Existence results for Schrödinger type double phase variable exponent problems with convection term in $\mathbb R^{N}$

1.   Introduction

The study of differential equations and variational problems with double phase operators is a new and interesting topic. Originally, in order to investigate the Lavrentiev phenomenon from strongly anisotropic materials, Zhikov ^[1] introduced the following functional

where the function $\mu(\cdot)$ was used as an aid to regulate the mixture between two different materials, with power hardening of rates $p$ and $q$, respectively. Since then, many scholars studied double phase problems and obtained abundant theoretical achievements.

In ^[2], Colasuonno and Squassina studied an eigenvalue problem of double phase variational integrals and proved some properties of the Musielak-Orlicz space for the first time. Liu and Dai ^[3] investigated the following problem

By variational methods, they verified various existence and multiplicity results. Furthermore, they also obtained some essential properties of double phase operators, which has been applied to many double phase problems. For Eq (1.1), the existence of solutions has also been studied by applying Morse theory ^[4]. In ^[5,6,7], the authors consider a double phase problem in $\mathbb{R}^{N}$ with reaction terms, which does not satisfy the Ambrosetti-Rabinowitz condition. They derived some existence results based on various variational methods. For more related results on the double phase problem, one can refer to ^{[8,9,10,11,12]} and references therein.

If nonlinearity $h$ also depends on the gradient $\nabla\upsilon$, such functions are usually called convection terms. Its presence increases the difficulty of the double phase problem because the gradient dependent term is non-variational. In ^[13], Gasinski and Winkert considered the following convection problem

They discussed the existence of weak solutions by using the theory of the pseudomonotone operator. The same methodology can be found in reference ^[14,15]. In addition, the methods for dealing with the existence of solutions to elliptic equations with convection terms also included Galerkin method ^[16,17], Brezis theorem ^[18], and Leray-Schauder alternative principle ^[19,20].

So far, there are only few results involving the variable exponent double phase operator. In ^[21], the authors considered double phase problems with variable exponent for the first time and established a suitable function spaces. Moreover, we refer to the recent results ^[22,23] for the existence of constant sign solutions and the existence results in complete manifolds, and to ^[24,25] for the study of the double phase problem with concave-convex nonlinearities or Baouendi-Grushin type operators. To our knowledge, no work has established the results for Schrödinger equations in $\mathbb R^{N}$, which involves double phase operators and convection terms. Enlightened by the above literatures, we discuss this kind of equation as follows

where $V:\mathbb R^{N}\rightarrow \mathbb R^{+}$ is a potential function, $h:\mathbb R^{N}\times \mathbb R \times \mathbb R^{N}\rightarrow \mathbb R$ is a Carath$\acute{\text{e}}$odory function, and $0\leq\lambda(\cdot)\in L^{1}(\mathbb{R}^{N})$. $p(x), q(x)\in C_{+}(\mathbb{R}^{N})$ such that $1 < p(x) < N$, $p(x) < q(x) < p^{*}(x)$ and $\frac{q(x)}{p(x)} < 1+\frac{1}{N}$. The Sobolev critical exponent is defined by

and also defines

For each $y(x)\in C_{+}(\mathbb R^{N})$, we denote

Throughout the paper, we consider equations $(H_{V})$ under some assumptions for the potential function $V$ and Carath$\acute{\text{e}}$odory function $h$.

$\rm{(V)}:$ $V(x)\in C(\mathbb{R}^{N})$ and there exists $V_{0} > 0$ such that

where $S_{1}(z) = \{x\in\mathbb{R}^{N}: |x-z|\leq1\}$, $S_{a}(z)$ denotes a ball of radius $a$ with center $z$.

$\rm{(H)}:$ There exist a nonnegative function $\gamma\in L^{1}(\mathbb{R}^{N})\cap L^{p'(x)}(\mathbb{R}^{N})$ and constants $d_{1}, d_{2}\geq0$ with $\max\left\{d_{2}-\frac{d_{2}}{p^{+}}, \frac{d_{1}p^{-}+d_{2}}{V_{0}p^{-}}\right\} < 1$ such that

The condition (V) was introduced by ^[26] to guarantee compactness of the embedding of the Sobolev space into Lebesgue space. Another condition on function $V$ is used in the literature ^[27], which satisfies

It is worth noting that the condition (1.3) is stronger than (V) (see ^[28]). In this paper, we will prove a new embedding theorem for the variable exponents Sobolev space in $\mathbb R^{N}$ under weaker assumption (V). In addition, we cannot implement the usual critical point theory due to the equation $(H_{V})$ not having a variational structure. Our main innovation is the first study of double phase variable exponent problems with convection terms by using Galerkin methods together with the topological degree theorem.

The outline of this article is as follows. In Section 2, we collect some necessary definitions and basic lemmas of Musielak-Orlicz space and corresponding Sobolev space. In Section 3, we present some classes of mappings and topological degree theory. We obtain the existence of strong generalized solutions and weak solutions in Sections 4 and 5, respectively. Finally, a conclusion is given in Section 6.

2.   Preliminaries

In this section, we first review some known results of Lebesgue and Sobolev spaces with the variable exponent, which will be used later.

Let the variable exponent Lebesgue spaces be defined as

endowed with the Luxemburg norm

where $\varrho_{p(x)}(\upsilon): = \int_{\mathbb R^{N}}|\upsilon|^{p(x)}dx$ is called modular and $p'(x)$ denotes the conjugate function of $p(x)$. Also, $W^{1, p(x)}(\mathbb R^{N})$ stands for the corresponding Sobolev spaces. Define a linear subspace of $W^{1, p(x)}(\mathbb R^{N})$ as

equipped with the norm

The spaces $L^{p(x)}(\mathbb R^{N})$, $W^{1, p(x)}(\mathbb R^{N})$, and $W$ are separable reflexive Banach spaces (see ^[27,29]).

Next, we introduce a new function space used in our study and give some of its properties. Let

Obviously, $H\in N(\mathbb{R}^{N})$ is locally integrable (see ^[24]).

The Musielak-Orlicz space $L^{H}(\mathbb{R}^{N})$ is given by

endowed with the Luxemburg norm

Lemma 2.1. ^[21] Suppose that $\varrho_{H}(\upsilon) = \int_{\mathbb{R}^{N}} (|\upsilon|^{p(x)}+\lambda(x)|\upsilon|^{q(x)})dx$. For $\upsilon\in L^{H}(\mathbb{R}^{N})$, we have

(i) $\; \chi = \|\upsilon\|_{H}$ if, and only if, $\varrho_{H}(\frac{\upsilon}{\chi}) = 1$;

(ii) $\; \|\upsilon\|_{H} < 1\Rightarrow \|\upsilon\|_{H}^{q^{+}}\leq \varrho_{H}(\upsilon) \leq \|\upsilon\|_{H}^{p^{-}}$;

(iii) $\; \|\upsilon\|_{H} > 1\Rightarrow \|\upsilon\|_{H}^{p^{-}}\leq \varrho_{H}(\upsilon) \leq \|\upsilon\|_{H}^{q^{+}}$;

(iv) $\; \|\upsilon\|_{H} < 1\; ( = 1; > 1) \Leftrightarrow \varrho_{H}(\upsilon) < 1\; ( = 1; > 1).$

The corresponding Sobolev spaces are given by

endowed with the norm

Moreover, in order to study problems $(H_{V})$, we consider the following space

with the equivalent norm

The modular $\varrho:E\rightarrow \mathbb{R}$ is given by

Analogy to the proof of Proposition 2.13 in ^[21], we have the following connection between modular and norm $\|\cdot\|$.

Lemma 2.2. Suppose that $\upsilon_{n}, \upsilon\in E$, then

(ⅰ) $\; \chi = \|\upsilon\|$ if, and only if, $\varrho(\frac{\upsilon}{\chi}) = 1$;

(ⅱ) $\; \|\upsilon\| < 1\Rightarrow \|\upsilon\|^{q^{+}}\leq \varrho(\upsilon) \leq \|\upsilon\|^{p^{-}}$;

(ⅲ) $\; \|\upsilon\| > 1\Rightarrow \|\upsilon\|^{p^{-}}\leq \varrho(\upsilon) \leq \|\upsilon\|^{q^{+}}$;

(ⅳ) $\; \|\upsilon\| < 1\; ( = 1; > 1) \Leftrightarrow \varrho(\upsilon) < 1\; ( = 1; > 1))$;

(ⅴ) $\; \lim_{n\rightarrow \infty}|\upsilon_{n}-\upsilon| = 0 \Leftrightarrow \lim_{n\rightarrow \infty}\varrho(\upsilon_{n}-\upsilon) = 0.$

Theorem 2.1. $L^{H}(\mathbb{R}^{N})$, $W^{1, H}(\mathbb{R}^{N})$, and $E$ are separable reflexive Banach spaces.

Proof. Since $H\in N(\mathbb{R}^{N})$ is locally integrable and the Lebesgue measure on $\mathbb{R}^{N}$ is $\sigma$-finite and separable, then $L^{H}(\mathbb{R}^{N})$ is a separable Banach space that follows from (^[30], Theorems 7.7 and 7.10). By Proposition 2.12 of ^[21], we know that $H$ is uniformly convex. Note that

which satisfies the condition $(\Delta_{2})$. Thus, $L^{H}(\mathbb{R}^{N})$ is uniformly convex, follows from (^[30], Theorem 11.6), and is a reflexive space based on the Milman-Pettis theorem. Similar to the proof of Theorem 2.7 (ⅱ) in ^[5], we obtain $W^{1, H}(\mathbb{R}^{N})$ as a separable reflexive Banach space and $E$ as a closed subspace of $W^{1, H}(\mathbb{R}^{N})$.

We present the following embedding results. For convenience, the notation $\rightharpoonup$$(\rightarrow)$ is means weak (strong) convergence and the symbol $\hookrightarrow$$(\hookrightarrow\hookrightarrow)$ denotes the continuous (compact) embedding, respectively.

Theorem 2.2. Assume that (V) holds and $\mu(x)\in C_{+}(\mathbb{R}^{N})$ satisfies $p(x)\leq\mu(x) < p^{*}(x)$, then the spaces $W$ are continuously compact embedded in $L^{\mu(x)}(\mathbb{R}^{N})$.

Proof. (ⅰ) First, we discuss the case $\mu(x) = p(x)$ and suppose that $\upsilon_{n}\rightharpoonup 0$ in $W$. If (V) holds, the embedding $W\hookrightarrow \hookrightarrow L^{p(x)}(S_{R}(0))$ holds (^[27], Proposition 2.4). So, we only show that for any $\varepsilon > 0$, there exists $R > 0$ such that

Note that $\{\upsilon_{n}\}_{n\in\mathbb{N}}$ is a bounded sequence in $W$. Set $\rho = \|\upsilon_{n}\|_{W}^{p^{+}}+\|\upsilon_{n}\|_{W}^{p^{-}}$ and choose an arbitrary number $s\in(1, \frac{N}{N-p^{-}})$, then $p(x) < sp(x) < p^{*}(x)$. Using Proposition 2.4 of ^[27] again, there is a constant $Q > 0$ such that

Let $\{z_{i}\}_{i}\subset \mathbb{R}^{N}$ such that $\bigcup_{i = 1}^{\infty}S_{1}(z_{i}) = \mathbb{R}^{N}$ and every $x\in \mathbb{R}^{N}$ is covered by at most $2^{N}$ such balls. Denote

Thus

By the Hölder inequality and (2.1), we get

Hence,

Now, we choose $b$ large enough such that $2^{N+1}\rho\leq b\varepsilon$. As is shown in ^[26], the $meas\left(Y(z)\right)\rightarrow0$ for $|z|\rightarrow \infty$, then we can find that $R' > 0$ satisfies

For the above $R'$,

Therefore, $\upsilon_{n}\rightarrow 0$ in $L^{p(x)}(\mathbb{R}^{N})$.

(ⅱ) For $p(x) < \mu(x) < p^{*}(x)$, there exists $\sigma(x)\in(0, 1)$ such that $\frac{1}{\mu(x)} = \frac{\sigma(x)}{p(x)}+\frac{1-\sigma(x)}{p^{*}(x)}$, then we have

According to the embedding $W\hookrightarrow L^{p^{*}(x)}(\mathbb{R}^{N})$ and $\{\upsilon_{n}\}$ is bounded in $W$, we get

From (ⅰ) and (2.2), we have

This means that $\upsilon_{n}\rightarrow 0$ in $L^{\mu(x)}(\mathbb{R}^{N})$. The proof is complete.

Theorem 2.3. Suppose that (V) holds and $p(x)\leq \theta(x)\leq p^{*}(x)$ for $x\in \mathbb{R}^{N}$. Thus, the embedding $L^{H}(\mathbb{R}^{N})\hookrightarrow L^{p(x)}(\mathbb{R}^{N})$ and $E\hookrightarrow L^{\theta(x)}(\mathbb{R}^{N})$ holds. Moreover, $E\hookrightarrow\hookrightarrow L^{\theta(x)}(\mathbb{R}^{N})$ also holds whenever $p(x)\leq\theta(x) < p^{*}(x)$. This implies there exists $C_{\theta} > 0$ such that

Proof. Let $H_{p} = t^{p}$, then $H_{p} < H$. Thus, applying Theorem 10.3 of ^[30], we obtain

From Theorem 2.2, we have $W\hookrightarrow\hookrightarrow L^{\theta(x)}(\mathbb{R}^{N})$, so $E\hookrightarrow\hookrightarrow L^{\theta(x)}(\mathbb{R}^{N})$ for $p(x)\leq \theta(x) < p^{*}(x)$. Using again the Theorem 10.3 of ^[30], we get $E\hookrightarrow W^{1, H}(\mathbb R^{N})\hookrightarrow W^{1, p(x)}(\mathbb R^{N})\hookrightarrow L^{p^{*}(x)}(\mathbb{R}^{N})$.

Before stating our main results, we need to present the corresponding definitions.

Definition 2.1. Let $E$ be a real reflexive Banach space with dual $E^{*}$. A mapping $\mathcal{L}:E\rightarrow E^{*}$ is said to be

(ⅰ) of class $(S_{+})$, if for each $\{\upsilon_{n}\}\in E$ with $\upsilon_{n}\rightharpoonup \upsilon$ and $\limsup_{n\rightarrow \infty}\langle \mathcal{L}\upsilon_{n}, \upsilon_{n}-\upsilon\rangle\leq0$, then $\upsilon_{n}\rightarrow \upsilon$ in $E$;

(ⅱ) quasimonotone, if for each $\{\upsilon_{n}\}\in E$ with $\upsilon_{n}\rightharpoonup \upsilon$, we have $\limsup\langle \mathcal{L}\upsilon_{n}, \upsilon_{n}-\upsilon\rangle\geq0$.

Definition 2.2. We say that $\upsilon\in E$ is a weak solution of problems $(H_{V})$, if

for any $\varphi \in E$, where $-\Delta_{\lambda}^{V}$ denotes the double phase type operator as

Define functional $J:E\rightarrow R$ as

Obviously, $J\in C^{1}(E, \mathbb{R})$ (see ^[21]). We denote $\mathcal{L} = J':E\rightarrow E^{*}$, then

Lemma 2.3. The operator $\mathcal{L}:E\rightarrow E^{*}$ has the properties, such as continuous, bounded, strictly monotone, homeomorphism, and of type $(S_{+})$.

Proof. (a) Since $\mathcal{L} = J'$ and $J\in C^{1}$, then $\mathcal{L}$ is continuous. For all $\eta_{1}, \eta_{2}\in \mathbb R^{N}$ with $\eta_{1}\neq\eta_{2}$, by the well-known inequality

we obtain

which implies that $\mathcal{L}$ is strictly monotone. Next, we show that $\mathcal{L}$ is bounded. Let $\chi_{1} = \|\upsilon\|$, $\chi_{2} = \|\phi\|$ and $L = max\left\{\chi_{1}^{p^{-}-1}, \chi_{1}^{q^{+}-1}\right\}$. From Hölder's inequality and Young's inequality, we obtain

thus, we have

Hence, $\mathcal{L}$ is bounded.

(b) Let $\{\upsilon_{n}\}_{n\in \mathbb{N}}\subseteq E$ such that

By the monotonicity of $\mathcal{L}$, we get

then

that is

In view of (2.6), $\nabla \upsilon_{n}$ and $\upsilon_{n}$ converge in measure to $\nabla \upsilon$ and $\upsilon$ in $\mathbb{R}^{N}$, respectively. Without loss of generality, let $\nabla \upsilon_{n}\rightarrow \nabla \upsilon$ and $\upsilon_{n}\rightarrow \upsilon$ a.e., on $\mathbb{R}^{N}$. Based on the Fatou lemma, we obtain

Noting that $\lim_{n\rightarrow \infty}\langle\mathcal{L}(\upsilon), \upsilon_{n}-\upsilon\rangle = 0$, then $\limsup_{n\rightarrow \infty}\langle\mathcal{L}(\upsilon_{n}), \upsilon_{n}-\upsilon\rangle\leq0$. According to Young's inequality, we also obtain

From this and (2.8), we get

Let $f(\upsilon) = \frac{1}{p(x)}|\nabla \upsilon|^{p(x)}+\frac{\lambda(x)}{q(x)}|\nabla \upsilon|^{q(x)}+ \frac{V(x)}{p(x)}|\upsilon|^{p(x)}+\frac{\lambda(x)V(x)}{q(x)}|\upsilon|^{q(x)}$. The Vitali theorem yields the uniform integrability of the sequence $\{f(\upsilon_{n})\}_{n\in \mathbb{N}}$. On the other hand,

which means the sequence $\left\{|\nabla \upsilon_{n}-\nabla \upsilon|^{p(x)}+\lambda(x)|\nabla \upsilon_{n}-\nabla \upsilon|^{q(x)}+ V(x)|\upsilon_{n}-\upsilon|^{p(x)}+\lambda(x)V(x)|\upsilon_{n}-\upsilon|^{q(x)}\right\}_{n\in \mathbb{N}}$ is also uniformly integrable. Applying the Vitali theorem, it follows that

Hence, $\upsilon_{n}\rightarrow \upsilon$ in $E$.

(c) Since $\mathcal{L}$ is strictly monotone, $\mathcal{L}$ is an injection, and

$\mathcal{L}$ is coercive. In view of the Minty-Browder Theorem, $\mathcal{L}$ has an inverse mapping $\mathcal{L}^{-1}:E^{*}\rightarrow E$. Next, we prove that $\mathcal{L}^{-1}$ is continuous to ensure $\mathcal{L}$ is homeomorphism.

If $\varpi_{n}, \varpi\in E^{*}$, $\varpi_{n}\rightarrow \varpi$, let $\upsilon_{n} = \mathcal{L}^{-1}(\varpi_{n})$, $\upsilon = \mathcal{L}^{-1}(\varpi)$, then $\mathcal{L}(\upsilon_{n}) = \varpi_{n}$, $\mathcal{L}(\upsilon) = \varpi$. Note that $\{\upsilon_{n}\}$ is bounded in $E$ by the coercivity of $\mathcal{L}$. Without loss of generality, assume that $\upsilon_{n}\rightharpoonup \upsilon_{0}$. It follows from $\varpi_{n}\rightarrow \varpi$ that

Thus, $\upsilon_{n}\rightarrow \upsilon_{0}$ in $E$, as $\mathcal{L}$ is of type $(S_{+})$. Moreover, form $\mathcal{L}(\upsilon_{0}) = \lim_{n\rightarrow \infty}\mathcal{L}(\upsilon_{n}) = \lim_{n\rightarrow \infty}\varpi_{n} = \varpi$, we have $\upsilon_{n}\rightarrow \upsilon$. Therefore, $\mathcal{L}^{-1}$ is continuous.

3.   Some classes of mappings and topological degree

Let $E$ be a real separable reflexive Banach space. $E^{*}$ is its dual space and denote by $\langle \cdot, \cdot \rangle$ its duality pairing. For a nonempty subset $M$ of $E$, $\overline{M}$ and $\partial M$ denote the closure and the boundary of $M$.

Definition 3.1. Let $F$ be another real Banach space. A mapping $\mathcal{L}:M\subset E\rightarrow F$ is called

(ⅰ) bounded, if it takes any bounded set into a bounded set,

(ⅱ) demicontinuous, if for any $\{\upsilon_{n}\}\in M$, $\upsilon_{n}\rightarrow \upsilon$ implies $\mathcal{L}(\upsilon_{n})\rightharpoonup \mathcal{L}(\upsilon)$.

Definition 3.2. Let $T:M_{1}\subset E\rightarrow E^{*}$ be a bounded operator such that $M\subset M_{1}$ and any operator $\mathcal{L}:M\subset E\rightarrow E$. If for any $\{\upsilon_{n}\}\in M$ with $\upsilon_{n}\rightharpoonup \upsilon$, $\omega_{n}: = T\upsilon_{n}\rightharpoonup \omega$ and $\limsup\langle \mathcal{L}\upsilon_{n}, \omega_{n}-\omega\rangle\leq0$, we have $\upsilon_{n}\rightarrow \upsilon$, then $\mathcal{L}$ satisfies condition $(S_{+})_{T}$.

For each $M\subset E$, define the following types of operators

$\mathcal{L}_{1}(M)$ : = $\{\mathcal{L}:M\rightarrow E^{*} |\mathcal{L}\text{ is demicontinuous}, \text{ bounded, and satisfies condition}(S_{+})\}$,

$\mathcal{L}_{T}(M)$ : = $\{\mathcal{L}:M\rightarrow E | \mathcal{L} \text{ is demicontinuous and satisfies condition}(S_{+})_{T}\}$,

$\mathcal{L}_{T, B}(M)$ : = $\{\mathcal{L}:M\rightarrow E | \mathcal{L} \text{ is demicontinuous}, \text{ bounded, and satisfies condition}(S_{+})_{T}\}$,

$\mathcal{L}(E)$ : = $\{\mathcal{L}\in\mathcal{L}_{T, B}(\overline{M}) | M\in \Theta, \ T\in \mathcal{L}_{1}(\overline{M})\}$,

where $\Theta$ denotes the collection of all bounded open sets in $E$ and $T\in \mathcal{L}_{1}(\overline{M})$ is called an essential inner map to $\mathcal{L}$.

Lemma 3.1. ^[31] Let $M\subset E$ be a bounded open set. Suppose that $T\in\mathcal{L}_{1}(\overline{M})$ is continuous and $K:E_{k}\subset E^{*}\rightarrow E$ is demicontinuous such that $T(\overline{M})\subset E_{k}$, then the following properties hold

(i) If $K$ is quasimonotone, then $I+K\circ T\in\mathcal{L}_{T}(\overline{M})$, where $I$ stands for the identity operator.

(ii) If $K$ is class of $(S_{+})$, then $K\circ T\in\mathcal{L}_{T}(\overline{M})$.

Definition 3.3. Assume that $M\subset E$ is a bounded open set, $T\in\mathcal{L}_{1}(\overline{M})$ is continuous, and $\mathcal{L}, K\in\mathcal{L}_{T}(\overline{M})$. Define affine homotopy $\mathcal{H}:[0, 1]\times \overline{M}\rightarrow E$ as

where it is called an admissible affine homotopy with the common continuous essential inner map $T$ and it satisfies condition $(S_{+})_{T}$ (see ^[31]).

Now, we give the topological degree for the class $\mathcal{L}(E)$.

Theorem 3.1. There exists a unique degree function

which satisfies the properties such as normalization, additivity, homotopy invariance, and existence (see ^[31,32]).

4.   Existence of weak solutions

Lemma 4.1. Under assumptions (H), the operator $K:E\rightarrow E^{*}$ given by

is compact.

Proof. Define an operator $\varphi:E\rightarrow L^{p'(x)}(\mathbb{R}^{N})$ as

We prove that $\varphi$ is bounded and continuous.

For every $\upsilon\in E$, using the embedding $E\hookrightarrow \hookrightarrow L^{p(x)}(\mathbb{R}^{N})$ and condition (H), we obtain

where $C > 0$ stands for arbitrary constant, which means that $\varphi$ is bounded on $E$.

Let $\upsilon_{n}\rightarrow \upsilon$ in $E$, then $\upsilon_{n}\rightarrow \upsilon$ in $L^{p(x)}(\mathbb{R}^{N})$ and $\nabla\upsilon_{n}\rightarrow \nabla\upsilon$ in $\left(L^{p(x)}(\mathbb{R}^{N})\right)^{N}$. Thus, there exist a subsequence $\{\upsilon_{k}\}_{k\in\mathbb{N}}$ of $\{\upsilon_{n}\}_{n\in\mathbb{N}}$, and measurable functions $\vartheta\in L^{p(x)}(\mathbb{R}^{N})$ and $\zeta\in\left(L^{p(x)}(\mathbb{R}^{N})\right)^{N}$ satisfy

for any $k\in\mathbb{N}$ and a.e. $x\in \mathbb{R}^{N}$. Since $h$ is a Carath$\acute{\text{e}}$odory function, we get

It follows from (H) that

for any $k\in\mathbb{N}$ and a.e. $x\in \mathbb{R}^{N}$. Note that

According to (4.2), (4.3), and the dominated convergence theorem, we have

that is,

Therefore, the entire sequence $\varphi\upsilon_{n}$ converges to $\varphi\upsilon$ in $L^{p'(x)}(\mathbb{R}^{N})$. Thus, $\varphi$ is continuous.

Recall that the embeding $I:E\hookrightarrow\hookrightarrow L^{p(x)}(\mathbb{R}^{N})$. It is known that the adjoint $I^{*}:L^{p'(x)}(\mathbb{R}^{N})\hookrightarrow\hookrightarrow E^{*}$. Hence, we conclude that the composition $K = I^{*}\circ \varphi$ is compact.

Theorem 4.1. Assume that condition (H) hold. Then problem $(H_{V})$ has a weak solution in $E$.

Proof. Due to the Lemma 4.1 and the definition of the operator $\mathcal{L}$, we have that $\upsilon\in E$ is a weak solution of problem $(H_{V})$ when, and only when,

By the proof of Lemmas 2.3 and 4.1, we known that the inverse operator $T = \mathcal{L}^{-1}$ is continuous, bounded, and of type $(S_{+})$, and the operator $K$ is continuous, bounded, and quasimonotone.

Therefore, Eq (4.4) is equivalent to

Next, we solve Eq (4.5) with degree theory. First, we prove that the set

is bounded. Indeed, let $\xi\in A$. Set $\upsilon = T\xi$, then $\|\upsilon\| = \|T\xi\|$.

(ⅰ) If $\|\upsilon\|\leq 1$, then $\|T\xi\|$ is bounded.

(ⅱ) If $\|\upsilon\| > 1$, then

Now, we choose $\varsigma = 1-\max\left\{d_{2}-\frac{d_{2}}{p^{+}}, \frac{d_{1}p^{-}+d_{2}}{V_{0}p^{-}}\right\} > 0$, then by embedding $E\hookrightarrow L^{p(x)}(\mathbb{R}^{N})$, we obtain

Thanks to the assumption $p^{-} > 1$, $\|T\xi\|$ is bounded, which means $\{T\xi|\xi\in A\}$ is bounded.

Moreover, the boundedness of operator $K$ and (4.5) implies the set $A$ is bounded in $E$. Therefore, there exists $a > 0$ such that

This means that

From Lemma 3.1, we conclude that

and $I+K\circ T$ is also bounded due to that the operators $I$, $K$, and $T$ are bounded. It follows that

Next, discuss a homotopy $\mathcal{H}:[0, 1]\times \overline{S_{a}(0)}\rightarrow E^{*}$ as

Based on the normalization property and homotopy invariance of the degree $d$ in Theorem 3.1, we have

Thus, there exists a point $\xi\in S_{a}(0)$ that satisfies equation

which means that $\upsilon = T\xi$ is a weak solution of problem $(H_{V})$.

5.   Existence of strong generalized solutions

Since the Banach space $E$ is separable, we can find a Galerkin basis of $E$, which means a sequence $\{E_{n}\}_{n\in\mathbb{N}}$ of vector subspaces of $E$ with

First, we introduce the notion of the strong generalized solution, then we derive the existence of strong generalized solutions for the problem $(H_{V})$ based on the Galerkin method. Our approach is largely inspired by ^[16].

Definition 5.1. A function $\upsilon\in E$ is a strong generalized solution to equation $(H_{V})$, if there exists a sequence $\{\upsilon_{n}\}_{n\in\mathbb{N}}\subseteq E$ satisfying the following statements

(ⅰ) $\upsilon_{n}\rightharpoonup \upsilon$ in $E$, as $n\rightarrow \infty$;

(ⅱ) $-\Delta_{\lambda}^{V}\upsilon_{n}-h(\cdot, \upsilon_{n}(\cdot), \nabla\upsilon_{n}(\cdot))\rightharpoonup 0$ in $E^{*}$, as $n\rightarrow \infty$;

(ⅲ) $\lim_{n\rightarrow \infty}\langle-\Delta_{\lambda}^{V}\upsilon_{n}, \upsilon_{n}-\upsilon\rangle = 0.$

Lemma 5.1. Assume that assumption (H) holds. One has the inequality

for any $\upsilon, \xi \in E$, where $C_{1}, C_{2}$ is shown below.

Proof. Using condition (H), we obtain

where

Lemma 5.. Let $(E, \|\cdot\|)$ be a normed finite dimensional space and $B:E\rightarrow E^{*}$ be a continuous map. Suppose that there exists some $\delta > 0$, which satisfies

then $B(\upsilon) = 0$ has a solution $\upsilon\in E$ with $\|\upsilon\|\leq \delta$.

Theorem 5.1. Suppose that condition (H) is satisfied, then for all $n\in \mathbb{N}$ and $\psi\in E_{n}$, there exists $\upsilon_{n}\in E_{n}$ such that

Proof. For every $n\in \mathbb{N}$, we define the operator $B_{n}: E_{n}\rightarrow E_{n}^{*}$ by

for every $\upsilon, \psi\in E_{n}$. From (H) and (4.6), we have the following estimate

If $\|\upsilon\| > 1$, then there exists $\delta > 0$ large enough. Whenever $\upsilon\in E_{n}$ with $\|\upsilon\| = \delta$, such that

since $p^{-} > 1$ and $\varsigma > 0$. In view of Lemma 5.2, the equation $B_{n}(\upsilon) = 0$ has an approximate solution $\upsilon_{n}\in E_{n}$, which is (5.1). The proof is complete.

Lemma 5.3. If condition (H) holds, then the sequence $\{\upsilon_{n}\}_{n\in\mathbb{N}}$ with $\upsilon_{n}\in E_{n}$ constructed in Theorem 5.1 is bounded in $E$.

Proof. If $\|\upsilon_{n}\|\leq1$ for any $n\in\mathbb{N}$, then $\{\upsilon_{n}\}_{n\in\mathbb{N}}$ is bounded in $E$. So, when $\|\upsilon_{n}\| > 1$ for any $n\in\mathbb{N}$, insert $\psi = \upsilon_{n}$ in (5.1), then we have

Based on hypotheses (H) and (4.6), we obtain

Recalling that $p^{-} > 1$ and $\varsigma > 0$, we conclude that the desired conclusion.

Theorem 5.2. If conditions (H) holds, then equation $(H_{V})$ has a strong generalized solution in $E$.

Proof. We know that $\{\upsilon_{n}\}$ is bounded in $E$ by Lemma 5.2. Since $E$ is reflexive, then

Lemma 5.1 indicates that the Nemytskii operator $N_{h}:E\rightarrow E^{*}$ given by

is well defined, and we can find that constants $C_{1}, C_{2} > 0$ satisfy

Thus, the Nemytskii operator $N_{h}$ is bounded. In association with (5.2), then

The boundedness of the operator $-\Delta_{\lambda}^{V}:E\rightarrow E^{*}$ implies that

Again, by the reflexivity of $E^{*}$, we obtain

for some $\kappa\in E^{*}$.

Let $\zeta\in\cup_{i = 1}^{\infty}E_{n}$, then we can find $m\in \mathbb{N}$ such that $\zeta\in E_{m}$. So, Theorem 5.1 implies that equality (5.1) is ture for any $n\geq m$. As $n\rightarrow \infty$ in (5.1), then

Since $\zeta\in\cup_{i = 1}^{\infty}E_{n}$ is dense in $E^{*}$, then we deduce that $\kappa = 0$. Therefore, (5.5) becomes

Next, choose $\psi = \upsilon_{n}$ in (5.1), that is,

In addition to this, from (5.6) we have

Combining (5.7) and (5.8), we get

From Lemma 5.1, choosing the test function $\xi = \upsilon_{n}-\upsilon$, we get

Due to the compact embeddings $E\hookrightarrow \hookrightarrow L^{p(x)}(\mathbb{R}^{N})$, we deduct that $\upsilon_{n}\rightarrow \upsilon$ in $L^{p(x)}(\mathbb{R}^{N})$. The $\{\upsilon_{n}\}_{n\in\mathbb{N}}$ is bounded in $E$ and, hence, in $L^{p(x)}(\mathbb{R}^{N})$. Also, $\{\nabla\upsilon_{n}\}_{n\in\mathbb{N}}$ is bounded in $L^{p(x)}(\mathbb{R}^{N})$. This implies that

Consequently, (5.4) gives us

Obviously, (5.2), (5.6) and (5.11) show that $\upsilon\in E$ is a strong generalized solution to equation $(H_{V})$. This completes the proof.

Corollary 5.1. Assume that the equation $(H_{V})$ has a strong generalized solution $\upsilon\in E$ stated in Theorem 5.2, then $\upsilon\in E$ is a weak solution to equation $(H_{V})$. The same holds in the opposite sense.

Proof. If $\omega\in E$ is a strong generalized solution to equation $(H_{V})$, then

which means $\upsilon_{n}\rightarrow \upsilon$ in $E$, since $-\Delta_{\lambda}^{V}$ fulfills the $(S_{+})$-property. Using again Definition 5.1, we have

and it follows that

Thus, $\upsilon\in E$ is a weak solution of equation $(H_{V})$ (see (2.3)). If posing $\upsilon = \upsilon_{n}$, it is clear that any weak solution is a strong generalized solution for equation $(H_{V})$.

Remark 5.1. Note that for the problem $(H_{V})$, each weak solution is a generalized solution. However, a generalized solution does not necessarily derive the notion of weak solution. For the definition of a generalized solution, one can refer to ^[16].

6.   Conclusions

In this article, we study a class of Schrödinger equations in $\mathbb{R}^{N}$. One of the main features of the paper is the presence of a new double phase operator with variable exponents. We give the corresponding Musielak-Orlicz Sobolev spaces and compact embedding result. Another significant characteristic of the paper is the presence of convection term. Based on the topological degree theory and Galerkin method, we not only obtain the existence of strong generalized solutions, but also the existence of weak solutions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Postgraduate Research Practice Innovation Program of Jiangsu Province under Grant (KYCX23-0669) and the Doctoral Foundation of Fuyang Normal University under Grant (2023KYQD0044).

Conflict of interest

The authors declare no conflicts of interest.