Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition

1 Introduction and main results

In this article, we mainly investigate the following quasilinear elliptic equation with Neumann boundary conditions:

where Ω ⊂ R n is a bounded domain with C 1 boundary, n ≥ 3 , p * = n p n − p , p ♯ = ( n − 1 ) p n − p , 1 < p < n , Δ p u = div ( ∣ ∇ u ∣ p − 2 ∇ u ) is the p -Laplacian operator, and ν denotes the unit outward normal to ∂ Ω . The function f ( x , u ) satisfies the following assumptions:

1. f ∈ C ( R , R + ) , f ( x , 0 ) = 0 , and f ( x , t ) ≡ 0 for all x ∈ Ω if t < 0 ;

2. lim u → 0 f ( x , u ) u p − 1 = 0 uniformly for x ∈ Ω ;

3. there exists a constant γ ∈ ( p , p ♯ ) such that lim u → ∞ f ( x , u ) u γ − 1 = 0 uniformly for x ∈ Ω ;

4. f ( x , t ) t − p F ( x , t ) ≥ 0 for all t ≥ 0 and x ∈ Ω , where F ( x , t ) = ∫ 0 t f ( x , s ) d s .

which is a C 1 functional in W 1 , p ( Ω ) , where d σ is the measure on the boundary. Critical points of J are weak solutions to (1.1), namely, they satisfy

for all v ∈ W 1 , p ( Ω ) .

In the last few decades, elliptic problems involving critical Sobolev nonlinearities with Neumann boundary conditions have been extensively investigated (see [3,4,6,8,10,13,15,16,19,22,24] and references therein). The majority of these studies focus on nonlinearities with subcritical growth in the boundary conditions [3,6,8,13,15,16,22,24]. A natural question is whether there exists a positive solution when the Sobolev critical nonlinearities appear in both the equation and the boundary condition.

The early work addressing this issue was conducted by Deng et al. [10]. They considered the Neumann problem

and proved that (1.4) admits a positive solution when λ < 0 . Pierotti and Terracini [20] investigated another type of Neumann problem:

in which the existence of a positive solution was also demonstrated when δ ≥ 0 and g ( x , u ) satisfies the growth condition ∣ g ( x , u ) ∣ ≤ C 1 + C 2 ∣ u ∣ θ − 1 for some θ < 2 ♯ . For further exploration, we refer readers to [4] and [19].

Recently, the following critical Neumann problem

was investigated in [7,9,14], where α , β ∈ { 0 , 1 } , λ ∈ R , and f ( u ) is a nonlinear term. The main characteristic of equation (1.6) is that at least one critical term is involved. If α = β = 1 , λ ∈ R , and f ( u ) = 0 , Ferreira et al. [14] obtained that equation (1.6) has a positive solution for λ ∈ ( n 2 − δ , n 2 ) and has no positive solutions for λ ∈ ( − ∞ , n 4 ) ∪ [ n 2 , + ∞ ) , where δ > 0 is a small constant. Deng and Shi complemented and improved the existence result in [7,14]. More precisely, they found a lower bound Λ * ∈ [ n 4 , n 2 ) depending on n such that equation (1.6) when α = β = 1 and f ( u ) = 0 , has a positive solution if λ ∈ ( Λ * , n 2 ) . Particularly, they also estimated that Λ * ∈ ( n 4 , n − 2 2 ) if n ≥ 5 and Λ * = n 4 if n = 4 , which gives the best lower bound of λ for the existence of a positive solution for (1.6) if n = 4 .

However, despite the fact that critical elliptic equations with critical Neumann boundary condition, for example (1.4) and (1.5) and (1.6), have been studied for 30 years, there are no existing results that address problems involving the p -Laplacian operator with critical exponents in both the equation and the boundary conditions. This gap in the literature is the main motivation for us to consider problem (1.1).

Let R + n = { x = ( x ′ , x n ) : x ′ ∈ R n − 1 , x n ≥ 0 } denote the upper half n -dimensional Euclidean space. Note that both p * and p ♯ are the critical Sobolev exponents for the embedding W 1 , p ( R + n ) ↪ L p * ( R + n ) and W 1 , p ( R + n ) ↪ L p ♯ ( ∂ R + n ) , respectively. For any given positive constants a , b , one can define a sharp constant S a , b p as follows:

where

for all p > 1 .

The sharp constant S a , b p has been extensively studied, including the search for its extremal functions. When b = 0 , a > 0 , and p > 1 , the extremal function of S a , 0 p can be deduced readily from the work of Aubin [1] and Talenti [21]. When a = 0 , b > 0 and p > 1 , the extremal function of S 0 , b p was obtained by Nazaret [18] and Escobar [12]. In the case a and b are positive, the situation for p = 2 was investigated by Nazaret [11] using conformal invariance argument. The general cases for a > 0 , b > 0 and p ≠ 2 remain open.

Recently, Maggi and Neumayer considered the following variational problem:

in [17]. Employing the mass transportation argument developed by Nazaret et al. [5], they have the following result:

Taking α ∈ ( 1 a + b ( T E ) p , 1 a ) and multiplying both the numerator and the denominator of (1.7) by it, we obtain

where T α = ( 1 − a α b α ) 1 ⁄ p . From Theorem A and (1.11), the following corollary follows.

Let u ( x ′ , x n ) = C U ε , x 0 , t 0 ( x ′ , x n ) for any given ε > 0 , x 0 ∈ R n − 1 and C > 0 . A standard variational argument demonstrates that u satisfies

where ν = ( 0 , − 1 ) ∈ R n − 1 × R denotes the unit outward normal to ∂ R + n . Multiplying the boundary equation in (1.12) on both sides by u and integrating over ∂ R + n , we then obtain

Equation (1.13) indicates that the constant t 0 < 0 depends only on a , b , n , and p . Particularly, if b = 0 , it follows from (1.13) that t 0 = 0 . We then obtain the well-known result [1,21] that the Sobolev constant

is achieved by the function

up to dilations for ε > 0 and translations for x 0 ∈ R n − 1 .

As an application, we shall employ the extremal function of S a , b p to establish the following result concerning the existence of weak solutions for equation (1.1).

To complete the proof of Theorem 1.1, we adopt the method established by Brézis and Nirenberg [2]. After obtaining the local compactness result for the functional J ( u ) , the difficulty is to show that the mountain pass level of the functional belongs to the compact range. This is performed by taking the minimizers of S a , b p as test functions and performing some delicate estimates on the asymptotic behavior of the test functions (see Lemmas 3.1 and 3.2 for further details).

For simplicity of notation, similar to (1.8), we will denote the integration over Ω and its boundary ∂ Ω by

respectively, for all p > 1 . We consider our problem in the Sobolev space W 1 , p ( Ω ) equipped with the norm defined by

We will omit p and write S a , b instead of S a , b p throughout this article.

The structure of this article is as follows. In Section 2, we determine the energy threshold for λ > 0 . In Section 3, we first verify that the energy functional J ( u ) satisfies the (PS) c condition, and then use the Mountain Pass Lemma to establish the existence of a positive solution for (1.1), provided λ > 0 . Finally, we demonstrate the nonexistence of a positive solution when λ ≤ 0 by a contradiction argument.

2 Energy threshold

In this section, we shall analyze the critical energy of functional J ( u ) to satisfy the classical Palais-Smale (PS) condition when λ > 0 . Toward this purpose, we require some additional notations. We set

where

and d being a constant large enough so that J ( t γ ( 1 ) ) < 0 for all t > 1 . It follows from ( H 1 ) – ( H 3 ) that for any ε > 0 , there exists a constant C ε > 0 such that

Applying the Sobolev inequalities, we then have

Taking ε > 0 small enough such that 1 ⁄ p − ε > 0 , we then conclude from (2.2) that c > 0 .

For θ ∈ ( 0 , 1 ] , in what follows, we define

then S 1 = S . It follows from Corollary 1.1 that S θ is achieved by

for any ε > 0 . For some t 0 < 0 , a standard argument shows that U ε , 0 , t 0 ( x ′ , x n ) solves the following system:

Analysis similar to (1.13) shows that

For any τ > 0 , we define

where

Simple computations show that ψ ε , τ satisfies

Define the constant

which is independent of ε . Then, ψ ε , τ is a minimizer of S θ with

In the following discussion, we denote

and

The following lemma shows that the constant A can be attained by some function.