We consider the existence of multiple solutions of the singular elliptic problem , u(x) → 0 as |x| → +∞, where x ∈ ℝ^N, 1 < p < N, a < (N − p)/p, a ≤ b ≤ a + 1, r, s > 1, p^* = Np/(N − pd), d = a + 1 − b. By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.

1. Introduction and Main Results

In this paper, we consider the existence of multiple solutions for the singular elliptic problem

(1.1)

where 1 < p < N, a < (N − p)/p, a ≤ b ≤ a + 1, r > 1, p^* = Np/(N − pd), d = a + 1 − b. f(x) and h(x) are nonnegative functions in ℝ^N.

In recent years, the existence of multiple solutions on elliptic equations has been considered by many authors. In [1], Assunção et al. considered the following quasilinear degenerate elliptic equation:

(1.2)

where x ∈ ℝ^N, 1 < p < N, q = Np/[N − p(a + 1 − b)]. When λ = 0, f = εg, where 0 < ε ≤ ε₀ and

; the authors proved that problem (1.2) has at least two positive solutions. Rodrigues in [2] studied the following critical problem on bounded domain Ω ∈ ℝ^N:

(1.3)

By the variational method on Nehari manifolds [3, 4], the author proved the existence of at least two positive solutions and the nonexistence of solutions when some certain conditions are satisfied. When p = 2 and a = −1, Miotto and Miyagaki in [5] considered the semilinear Dirichlet problem in infinite strip domains

(1.4)

The authors also proved that problem (1.4) has at least two positive solutions by the methods of Nehari manifold. For other references, we refer to [6–11] and the reference therein. In fact, motivated by [1, 2, 5], we consider the problem (1.1). Since our problem is singular and is studied in the whole space ℝ^N, the loss of compactness of the Sobolev embedding renders a variational technique that is more delicate. By the variational method and the theory of genus, we prove that problem (1.1) has infinitely many solutions when some suitable conditions are satisfied.

In order to state our result, we introduce some weighted Sobolev spaces. For r, s ≥ 1 and g = g(x) > 0 in R^N, we define the spaces L^r(ℝ^N, g) and L^s(ℝ^N, g) as being the set of Lebesgue measurable functions u : ℝ^N → ℝ¹, which satisfy

(1.5)

Particularly, when g(x) ≡ 1, we have

(1.6)

We denote the completion of

with the norm of

(1.7)

where 1 < p < N and a < (N − p)/p. It is easy to find that X is a reflexive and separable Banach space with the norm ∥u∥_X.

The following Hardy-Sobolev inequality is due to Caffarelli et al. [12], which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants S₁, S₂ > 0 such that

(1.8)

(1.9)

where p^* = Np/(N − pd) is called the Sobolev critical exponent.

In the present paper, we make the following assumptions:

(A₁) for 1 < r < p, where ;
(A₂) for p < r < p^*, where .
(A₃) for p < s < p^*, where .

Then, we give some basic definitions.

Definition 1.1. u ∈ X is said to be a weak solution of (1.1) if for any there holds

(1.10)

Let I(u) : X → ℝ¹ be the energy functional corresponding to problem (1.1), which is defined as

(1.11)

for all u ∈ X. Then the functional I ∈ C¹(X, R¹) and for all φ ∈ X, there holds

(1.12)

It is well known that the weak solutions of problem (1.1) are the critical points of the functional I(u), see [13]. Thus, to prove the existence of weak solutions of (1.1), it is sufficient to show that I(u) admits a sequence of critical points in X.

Our main result in this paper is the following.

Theorem 1.2. Let 1 < p < N, a < (N − p)/p, a ≤ b ≤ a + 1, r > 1, p^* = Np/(N − pd), d = a + 1 − b, max {r, p} < s < p^*. Assume (A₁)–(A₃) are fulfilled. Then problem (1.1) has infinitely many solutions in X.

2. Preliminary Results

Our proof is based on variational method. One important aspect of applying this method is to show that the functional I(u) satisfies (PS) _c condition which is introduced in the following definition.

Definition 2.1. Let c ∈ R¹ and X be a Banach space. The functional I(u) ∈ C¹(X, R) satisfies the (PS) _c condition if for any {u_n} ⊂ X such that

(2.1)

contains a convergent subsequence in X.

The following embedding theorem is an extension of the classical Rellich-Kondrachov compactness theorem, see [14].

Lemma 2.2. Suppose Ω ⊂ ℝ^N is an open bounded domain with C¹ boundary and 0 ∈ Ω. N ≥ 3, a < (N − p)/p. Then the embedding is continuous if 1 ≤ r ≤ Np/(N − p) and 0 ≤ α ≤ (1 + a)r + N(1 − r/p), and is compact if 1 ≤ r < Np/(N − p) and 0 ≤ α < (1 + a)r + N(1 − r/p).

Now we prove an embedding theorem, which is important in our paper.

Lemma 2.3. Assume (A₁)-(A₂) and 1 < r < p^*. Then the embedding X↪L^r(ℝ^N, f) is compact.

Proof. We split our proof into two cases.

(i) Consider 1 < r < p.

By the Hölder inequality and (1.9) we have that

(2.2)

where

. Then the embedding is continuous. Next, we will prove that the embedding is compact.

Let B_R be a ball center at origin with the radius R > 0. For the convenience, we denote L^r(ℝ^N, f) by Z, that is, Z = L^r(ℝ^N, f). Assume {u_n} is a bounded sequence in X. Then {u_n} is bounded in X(B_R). We choose α = 0 in Lemma 2.2, then there exist u ∈ Z(B_R) and a subsequence, still denoted by {u_n}, such that as n → ∞. We want to prove that

(2.3)

where

. In fact, we obtain from (2.2) that

(2.4)

The fact

shows that

(2.5)

Then (2.4) and (2.5) imply that

(2.6)

which gives (2.3).

In the following, we will prove that u_n → u strongly in Z(R^N).

Since X is a reflexive Banach space and {u_n} is bounded in X. Then we may assume, up to a subsequence, that

(2.7)

In view of (2.3), we get that for any ε > 0 there exists R_ε > 0 large enough such that

(2.8)

On the other hand, due to the compact embedding

in Lemma 2.2, we have that

(2.9)

Therefore, there is N₀ > 0 such that

(2.10)

for n > N₀. Thus, the inequalities (2.8) and (2.10) show that

(2.11)

This shows that {u_n} is convergent in Z = L^r(ℝ^N, f).

(ii) Consider p ≤ r < p^*.

It follows from (1.8) and the Hölder inequality that

(2.12)

where

. Thus, the fact of

and (2.12) imply that the embedding is continuous. Similar to the proof of (i) we can also prove that the embedding X↪L^r(ℝ^N, f) is compact for p ≤ r < p^*.

Similarly, we have the following result of compact embedding.

Lemma 2.4. Assume 1 < p < s < p^* and (A₃), then the embedding X↪L^s(ℝ^N, h) is compact.

The following concentration compactness principle is a weighted version of the Concentration Compactness Principle II due to Lions [15–18], see also [19, 20].

Lemma 2.5. Let 1 < p < N, − ∞ < a < (N − p)/p, a ≤ b ≤ a + 1, p^* = Np/(N − pd), d = a + 1 − b. Suppose that is a sequence such that

(2.13)

where μ, η are measures supported on Ω and ℳ(ℝ^N) is the space of bounded measures in ℝ^N. Then there are the following results.

(1)
There exists some at most countable set J, a family {x_j ∈ Ω∣j ∈ J} of distinct points in ℝ^N, and a family {η_j∣j ∈ J} of positive numbers such that
(2.14)
where is the Dirac measure at x_j.
(2)
The following equality holds
(2.15)
for some family {μ_j > 0∣j ∈ J} satisfying
(2.16)
(3)
There hold
(2.17)
where
(2.18)

Lemma 2.6. Let 1 < p < r < s < p^*. Then I(u) satisfies the (PS) _c condition with , where S₁ is as in (1.8).

Proof. We will split the proof into three steps.

Step 1. {u_n} is bounded in X.

Let {u_n} be a (PS) _c sequence of I(u) in X, that is,

(2.19)

Then, we have

(2.20)

Since p > 1, (2.20) shows that {u_n} is bounded in X.

Step 2. There exists {u_n} in X such that u_n → u in .

The inequality (1.8) shows that {u_n} is bounded in . Then the above argument and the compactness embedding in Lemma 2.2 mean that the following convergence hold:

(2.21)

It follows from Lemma 2.5 that there exist nonnegative measures μ and η such that

(2.22)

(2.23)

Thus, in order to prove

it is sufficient to prove that η_j = η_∞ = 0.

For the proof of η_j = 0, we define the functional such that

(2.24)

where x_j belongs to the support of dη. It follows from (2.1) that

(2.25)

Since

is bounded, we can get from (1.8)-(1.9), Lemmas 2.3 and 2.5 that

(2.26)

On the other hand,

(2.27)

where B_2ε≜B(x_j, 2ε). Then μ_j = η_j; furthermore, (2.16) implies that μ_j = η_j = 0 or

. We will prove that the later does not hold. Suppose otherwise, there exists some j₀ ∈ J such that

. Then (2.19) and Lemma 2.4 show that

(2.28)

which contradicts the hypothesis of c. Then μ_j = η_j = 0.

Similarly, we define the functional as

(2.29)

Then, the similar proof as above shows that η_∞ = μ_∞ = 0. Thus, we can deduce from (2.22) that

(2.30)

which implies that u_n → u in

Step 3. {u_n} converges strongly in X.

The following inequalities [21] play an important role in our proof:

(2.31)

Our aim is to prove that {u_n} is a Cauchy sequence of X. In fact, let ψ = u_n − u_m in (1.12), it follows from (2.19) that

(2.32)

where

(2.33)

Using the inequalities (2.31), we can get by direct computation that

(2.34)

with some constant c > 0, independent of n and m.

Then the Hölder inequality together with (1.8) and (2.30) yield that

(2.35)

Similarly, we have from the Hölder inequality, Lemmas 2.3 and 2.4 that

(2.36)

Therefore, the above estimates imply that

, that is, {u_n} is a Cauchy sequence of X. Then {u_n} converges strongly in X and we complete the proof.

Similarly, we have the following lemma.

Lemma 2.7. Let 1 < r < p < s < p^*. Then I(u) satisfies the (PS) _c condition with , where S₁, S₂ are as in (1.8), and (1.9) respectively.

Proof. Step 1. {u_n} is bounded in X.

Let {u_n} be a (PS) _c sequence of I(u) in X. Then we have from Lemma 2.3 that

(2.37)

Since 1 < r < p < s, (2.37) shows that ∥u_n∥ is bounded in X.

Step 2. There exists {u_n} in X such that u_n → u in .

Similar to the proof of Lemma 2.5, we can get that μ_j = η_j = 0 or by applying the functional ψ. Now we prove that there is no j₀ ∈ J such that . Suppose otherwise, then

(2.38)

Let

(2.39)

Then q(t) has the unique minimum point at

(2.40)

Then it follows from (2.38) that

(2.41)

which contradicts the hypothesis of c.

Step 3. {u_n} converges strongly in X.

By Lemma 2.4, this result can be similarly obtained by the method in Lemma 2.6, so we omit the proof.

3. Existence of Infinitely Solutions

In this section, we will use the minimax procedure to prove the existence of infinity many solutions of problem (1.1). Let 𝒜 denotes the class of A ⊂ X∖{0} such that A is closed in X and symmetric with respect to the origin. For A ∈ 𝒜, we recall the genus γ(A) which is defined by

(3.1)

If there is no mapping ϕ as above for any m ∈ N, then γ(A) = +∞, and γ(∅) = 0. The following proposition gives some main properties of the genus, see [13, 22].

Proposition 3.1. Let A, B ∈ 𝒜. Then

(1)
if there exists an odd map g ∈ C(A, B), then γ(A) ≤ γ(B),
(2)
if A ⊂ B, then γ(A) ≤ γ(B),
(3)
γ(A⋃ B) ≤ γ(A) + γ(B).
(4)
if S is a sphere centered at the origin in ℝ^N, then γ(S) = N,
(5)
if A is compact, then γ(A) < ∞ and there exists δ > 0 such that N_δ(A) ∈ 𝒜 and γ(N_δ(A)) = γ(A), where N_δ(A) = {x ∈ X : ∥x − A∥≤δ}.

Lemma 3.2. Assume (A₁)–(A₃). Then for any m ∈ N, there exists ε = ε(m) > 0 such that

(3.2)

Proof. For given m ∈ ℕ⁺, let X_m be a m-dimensional subspace of X. If p < r < s < p^*, then for u ∈ X_m we have

(3.3)

The fact that all the norms on finite dimensional space are equivalent implies that for all u ∈ X_m

(3.4)

for some constant c > 0. Then there exist large ρ > 0 and small ε > 0 such that

(3.5)

Denote

(3.6)

Then S_ρ is a sphere centered at the origin with radius of ρ and

(3.7)

Therefore, Proposition 3.1 shows that γ(I^−ε) ≥ γ(S_ρ) = m.

If r < p < s < p^*, we have

(3.8)

Since

is also a norm and all norms on the finite dimensional space X_m are equivalent, we have

(3.9)

Then there exist large σ > 0 and small ε > 0 such that

(3.10)

Denote

(3.11)

Then S_σ is a sphere centered at the origin with radius of σ and

(3.12)

Therefore, Proposition 3.1 shows that γ(I^−ε) ≥ γ(S_σ) = m.

Let 𝒜_m = {A ∈ 𝒜 : γ(A) ≥ m}. It is easy to check that 𝒜_m+1 ⊂ 𝒜_m(m = 1,2, …). We define

(3.13)

It is not difficult to find that

(3.14)

and c_m > −∞ for any m ∈ ℕ since I(u) is coercive and bounded below. Furthermore, we define the set

(3.15)

Then, K_c is compact and we have the following important lemma, see [22].

Lemma 3.3. All the c_m are critical values of I(u). Moreover, if c = c_m = c_m+1 = ⋯ = c_m+τ, then γ(K_c) ≥ 1 + τ.

Proof of Theorem 1.2. In view of Lemmas 2.6 and 2.7, I(u) satisfies the (PS) _c condition in X. Furthermore, as the standard argument of [13, 22, 23], Lemma 3.3 gives that I(u) has infinity many critical points with negative values. Thus, problem (1.1) has infinitely many solutions in X, and we complete the proof.

Acknowledgments

The author would like to express his sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.

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Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

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1. Introduction and Main Results

2. Preliminary Results

3. Existence of Infinitely Solutions

Acknowledgments

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Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

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1. Introduction and Main Results

2. Preliminary Results

3. Existence of Infinitely Solutions

Acknowledgments

References

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