⁡ ( M ⁢ ( x ) ⁢ ∇ ⁡ u n ) = f n ⁢ ( x ) ( | u n | + 1 n ) γ ⁢ ( x ) in ⁢ Ω , u n = 0 on ⁢ ∂ ⁡ Ω .

Now, taking un- as test function, using (1.2) and taking into account that fn⁢(x)(|un|+1n)γ⁢(x)≥0, we get

α ∫ Ω | ∇ u n - | 2 ≤ ∫ Ω M ( x ) ∇ u n ∇ u n - = ∫ Ω f n ⁢ ( x ) ( | u n | + 1 n ) γ ⁢ ( x ) u n - ≤ 0 .

Therefore, un-≡0 and in particular, un≥0 and solves (2.1). As the right-hand side of (2.1) belongs to L∞⁢(Ω), we can use [15, Theorem 4.2] to deduce that un belongs to L∞⁢(Ω). ∎

The sequence un is increasing with respect to n, un>0 in Ω and, for every ω⊂⊂Ω, there exists cω>0 (independent on n) such that

Proof.

Observe that taking (un-un+1)+ as test function in the equations satisfied by un and un+1 and subtracting, and then taking into account (1.2) and that 0≤fn⁢(x)≤fn+1⁢(x), we get that

The last inequality is due to the fact that fn+1⁢(x)≥0, (un-un+1)+≥0 and

( 1 ( u n + 1 n + 1 ) γ ⁢ ( x ) - 1 ( u n + 1 + 1 n + 1 ) γ ⁢ ( x ) ) ≤ 0   in ⁢ { x ∈ Ω : u n ⁢ ( x ) ≥ u n + 1 ⁢ ( x ) } .

Therefore, (un-un+1)+≡0, and thus

On the other hand, we know that

∫ Ω M ⁢ ( x ) ⁢ ∇ ⁡ u 1 ⁢ ∇ ⁡ ϕ = ∫ Ω f 1 ⁢ ( x ) ( u 1 + 1 ) γ ⁢ ( x ) ⁢ ϕ ≥ ∫ Ω f 1 ⁢ ( x ) ( ∥ u 1 ∥ L ∞ ⁢ ( Ω ) + 1 ) γ ⁢ ( x ) ⁢ ϕ ,

and since

f 1 ⁢ ( x ) ( ∥ u 1 ∥ L ∞ ⁢ ( Ω ) + 1 ) γ ⁢ ( x ) ≢ 0 ,

we deduce, using the strong maximum principle, that u1>0 in Ω. Thus, u1⁢(x)≥cω>0 for every x∈ω and every n∈ℕ. By (2.4) the proof is completed. ∎

Let f∈Lm⁢(Ω) for some m>N2. The sequence {un} of solutions of problem (2.1) is bounded in L∞⁢(Ω), i.e., there exists C>0 independent of n and γ⁢(x) with

∥ u n ∥ ∞ ≤ C   for all ⁢ n ∈ ℕ .

Proof.

To prove an a priori estimate in L∞⁢(Ω), let k>1, we take ϕ=Gk⁢(un) as test function in (2.1) and using (1.2) we obtain

α ∫ Ω | ∇ G k ( u n ) | 2 ≤ ∫ Ω M ( x ) ∇ G k ( u n ) ∇ G k ( u n ) = ∫ Ω f n ⁢ ( x ) ( u n + 1 n ) γ ⁢ ( x ) G k ( u n ) .

Using the fact that un+1n≥k≥1 on the set {un≥k}, where Gk⁢(un)≠0, we deduce that

α ∫ Ω | ∇ G k ( u n ) | 2 ≤ ∫ Ω f ( x ) G k ( u n ) .

Now, by Stampacchia’s method [15], from the last inequality follows the existence of C>0 such that

∥ u n ∥ ∞ ≤ C ,

where the constant C does not depend on γ⁢(x). ∎

Let f∈L2⁢N/(N+2)⁢(Ω) and assume that there exists δ>0 with γ⁢(x)≤1 in Ωδ and that (1.2) holds. Then, the sequence {un} of solutions of the problem (2.1) is bounded in H01⁢(Ω), i.e., there exists C>0, independent of n, with

∥ u n ∥ H 0 1 ⁢ ( Ω ) ≤ C   for all ⁢ n ∈ ℕ .

In order to show the existence of a solution, we will use the fact that γ⁢(x)≤1 for every x in a strip around ∂⁡Ω and inside Ω. Our hypothesis on γ⁢(x) in Theorem 1.1 guarantees this fact. Note that we can extend the result to functions γ⁢(x) such that γ⁢(x)<1 on A⊂∂⁡Ω and γ⁢(x)=1 on ∂⁡Ω∖A with ∂⁡γ⁢(x)∂⁡ne≥0 there.

Proof.

Let us denote ωδ=Ω∖Ω¯δ and recall that un≥cωδ in ωδ, where 0<cωδ is given by Lemma 2.3. Thus, taking un as test function in (2.1) and using (1.2), it follows that

Using Hölder’s and Sobolev’s inequalities, we deduce that

α ⁢ ∥ u n ∥ H 0 1 ⁢ ( Ω ) 2 ≤ ∥ f ∥ L 1 ⁢ ( Ω ) + 𝒮 ⁢ ( 1 + ∥ c ω δ - γ ⁢ ( x ) ∥ L ∞ ⁢ ( Ω ) ) ⁢ ∥ f ∥ L 2 ⁢ N / ( N + 2 ) ⁢ ( Ω ) ⁢ ∥ u n ∥ H 0 1 ⁢ ( Ω ) ,

and this implies the existence of C>0 such that

∥ u n ∥ H 0 1 ⁢ ( Ω ) ≤ C   for all ⁢ n ∈ ℕ .

Hence, we conclude that the sequence un is bounded in H01⁢(Ω). ∎

Assume that for some γ*>1 and δ>0 we have that ∥γ∥L∞⁢(Ωδ)≤γ* and that (1.2) holds. Assume also that f∈Lm⁢(Ω) with m=N⁢(γ*+1)N+2⁢γ*. Then, un(γ*+1)/2 is bounded in H01⁢(Ω) and un is bounded in Hloc1⁢(Ω). Moreover, if for some Γ⊂∂⁡Ω we have that γ⁢(x)≤1 in the set {x∈Ω:dist⁢(x,Γ)<δ}, then un is bounded in HΓ1⁢(ω) for every open set ω⊂Ω with ω¯⊂Ω∪Γ.

We point out that m=N⁢(γ*+1)N+2⁢γ* tends to 2⁢NN+2 if γ*→1 and tends to N2 if γ*→∞.

Proof.

We take unγ* as test function in (2.1), and then from (1.2) and Lemma 2.3 we obtain

This completes the first part. Observe that the Sobolev embedding implies that un is also bounded in L2*⁢(γ*+1)/2⁢(Ω).

In order to prove that un is bounded in H1⁢(ω) for every ω⊂⊂Ω we follow closely [2]. In fact, a careful analysis of the proof allow us to prove that if for some Γ⊂∂⁡Ω we have that γ⁢(x)≤1 in the set {x∈Ω:dist⁢(x,Γ)<δ}, then un is bounded in HΓ1⁢(ω) for every open set ω⊂Ω with ω¯⊂Ω∪Γ. Indeed, we take ϕ∈C1⁢(Ω¯) with supp⁡ϕ⊂Ω if γ⁢(x)>1 on ∂⁡Ω and supp⁡ϕ⊂Ω∪Γ otherwise.

We use the notation Ω*={ϕ≠0} and Ωδ,Γ*={x∈Ω*:dist⁢(x,∂⁡Ω*∩Γ)<δ}. Recall that Ω¯*⊂Ω∪Γ, and thus ωδ,Γ≡Ω*∖Ω¯δ,Γ* is compactly embedded in Ω and γ⁢(x)≤1 in Ω¯δ,Γ*.

Taking un⁢ϕ2 as test function, then by (1.2) and Lemma 2.3, we obtain

Using Young’s inequality and (1.2), we also have that

2 ∫ Ω M ( x ) u n ϕ ∇ u n ∇ ϕ ≥ - 2 β ∫ Ω | u n ϕ | | ∇ u n | | ∇ ϕ | ≥ - α 2 ∫ Ω | ∇ u n | 2 ϕ 2 - 2 ⁢ β 2 α ∫ Ω | ∇ ϕ | 2 u n 2 .

Combining both inequalities and taking into account that m≥(2*⁢(γ*+1)2)′≡s′ yields

α 2 ∫ Ω | ∇ u n | 2 ϕ 2 ≤ ∥ f ϕ 2 ∥ L 1 ⁢ ( Ω ) + ( 1 + ∥ c ω δ , Γ - γ ⁢ ( x ) ∥ L ∞ ⁢ ( Ω ) ) ∥ ϕ 2 ∥ L ∞ ⁢ ( Ω ) ∥ f ∥ L s ′ ⁢ ( Ω ) ∥ u n ∥ L s ⁢ ( Ω ) + 2 ⁢ β α ∥ | ∇ ϕ | 2 ∥ L ∞ ⁢ ( Ω ) ∥ u n ∥ L 2 ⁢ ( Ω ) 2 ,

which allow us to complete the proof using that s>2 and the fact that ∥un∥L2⁢(Ω) and ∥un∥Ls⁢(Ω) are bounded sequences. ∎

Proof of Theorem 1.1.

Since un is bounded in H01⁢(Ω), then by Proposition 3.1, up to a subsequence, un⇀u for some u∈H01⁢(Ω). Thus, un→u strongly in Lt⁢(Ω) with t<2* and un⁢(x)→u⁢(x) almost everywhere in Ω.

Therefore, we have

lim n → + ∞ ⁡ ∫ Ω M ⁢ ( x ) ⁢ ∇ ⁡ u n ⁢ ∇ ⁡ ϕ = ∫ Ω M ⁢ ( x ) ⁢ ∇ ⁡ u ⁢ ∇ ⁡ ϕ   for every ϕ ∈ C 0 1 ⁢ ( Ω ) .