Existence and Regularizing Effect of Degenerate Lower Order Terms in Elliptic Equations Beyond the Hardy Constant

Theorem 2.1.

Assume that p>1, f∈Lh(p+1)/p⁢(Ω) and that there exists δ≥0 such that ∂⁡Ωδ is smooth, 0∉Ω¯δ and h>0 a.e. in Ω∖Ωδ.

1. If condition ( 1.2 ) holds true, then there exists Λ ⁢ ( δ ) such that ( 1.1 ) has a solution u ∈ E for every λ ≤ Λ ⁢ ( δ ) . In addition, Λ ⁢ ( δ ) → ∞ as δ → 0 .

2. If, in addition, there exists s ¯ ∈ ( 2 , p + 1 ) such that condition ( 1.3 ) holds true, then u is a minimum of functional I λ given by ( 2.1 ).

Remark 2.2.

(i) As it has been previously observed, every function f∈Lh(p+1)/p⁢(Ω) can be considered as an element of the dual space E* of E. We will see in the proof that for the above existence result the hypothesis f∈Lh(p+1)/p⁢(Ω) can be relaxed to f∈E*.

(ii) Observe that condition (1.2) is equivalent to

while condition (1.3) means that

Observe that if 2<s¯<p+1, then 2<2⁢(p+1)p-1<2⁢s¯s¯-2 and it follows that (1.3) implies (1.2).

(iii) Moreover, (1.3) is clearly satisfied in the case in which h⁢(x) is a Hardy potential term of order p+1 on the left-hand side of equation (1.1), i.e. h⁢(x)=1/|x|p+1. Indeed, in this context, condition (1.3) holds true due to the boundedness of the domain Ω.

(iv) In the case h≡0, part (b) of Theorem 2.1 has to be compared with [8, Theorem 3.4] where García Azorero and Peral proved the existence of a minimum of the functional by using an argument that does not require the weak lower semicontinuity of the functional Iλ, leaving this semicontinuity as an open problem. As for us, we prove that the hypothesis (1.3) implies that Iλ is weakly lower semicontinuous.

Proof of Theorem 2.1.

(a) By (1.2), using the Hölder inequality with exponent p+12, we obtain for every u∈E,

where ρ⁢(δ):=dist⁢(0,Ωδ)>0.

Moreover, since u=0 in ∂⁡Ω and ∂⁡Ω⊂∂⁡Ωδ, we can use a Poincaré inequality in Ωδ (see e.g. [9], [13, Section 4.6] or [1, Section 8]) to assert that

with the positive constant C⁢(δ) satisfying

where C1,2⁢(∂⁡Ω) denotes the capacity of ∂⁡Ω.

Hence, the functional Iλ given by (2.1) satisfies for every u∈E that

Thus, since 2p+1<1, we obtain that Iλ is coercive and bounded from below provided that

As a consequence, by the Variational Principle of Ekeland [7], there is a bounded minimizing sequence {un}⊂E such that

and Iλ′⁢(un)→0 in E*, i.e., there exists a sequence of positive numbers {εn} converging to zero such that

We are going to pass to the limit in this inequality as n tends to infinity. The boundedness of {un} in E implies that, up to a subsequence, we have the weak convergence of un in E to some u∈E. In particular, up to a subsequence, we can assume that

1. u n ⇀ u in W01,2⁢(Ω),

2. u n ⁢ h 1 p + 1 ⇀ u ⁢ h 1 p + 1 in Lp+1⁢(Ω),

3. u n → u in Lq⁢(Ω) (1≤q<2*),

4. u n ⁢ ( x ) → u ⁢ ( x ) a.e. in Ω,

5. there exists g∈Lq⁢(Ω) (1≤q<2*) such that |un⁢(x)|≤g⁢(x).

Obviously, by (A),

and by (B) the sequence |un|p-1⁢un is bounded in Lhp+1⁢(Ω). Due to the a.e. convergence (D), it follows that |un|p-1⁢un⇀|u|p-1⁢u in Lp+1⁢(Ω;h⁢d⁢x). Hence, by (E), the Lebesgue dominated convergence theorem implies that

In order to get the convergence of the term with Hardy potential, i.e., ∫Ωun|x|2⁢v, we point out that for each v∈W01,2⁢(Ω) the operator Tv:W01,2⁢(Ω)→ℝ defined as

is linear and continuous since (by using Hölder and Hardy inequalities)

for every v∈W01,2⁢(Ω), where ℋ is the Hardy constant.

In particular, since Tv has finite range, it is also compact and hence Tv⁢(un) strongly converges to Tv⁢(u), i.e.

In conclusion, taking limits in (2.3), we obtain that u∈E is a solution of problem (1.1) for λ<Λ⁢(δ).

In addition, since ρ⁢(δ)→dist⁢(0,∂⁡Ω)>0 as δ→0, equation (2.2) implies that Λ⁢(δ)→∞ as δ→0.

(b) As we have seen in the proof of part (a), for every λ≤Λ⁢(δ) the functional Iλ is bounded from below and coercive. Thus, in order to deduce that Iλ attains its minimum, it suffices to show that it is weakly lower semicontinuous. Assume hence that {un} is a sequence weakly convergent in E. As before, up to a subsequence, we can assume that {un} verifies the convergences (A)–(E). In addition, we note that the boundedness of un⁢h1/(p+1) in Lp+1⁢(Ω) and the a.e. convergence (D) of un imply the strong convergence of un⁢h1/(p+1) in Ls⁢(Ω) for every 1≤s<p+1. As a consequence, there exists G∈Ls⁢(Ω) such that (again up to a subsequence)

We claim that

Indeed, if we consider the function g∈L2⁢(Ω) given in (E) with q=2 which satisfies that |un⁢(x)|≤g⁢(x) for every n∈ℕ and almost every x∈Ω, then

where the function H is defined in Ω as

By (D) we also have the convergence of un⁢(x)2/|x|2 to u⁢(x)2/|x|2 for almost every x∈Ω. Therefore, by the dominated convergence theorem, the claim will be proved if we show that H∈L1⁢(Ω). For this purpose, observe that taking into account that 0∉Ω¯δ, we deduce that g2⁢(x)/|x|2∈L1⁢(Ω¯δ), i.e., H∈L1⁢(Ω¯δ). To prove the integrability in Ω∖Ω¯δ, we use the Hölder inequality with exponent s2>1 to obtain

The last two integral terms are finite due to hypothesis (1.3) and the fact that G∈Ls⁢(Ω). Consequently, we also have H∈L1⁢(Ω∖Ω¯δ). Thus, the claim is proved.

On the other hand, Theorem 2.1 of [5] implies that (up to a subsequence) ∇⁡un→∇⁡u strongly in (Lq⁢(Ω))N (1<q<2) and in particular (up to a subsequence) it converges almost everywhere in Ω. Then, applying the Fatou lemma, we have

Summarizing (2.4) and (2.5), we obtain

i.e. the functional Iλ is weakly lower semicontinuous and the proof is concluded. ∎

Corollary 2.3.

Assume p>2*-1, f∈L(p+1)/p⁢(Ω) and h⁢(x)≡h0>0 in Ω. There exists a solution u∈E of problem (1.1) for every λ∈R.

Remark 2.4.

In particular, we recover the existence result of [2, 10]: there exists a solution in

Corollary 2.5.

Let p>2*-1, 0<δ≤δ0, f∈L(p+1)/p⁢(Ω∖Ωδ) and h≡h0⁢χΩ∖Ωδ for some h0>0. Then, there is a solution of (1.1) in E for λ≤Λ⁢(δ).

Corollary 2.6.

If p>1, the function h satisfies (1.3) with h>0 a.e. in Ω∖Ωδ and H<λ≤Λ⁢(δ), then the problem

has at least one nonzero solution.

Remark 2.7.

As usual, by considering instead of Iλ the functional Jλ given by

it is possible to deduce the existence of a positive solution of problem (2.6). Therefore we improve the corresponding existence result of [12], where it is required additionally that h is a continuous and positive function in Ω¯, and of [11], where the case h⁢(x)=1/|x|β with β<2 is studied. (Observe that in both cases considered in those papers, Λ⁢(δ)=∞ in the above corollary.)

Theorem 3.1.

Assume that h∈L1⁢(Ω) with h⁢(x)>0 a.e. in Ω and that there exists m≥p+1p such that

1. f ∈ L h m ⁢ ( Ω ) ,

2. | x | 2 ⁢ p ⁢ m / ( 1 - p ) ⁢ h 1 - p ⁢ m / ( p - 1 ) ∈ L 1 ⁢ ( Ω ) .

If u is a solution of (1.1), then u∈Lhp⁢m⁢(Ω).

Remark 3.2.

If instead of assuming that h∈L1⁢(Ω) we only assume that h∈Lloc1⁢(Ω), then the above hypothesis (i) should be replaced by f∈Lh(p+1)/p⁢(Ω)∩Lhm⁢(Ω).

Proof.

For every k>0, we define the auxiliary function Tk:ℝ→ℝ as usual by

Let u∈E be a solution of (1.1). Since m≥p+1p, we have γ:=p⁢m-1-p>0 and we can choose |Tk⁢(u)|γ⁢Tk⁢(u) as a test function in problem (1.1) to obtain, by dropping the positive term coming from the principal part, that

Next, we estimate each term of the above inequality. For this purpose, we define

where

Using that |Tk⁢(s)|≤|s| for all s∈ℝ, we deduce that

and thus

On the other hand, using the Hölder inequality with exponent p-δ>1 and the fact that 1+δ+γ=(1+γ)⁢(p-δ), we get

where the last inequality is a consequence of hypothesis (ii).

In addition, using the Hölder inequality with exponent m and taking into account that

we obtain, by (i),

In conclusion, substituting (3.2), (3.3) and (3.4) into (3.1), we deduce that

Since 1p-δ and m-1m are less than 1, inequality (3.5) implies the existence of k0>0 and C3>0 (independent of k and u) such that

Fatou’s lemma when k tends to ∞ and the fact that γ+1+p=p⁢m imply that

as we desired. ∎

Corollary 3.3.

Assume that f∈Lhm⁢(Ω) for m≥p+1p, and that there exist μ>0 and β≥0 such that the function h∈L1⁢(Ω) satisfies

If u is a solution of (1.1), then u∈Lp⁢m⁢(Ω;d⁢x|x|β) for every

Remark 3.4.

(i) The integrability of h implies that necessarily β<N.

(ii) Observe that if β∈[0,2), then the interval [p+1p,(N-β)⁢(p-1)(2-β)⁢p] of the possible values of m is not empty (i.e., p+1p<(N-β)⁢(p-1)(2-β)⁢p) if and only if h satisfies condition (1.2).

(iii) We note that in the particular case β=0, the regularity result is proved in [2] only for a solution obtained as limit of solutions of a sequence of suitable approximate problems, but not for every solution as in the previous result.