In this paper, we study the following double phase problem with indefinite weight:

where N ≥ 3 , 1 < q < N , q < p < q * = N q N − q , a ∈ C 0 , α ( ℝ N , [ 0 , ∞ ) ) , a ≡ 0 , α ∈ ( 0 , 1 ] , V , K : ℝ N → ℝ are nonnegative continuous functions, and f : ℝ → ℝ is a continuous function with quasicritical growth.

In recent years, quasilinear elliptic equations with nonstandard growth conditions have been widely applied in fields such as nonlinear elasticity, non-Newtonian fluids, strongly anisotropic materials, and phase transition theory. The study of such problems has long been an important direction in the fields of nonlinear partial differential equations and variational analysis, see [1]-[9]. Among various problems with nonstandard growth, the double phase operator initially introduced and systematically investigated by Zhikov has attracted considerable attention, as it can characterize composite materials with both p-growth and q-growth behavior in [10]-[13]. In 1986, he introduced the energy functional

which serves as a prototype for functionals whose ellipticity behavior changes according to the vanishing set of the coefficient a ( x ) , we refer the reader to [14][15]. This model describes strongly anisotropic materials, where the modulation coefficient a ( x ) determines a composite material with two distinct power hardening exponents p and q . In this context, Zhikov discovered a gap phenomenon with further mathematical applications in homogenization theory.

Li et al. [16] investigated ground states for nonlinear Schrödinger equations in ℝ N :

weakened the standard Ambrosetti-Rabinowitz super-linear condition and relaxed the superlinear conditions of nonlinearities to guarantee ground states via Nehari manifold arguments. Later, Alves and Souto [17] further studied Schrödinger equations with weighted nonlinearity K ( x ) f ( u ) and potentials vanishing at infinity in ℝ N :

where N ≥ 3 , the potentials V and K are nonnegative continuous functions, and the nonlinearity f satisfies the quasicritical growth condition. They proved the existence of positive ground state solutions via Hardy-type inequalities and variational arguments. However, the above works only deal with linear Laplace operator and nonnegative weight functions, and ignored the nonstandard growth induced by double phase diffusion.

Ge et al. [18] investigated double phase equations on bounded domains:

where N ≥ 2 , 1 < p < q < N and λ > 0 , they derived the existence of infinitely many solutions via the fountain theorem, dual fountain theorem and Krasnoselskii’s genus theory. Furthermore, researchers have investigated double phase elliptic problems with indefinite weights in ℝ N , and establish the existence and multiplicity of solutions.

Liu and Dai [19] studied multiplicity results for double phase problems in ℝ N :

where 1 < p < q < N , q p ≤ 1 + α N , 0 ≤ a ∈ C 0 , α ( ℝ N ) , α ∈ ( 0 , 1 ] . After weakening the Ambrosetti-Rabinowitz condition, they obtained positive and negative ground state solutions as well as one sign-changing solution with a single sign transition. Meanwhile, they established the existence of infinitely many solutions for even functionals.

Motivated by the above analysis, we discuss the existence and multiplicity results for the problem (1.1). Our primary difficulty of the problem (1.1) is the loss of compactness in ℝ N . To overcome this, we establish the compactness by applying the Alves’s some proof and complete relevant verification with Hardy-type inequalities together with variational methods. Furthermore, we establish the existence of infinitely many solutions via the symmetric mountain pass theorem.

Let us now state the precise hypotheses to problem (1.1) and the main results. Throughout of this paper, we suppose that the potential ( V , K ) ∈ K satisfy the following conditions:

(I) V ( x ) , K ( x ) > 0 ∀ x ∈ ℝ N and K ∈ L ∞ ( ℝ N ) .

(II) If { A n } ⊂ ℝ N is a sequence of Borel sets such that | A n | ≤ R , for all n and some R > 0 , then

lim r → + ∞ ∫ A n ∩ B r c ( 0 ) K ( x ) d x = 0 ,     uniformly   in   n ∈ ℕ . (K₁)

(III) One of the below conditions occurs:

K V ∈ L ∞ ( ℝ N ) (K₂)

or there is s ∈ ( q , q * ) such that

K ( x ) [ V ( x ) ] q * − s q * − q → 0     as   | x | → + ∞ . (K₃)

Related to the function f , we assume the following conditions:

(f₁)

lim sup t → 0 | f ( t ) | t q − 1 = 0 if (K₂) holds,

or

lim sup t → 0 + | f ( t ) | t p − 1 < + ∞ if (K₃) holds.

(f₂) f has a quasicritical growth, that is

(f₃) the cap t ↦ f ( t ) t p − 1 is a non-decreasing function in ( 0 , + ∞ ) and its primitive F is super-p-power at infinity, that is

Now, we present the main results of this paper.

Theorem 1.1Suppose that f satisfies(f₁)-(f₃)and ( V , K ) ∈ K , then problem(1.1)possesses a positive ground state solution.

Theorem 1.2Suppose that f satisfies(f₁)-(f₃)and ( V , K ) ∈ K , and that f ( x , − t ) = − f ( x , t ) , then problem(1.1)admits infinitely many solutions with negative energy values.

This paper is organized as follows. Section 2 lists necessary preliminary results. Section 3 is devoted to prove the energy functional of problem (1.1) satisfying the Cerami condition, further obtain the existence of a positive solution. In Section 4, we prove the existence of infinite many solutions by the symmetric mountain pass theorem.

This section first introduces some notations, and proves that the space X is a separable and uniformly convex Banach space. Similar to reference [20], the space X is given by

endowed with the norm

where

the space D 1 , ξ ( ℝ N ) is defined as the completion of C 0 ∞ ( ℝ N ) under the norm ‖ ∇ u ‖ ξ , see Proposition 2.18(ii) of [21]. Furthermore, we will prove that X is compactly embedded into the weighted Lebesgue space

Next we present the energy functional associated with problem (1.1) is given by

for all u ∈ X . Thanks to the regularity assumptions imposed on f , we obtain J ∈ C 1 ( X , ℝ ) , and its Gateaux derivative is

It is easy to check that critical points of J are weak solutions of problem (1.1). Once that we intend to find positive solution, we will assume that

Accordingly, there is a sequence { u n } ⊂ X such that

where c is the mountain pass level given by

and construct the set

The above sequence is the so-called Cerami sequence for J at level c , arguing as in Willem [22].

Now, we are ready to show our main results of continuous embedding theorem, see the reference [23] for more details.

Lemma 2.1Let ξ : ℝ N × [ 0 , ∞ ) → [ 0 , ∞ ) , and ρ ξ ( u ) = ∫ ℝ N ( a ( x ) | u | p + | u | q ) d x . Then, the embedding L ξ ( ℝ N ) ↪ L q ( ℝ N ) is continuous.

As a consequence of Lemma 2.1 and Sobolev embedding in W 1 , q ( ℝ N ) and D 1 , q ( ℝ N ) for 1 < q < N , we have the following embedding result.

Lemma 2.2Let ξ : ℝ N × [ 0 , ∞ ) → [ 0 , ∞ ) , ρ ξ ( u ) = ∫ ℝ N ( a ( x ) | u | p + | u | q ) d x . Then, the embedding W 1 , ξ ( ℝ N ) ↪ W 1 , q ( ℝ N ) ↪ L r ( ℝ N ) is continuous, for any q ≤ r ≤ q * . Moreover, the embedding D 1 , ξ ( ℝ N ) ↪ D 1 , q ( ℝ N ) ↪ L q * ( ℝ N ) is continuous.

Next, we present some properties of space X .

Lemma 2.3The space X = ( X , ‖   ⋅   ‖ X ) is a separable and uniformly convex Banach space.

Proof. We first verify that X is complete. Let { u n } be a Cauchy sequence in X , then for any ε > 0 , there exists N > 0 such that if n , m ≥ N , we have

By the definition of X , we know that

By the completeness of D 1 , ξ ( ℝ N ) , there exists u ∈ D 1 , ξ ( ℝ N ) such that u n → u in D 1 , ξ ( ℝ N ) , we can extract a subsequence { u n k } such that u n k → u a.e. in ℝ N as k → ∞ . Therefore, by the Fatou’s Lemma and (2.2), we obtain

which implies that u ∈ X . Let n ≥ N , then we get

that is u n → u in X . Hence, the space X is complete.

Next, we prove that X is uniformly convex. Fix ε > 0 , let u , v ∈ X with ‖ u ‖ X = ‖ v ‖ X = 1 and ‖ u − v ‖ X ≥ ε . For q ≥ 2 , by the following inequality (see [24])

we have

Then we have

Taking δ = 1 − [ 1 − ( ε 2 ) q ] 1 / q > 0 , we obtain that

For 1 < q < 2 , let q ′ = q q − 1 . We can write

In view of (2.4) and the reverse Minkowski inequality (see [24] Theorem 2.13), we deduce

By the following inequality (see [24])

it follows from (2.5) that

Then we have

Taking δ = 1 − [ 1 − ( ε 2 ) q ′ ] 1 / q ′ , we can get

Hence, the space X is uniformly convex.

Finally, we show that X is separable in four steps.

Step 1: We establish the basic density property. The space C 0 ∞ ( ℝ N ) is dense in X ,i.e., every element in X can be approximated by mollifiers:

That is, for any u ∈ X , there exists a sequence { φ n } ⊂ C 0 ∞ ( ℝ N ) such that

Step 2: Proof of separability. Consider the map T : X → L a p ( ℝ N ) × L q ( ℝ N ) × L V q ( ℝ N ) , defined by

where L a p ( ℝ N ) = { v : ∫ ℝ N a ( x ) | v | p d x < ∞ } , and L V q ( ℝ N ) = { v : ∫ ℝ N V ( x ) | v | q d x < ∞ } . Then T is an isometric embedding of X into the product space L a p ( ℝ N ) × L q ( ℝ N ) × L V q ( ℝ N ) . Since 1 < p , q < ∞ , each of the weighted Lebesgue spaces L a p , L q , and L V q is separable. Therefore their product is separable. Any subspace of a separable metric space is separable, and hence X is separable. The uniform convexity follows from the uniform convexity of L a p , L q , and L V q together with the fact that the norm ‖   ⋅   ‖ X is equivalent to the product norm. This completes the proof.

Proposition 3.1(Hardy-type inequality)Assume that ( V , K ) ∈ K hold. Then, X is compactly embedded in L K m ( ℝ N ) for all m ∈ ( q , q * ) if(K₂)holds. If(K₃)holds, X is compactly embedded in L K s ( ℝ N ) .

Proof. The proof will be made into two parts, we first derive the conclusion under condition (K₂), then handle the case for (K₃).

Suppose (K₂) holds. Fix m ∈ ( q , q * ) , for any ε > 0 , there exist 0 < t 0 < t 1 and C > 0 such that

Hence,

where

and

If { v n } is a sequence such that v n ⇀ v in X , then there exists a constant M 1 > 0 satisfying

this implies that the sequence { Q ( v n ) } is bounded. On the other hand, let

the last inequality implies that

showing that sup n ∈ ℕ | A n | < + ∞ . Therefore, by virtue of (K₁), there exists r > 0 such that

Now, (3.2) and (3.3) lead to

Since m ∈ ( q , q * ) and K is a continuous function, Sobolev embeddings imply that

Combining (3.4) and (3.5), we have

which gives

We now assume that (K₃) holds. First of all, it is important to observe that for any fixed x ∈ ℝ N , the function

attains its minimum value C p V ( x ) q * − s q * − q , where

Hence,

Combining the last inequality with (K₃), for any ε ∈ ( 0 , C p ) , there exists r > 0 sufficiently large, such that

which implies

If { v n } is a sequence such that v n ⇀ v in X , there is M 1 > 0 such that

consequently,

Since p ∈ ( q , q * ) and K is a continuous function, the Sobolev embeddings yield

Combining (3.8) and (3.9), we obtain

which implies

Thus, we completes the proof of the proposition.

The following lemma provides a key step for establishing the boundedness of the Cerami sequence given in (2.1).

Lemma 3.2Assume that f satisfies(f₁)-(f₃)and ( V , K ) ∈ K . Let { v n } be a sequence such that v n ⇀ v in X . Then

and

Proof. We start the argument under condition (K₂). Given fixed m ∈ ( q , q * ) and ε > 0 , there is C > 0 , the assumptions (f₁)-(f₃) yield that

By Proposition 3.1,

then there is r > 0 such that

Since { v n } is bounded in X , there is M 1 > 0 such that

Combining the last inequalities with (3.10) and (3.11),

Now, if (K₃) holds, we employ the repeating the same arguments explored in the proof of Proposition 3.1. For sufficiently small ε > 0 , taking r > 0 large enough such that

Hence,

From (f₁) and (f₂), there are C , t 0 , t 1 > 0 verifying

where I = { t ∈ ℝ : | t | < t 0   or   | t | > t 1 } . Thus, for any u ∈ X , we have the following estimate

with

and

Once that { v n } is bounded in X , there is M 1 > 0 such that