On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth

1 Introduction and main results

In this article, we are concerned with the existence and multiplicity of solutions to the following problem:

where Ω is a bounded domain in R N with a Lipschitz boundary ∂ Ω ; a : Ω × R N → R N is a Carathéodory function which generalizes ∣ ξ ∣ p ( x ) − 2 ξ , where p ∈ C + ( Ω ¯ ) with

f : Ω × R → R is a Carathéodory function having subcritical growth; t ∈ C + ( Ω ¯ ) with p + < t − ≤ t ( x ) ≤ p ∗ ( x ) for all x ∈ Ω ¯ such that

where p ∗ ( x ) ≔ N p ( x ) N − p ( x ) if p + < N and p ∗ ( x ) ≔ p 0 ( x ) for any fixed p 0 ∈ C + ( Ω ¯ ) with p ( x ) < p 0 ( x ) for all x ∈ Ω ¯ if p + ≥ N ; b ∈ L ∞ ( Ω ) and b ( x ) > 0 for a.e. x ∈ Ω ; and λ , θ are parameters.

Throughout this article, we always assume the following conditions on a :

1. a ( x , − ξ ) = − a ( x , ξ ) for a.e. x ∈ Ω and all ξ ∈ R N .

2. There exists a Carathéodory function A : Ω × R N → R , continuously differentiable with respect to its second variable, such that A ( x , 0 ) = 0 for a.e. x ∈ Ω and a ( x , ξ ) = ∇ ξ A ( x , ξ ) for a.e. x ∈ Ω and all ξ ∈ R N .

3. ∣ a ( x , ξ ) ∣ ≤ C [ 1 + ∣ ξ ∣ p ( x ) − 1 ] for a.e. x ∈ Ω and all ξ ∈ R N , where C is a positive constant.

4. ∣ ξ ∣ p ( x ) ≤ a ( x , ξ ) ⋅ ξ ≤ p ( x ) A ( x , ξ ) for a.e. x ∈ Ω and all ξ ∈ R N .

5. 0 < [ a ( x , ξ ) − a ( x , η ) ] ⋅ ( ξ − η ) for a.e. x ∈ Ω and all ξ , η ∈ R N , ξ ≠ η .

which we call the p ( ⋅ ) -Laplacian and the generalized mean curvature operator, respectively. Another remarkable example is

which satisfies (A0)–(A4) with p ( x ) ≔ max 1 ≤ i ≤ n p i ( x ) , and we call this operator the multiple p ( ⋅ ) -Laplacian. A special case of the multiple p ( ⋅ ) -Laplacian is the operator div ( ∣ ∇ u ∣ p − 2 ∇ u + ∣ ∇ u ∣ q − 2 ∇ u ) with 1 < q ≤ p < ∞ , which is called the ( p , q ) -Laplacian and has received a great attention from mathematicians in the last decade.

The study of p ( ⋅ ) -Laplacian was initiated to describe some materials (e.g., electrorheological fluids) that are inadequate to deal with in a constant p ( ⋅ ) . Indeed, Rǎdulescu [27] gave a couple of examples that required the setting of p ( ⋅ ) -Laplacian problems that portray phenomenon strongly depending on the position. Thus, it is natural to ask for sufficient effects of the ration of p ( ⋅ ) and growth of a nonlinear term in a local sense. It is worth mentioning that the set C is the other condition of locality. Meanwhile, a local effect was also assumed in this study when p ( ⋅ ) is even constant in the sequel papers [4,5,6]. More precisely, the authors were concerned with the “local” growth of nonlinearities, investigating the existence, nonexistence, and multiplicity of solutions for the family of problems

where f is locally “ p -sublinear” at 0 and is locally “ p -superlinear” at ∞ , namely (roughly speaking):

for x in a subdomain Ω 1 of Ω and

for x in a subdomain Ω 2 of Ω . We emphasize that such local assumptions are effective for a nonautonomous nonlinear term and does not allow higher or lower growth than p , respectively. Recently, Komiya and Kajikiya [20] studied the existence of infinitely many solutions to problem (1.1) when a ( x , ξ ) = ∣ ξ ∣ p − 2 ξ + ∣ ξ ∣ q − 2 ξ with 1 < q ≤ p < ∞ and θ = 0 . Indeed, they obtained a sequence of solutions with C 1 -norms converging to 0 (resp. ∞ ) when f satisfies a locally q -sublinear condition, i.e., inf x ∈ B ε ( x 0 ) F ( x , τ ) ∣ τ ∣ q → ∞ as τ → 0 (resp. a locally p -superlinear condition, i.e., inf x ∈ B ε ( x 0 ) F ( x , τ ) ∣ τ ∣ p → ∞ as ∣ τ ∣ → ∞ ) for some ball B ε ( x 0 ) ⊂ Ω . Here and in what follows, F ( x , τ ) ≔ ∫ 0 τ f ( x , s ) d s and B ε ( x 0 ) denotes a ball in R N centered at x 0 with radius ε . It is worth mentioning that they did not treat the case of growth between p and q , which we call sandwich-type growth for the ( p , q ) -problem. These results have recently been obtained for problem (1.1) under (A0)–(A4) in [12]. Note that both [20] and [12] only considered subcritical problems. The existence of infinitely many solutions to critical problems involving the p -Laplacian for this kind of locally p -superlinear condition was investigated in [28].

Regarding critical and sandwich-type growth, in [15] we considered problem (1.1) of the form:

where 1 < q < s < p and the weight m is possibly sign-changing. We obtained the existence of nontrivial nonnegative solutions to [1] for a fixed λ and some range of parameter θ (depending on λ ). In [1], Baldelli et al. obtained the multiplicity of solutions for problem (1.2) when Ω = R N in a reverse way, that is, for θ fixed and small, they find some range of λ . The subcritical problem of sandwich-type growth involving variable exponent was studied by Mihailescu-Rǎdulescu in [25]. In that article, the authors studied (1.1) of the form:

with p , q , s ∈ C + ( Ω ¯ ) satisfying q + < s − ≤ s + < p − and s + < q ∗ ( x ) for all x ∈ Ω ¯ and showed that there exists λ ∗ > 0 such that problem (1.3) admits a nontrivial solution for any λ ∈ [ λ ∗ , ∞ ) .

Motivated by [1,4,5, 6,11,15,20,25,28], our aim is to show the existence of multiple solutions to problem (1.1), which may include a critical term and generalize previous results, in particular [15,25,28] when f satisfies one of three types of growth conditions: locally p ( ⋅ ) -sublinear, locally p ( ⋅ ) -superlinear, and sandwich-type. Moreover, we also obtain a nontrivial nonnegative solution for the sandwich-type case by changing the role of the parameters. We look for solutions to problem (1.1) in W 0 1 , p ( ⋅ ) ( Ω ) , that is the completion of C c ∞ ( Ω ) with respect to the norm

Note that if p + < N and p is logarithmic Hölder continuous (notation: p ∈ C 0 , 1 ∣ log t ∣ ( Ω ¯ ) ), namely,

then the following critical imbedding holds

(see Section 2 for the definitions and properties of the variable exponent Lebesgue-Sobolev spaces). Since we mainly focus on critical problems, throughout this article we always assume p ∈ C 0 , 1 ∣ log t ∣ ( Ω ¯ ) when p + < N in our main results.

Before giving a statement of the main results, we note that under ( A1 ) , we have that for a.e. x ∈ Ω and for all ξ ∈ R N ,

Thus, ( A2 ) yields

where C ˜ is a positive constant. Also, from ( A2 ) we obtain

for a.e. x ∈ Ω and all ξ ∈ R N . In what follows, for r ∈ C ( Ω ¯ ) with r ( x ) ≥ 1 for all x ∈ Ω and an open set D ⊆ Ω , denote

and

First, let us start with the locally p ( ⋅ ) -sublinear case. In this case, we additionally assume for the operator a that:

1. For each ball B ⊂ Ω , there exist constants C B , δ B > 0 , and q B ∈ [ 1 , ∞ ) such that

   ∣ a ( x , ξ ) ∣ ≤ C B ∣ ξ ∣ q B − 1 for a.e. x ∈ B and all ∣ ξ ∣ < δ B .

Assume that f satisfies the following conditions.

1. f : Ω × [ − ε 0 , ε 0 ] → R is a Carathéodory function with an ε 0 > 0 such that f is odd with respect to the second variable and d ( ⋅ ) ≔ sup ∣ τ ∣ ≤ ε 0 ∣ f ( ⋅ , τ ) ∣ ∈ L + σ ( ⋅ ) ( Ω ) with σ ∈ C + ( Ω ¯ ) satisfying σ ( x ) > max 1 , N p ( x ) for all x ∈ Ω ¯ .

2. There exist a ball B ⊂ Ω and m ∈ L + 1 ( B ) such that

   lim τ → 0 F ( x , τ ) m ( x ) ∣ τ ∣ q B = ∞ uniformly for a.e. x ∈ B ,

   where q B is determined as in ( A5 ) .

Next, we study problem (1.1) with a nonlinearity f of locally p ( ⋅ ) -superlinear-type growth. In this case, we need the following additional assumption on the exponent p :

1. There exists w ∈ L + p ∗ ( ⋅ ) p ∗ ( ⋅ ) − p ( ⋅ ) ( Ω ) such that

   (1.9) μ 1 ≔ inf φ ∈ C c ∞ ( Ω ) ∫ Ω ∣ ∇ φ ∣ p ( x ) d x ∫ Ω w ( x ) ∣ φ ∣ p ( x ) d x ∈ ( 0 , ∞ ) .

Now, we state conditions on f for the locally p ( ⋅ ) -superlinear case.

1. f : Ω × R → R is a Carathéodory function such that f is odd with respect to the second variable.

2. There exist functions r j , a j with r j ∈ C + ( Ω ¯ ) , inf x ∈ Ω ¯ [ t ( x ) − r j ( x ) ] > 0 , a j ∈ L + t ( ⋅ ) t ( ⋅ ) − r j ( ⋅ ) b − r j t − r j , Ω ( j = 1 , … , m 0 ), and max 1 ≤ j ≤ m 0 r j + > p − such that

   ∣ f ( x , τ ) ∣ ≤ ∑ j = 1 m 0 a j ( x ) ∣ τ ∣ r j ( x ) − 1 for a.e. x ∈ Ω and all τ ∈ R .

3. There exist an open ball B ⊂ Ω and a function m ∈ L + 1 ( B ) such that

   lim ∣ τ ∣ → ∞ F ( x , τ ) m ( x ) ∣ τ ∣ p B + = ∞ uniformly for a.e. x ∈ B .

4. For w given by (P) , there exist α ∈ [ p + , t − ) and e ∈ L + 1 ( Ω ) such that

   α F ( x , τ ) − τ f ( x , τ ) ≤ α − p + p + w ( x ) ∣ τ ∣ p ( x ) + e ( x ) for a.e. x ∈ Ω and all τ ∈ R .

In the last part, we study problem (1.1) with the nonlinearity f of locally sandwich-type growth. Precisely, we consider problem (1.1) of the form:

where a satisfies (A0)–(A5) and the following assumptions hold:

1. s ∈ C + ( Ω ¯ ) with s + < p − and t p + < t s − .

2. m ∈ L t ( ⋅ ) t ( ⋅ ) − s ( ⋅ ) b − s t − s , Ω .

1. There exists a ball B ⊂ Ω such that q B < s B − and meas { x ∈ B : m ( x ) > 0 } > 0 , where q B is defined by B as in (A5).

Let us start to study the sandwich case by investigating the multiplicity of solutions to problem (1.10). The following multiple existence result is a counterpart of [1, Theorem 1] for the bounded domain case involving Leray-Lions-type operators.

It is worth pointing out that [1, Theorem 1] actually asserted the existence of a sequence of solutions provided the L ∞ norm of K ( x ) ≔ θ b ( x ) is sufficiently small. However, we found that their argument can only produce a finite number of solutions in the same way as the statement of Theorem 1.3 and discussed it with the authors of [1] about this issue; please see Remark 4.4 for more details.

Finally, we investigate a nontrivial nonnegative solution to problem (1.10) changing the role of parameters. For this purpose, in addition to (L1) we assume that

1. There exists ϕ ∈ C c ∞ ( B ) such that ∫ B m ( x ) s ( x ) ϕ + s ( x ) d x > 0 with ϕ + ≔ max { ϕ , 0 } and for each τ > 0 , there exists s B ( τ ) ∈ [ s B − , s B + ] such that

   ∫ B m ( x ) s ( x ) ϕ + s ( x ) τ s ( x ) d x = τ s B ( τ ) ∫ B m ( x ) s ( x ) ϕ + s ( x ) d x .

where

where C p and C B are constants taken from (1.11). Clearly, λ ⋆ is well defined and λ ⋆ ∈ [ 0 , ∞ ) . The existence of a nontrivial nonnegative solution for the sandwich case is shown in the next theorem.

Outline of the article. We organize the article as follows: In Section 2, we review the definition and properties of Lebesgue-Sobolev spaces with variable exponents and introduce notations that are used throughout this article. In Section 3, we investigate the existence of infinitely many solutions for the locally p ( ⋅ ) -sublinear case and the multiplicity of solutions for the locally p ( ⋅ ) -superlinear case by giving proofs of Theorems 1.1 and 1.2. Section 4 is devoted to the study of the existence of k pairs of solutions with an arbitrarily given number k and a nontrivial nonnegative solution to problem (1.1) when the nonlinear f is of a (locally) sandwich type by giving a proof of Theorems 1.3 and 1.4. In this section, we also discuss problem (1.10) when the growth is subcritical. Finally, in Section 5, we comment on the regularity of solutions to problem (1.1) obtained in the previous sections.

2 Notions and preliminary results

Let Ω be a bounded Lipschitz domain in R N . Denote

and for h ∈ C + ( Ω ¯ ) , denote

For p ∈ C + ( Ω ¯ ) and for a Lebesgue measurable and positive a.e. function w : Ω → R , define the weighted variable exponent Lebesgue space L p ( ⋅ ) ( w , Ω ) as

endowed with the Luxemburg norm

Set L + p ( ⋅ ) ( w , Ω ) ≔ { u ∈ L p ( ⋅ ) ( w , Ω ) : u > 0 a.e. in Ω } . When w ≡ 1 , we write L p ( ⋅ ) ( Ω ) , L + p ( ⋅ ) ( Ω ) , and ‖ u ‖ L p ( ⋅ ) ( Ω ) in place of L p ( ⋅ ) ( w , Ω ) , L + p ( ⋅ ) ( w , Ω ) , and ‖ u ‖ L p ( ⋅ ) ( w , Ω ) , respectively. Some basic properties of L p ( ⋅ ) ( w , Ω ) are listed in the next three propositions.

Define