The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

1 Introduction and main results

In this article, we investigate the existence and nonexistence of positive solution for the following Brézis-Nirenberg problem with a logarithmic term:

where Ω ⊂ R N is a bounded open domain, λ , μ ∈ R , N ≥ 3 , and 2 ∗ = 2 N N − 2 is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) . Here, H 0 1 ( Ω ) denotes the closure of C 0 ∞ ( Ω ) equipped with the norm ∥ u ∥ ≔ ( ∫ Ω ∣ ∇ u ∣ 2 d x ) 1 2 .

Our motivation to consider (1.1) is that it resembles some variational problems in geometry and physics, which is lack of compactness. The most notorious example is Yamabe’s problem: finding a function u satisfying

where R ′ is a constant, ℳ is an N -dimensional Riemannian manifold, and Δ denotes the Laplacian and R ( x ) represents the scalar curvature. For some other examples, we refer to [2,8,10, 13–15] and the references therein.

When λ = μ = 0 , (1.1) is reduced to

Pohoǎev [11] asserts that (1.2) has no nontrivial solutions when Ω is starshaped. But, as Brézis and Nirenberg have shown in [3], a lower-order terms can reverse this circumstance. Indeed, they considered the following classical problem:

with λ ∈ R , N ≥ 3 and Ω ⊂ R N is a bounded domain. They found out that the existence of a solution depends heavily on the values of λ and N . Precisely, they showed that:

1. when N ≥ 4 and λ ∈ ( 0 , λ 1 ( Ω ) ) , there exists a positive solution for problem (1.3);

2. when N = 3 and Ω is a ball, problem (1.3) has a positive solution if and only if λ ∈ 1 4 λ 1 ( Ω ) , λ 1 ( Ω ) ;

3. problem (1.3) has no solutions when λ ≤ 0 and Ω is starshaped;

where f ( x , u ) satisfies some of the following assumptions:

1. f ( x , u ) = a ( x ) u + g ( x , u ) , a ( x ) ∈ L ∞ ( Ω ) ;

2. lim u → 0 + g ( x , u ) u = 0 , uniformly in x ∈ Ω ;

3. lim u → + ∞ g ( x , u ) u 2 ∗ − 1 = 0 , uniformly in x ∈ Ω ;

4. ∃ α > 0 such that ∫ ( ∣ ∇ v ∣ 2 − a ( x ) v 2 ) d x ≥ α ∫ v 2 d x for all v ∈ H 0 1 ( Ω ) ;

5. f ( x , u ) ≥ 0 for a.e x ∈ ω 0 and for all u ≥ 0 , where ω 0 is some nonempty open subset of Ω ;

6. f ( x , u ) ≥ δ 0 > 0 for a.e x ∈ ω 0 and for all u ∈ I , where ω 0 is given in ( f 5 ) , I ⊂ ( 0 , + ∞ ) is some nonempty open interval and δ 0 > 0 is some constant;

7. f ( x , u ) ≥ δ 1 u for a.e x ∈ ω 1 and for all u ∈ [ 0 , A ] , or, f ( x , u ) ≥ δ 1 u for a.e x ∈ ω 1 and for all u ∈ [ A , + ∞ ] , where ω 1 is some nonempty open subset of Ω and δ 1 , A are two positive constants;

8. lim u → + ∞ f ( x , u ) u 3 = + ∞ uniformly in x ∈ ω 2 , where ω 2 is some nonempty open subset of Ω .

1. If N ≥ 5 , (1.4) has a positive solution provided ( f 1 ) – ( f 6 ) ;

2. If N = 4 , (1.4) has a positive solution provided ( f 1 ) – ( f 5 ) , and ( f 7 ) ;

3. If N = 3 , (1.4) has a positive solution provided ( f 1 ) – ( f 5 ) , and ( f 8 ) .

To find a positive solution to equation (1.1), we define a modified functional:

which can be rewritten by

where u + = max { u , 0 } , u − = − max { − u , 0 } . It is easy to see that I is well defined in H 0 1 ( Ω ) , and any nonnegative critical point of I corresponds to a solution of equation (1.1).

Before stating our results, we introduce some notations. Hereafter, we use ∫ to denote ∫ Ω d x , unless specifically stated, and let S and λ 1 ( Ω ) be the best Sobolev constant of the embedding H 1 ( R N ) ↪ L 2 ∗ ( R N ) and the first eigenvalue of − Δ with zero Dirichlet boundary value, respectively, i.e.,

and

We also set

and

where

and

Let

Here, comes our main results.

Denote f ( s ) ≔ ∣ s ∣ 2 ∗ − 2 s + λ s + μ s log s 2 , which is of odd. It is easy to see that N ≠ ∅ and c M ≥ c g if problem (1.1) has a positive mountain pass solution. On the other hand, when λ ∈ R and μ > 0 , f ( s ) s is strictly increasing in ( 0 , + ∞ ) and strictly decreasing in ( − ∞ , 0 ) , which enable one to show that c M ≤ c g (see [18, Theorem 4.2]). Therefore, the ground state energy c g equals to the Mountain pass level energy c M , which implies that the mountain pass solution must be a ground state solution. So, in Theorem 1.2, we only need to show that problem (1.1) has a positive mountain pass solution.

The case of μ < 0 is thorny. Indeed for ( λ , μ ) ∈ B 0 ∪ C 0 , I ( u ) still has the mountain pass geometry (see Lemma 2.1). However, in such a case, it holds that c g < c M . Since we can not check the ( PS ) c M condition for I ( u ) , we apply the mountain pass theorem without the ( PS ) c M condition to gain a positive solution for equation (1.1) when ( λ , μ ) ∈ B 0 ∪ C 0 . However, we do not know whether this solution is of mountain pass type.

For the nonexistence of positive solutions for problem (1.1), we have the following partial result.

The existence and nonexistence results given by Theorems 1.2–1.4 can be described on the ( λ , μ ) plane in Figure 1. The pink regions stand for the existence of positive solution, while the blue regions correspond the nonexistence of the positive solution. Here, τ 1 , η 1 , η 2 , and η 3 are curves given by

Figure 1

Existence and nonexistence. (a) N = 3 . (b) N = 4 . (c) N ≥ 5 .

Before closing the introduction, we give the outline of this article. In Section 2, we will check the mountain pass geometry structure for I ( u ) , under different specific situations. We also give some other preliminaries. In Section 3, we are devoted to estimate the mountain pass level c M for different parameters λ , μ , and N . The proofs of our main Theorems 1.2, 1.3, and 1.4 are given in Section 4.

2 Preliminaries

In this section, first we verify the mountain pass geometry structure for I ( u ) when ( λ , μ ) ∈ A 0 ∪ B 0 ∪ C 0 . Second, we show that I ( u ) satisfies ( PS ) d condition provided d < 1 N S N 2 and μ > 0 . Finally, we deduce a existence result for equation (1.1) when c M ∈ 0 , 1 N S N 2 and ( λ , μ ) ∈ B 0 ∪ C 0 .