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Existence and multiplicity of solutions for fractional \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations with Robin boundary conditions
Boundary Value Problems volume 2024, Article number: 37 (2024)
Abstract
In this paper, we study fractional \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland’s variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in \(\mathbb{R}^{N}\setminus \overline{\Omega}\) for fractional order \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.
1 Introduction and the main results
In this paper, we consider fractional \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations, including concave-convex nonlinearities with nonlocal Robin boundary conditions
where \(V_{i}(x)\) (\(x\in \Omega \), \(i=1,2\)) is a potential function, \(\Omega \subset \mathbb{R}^{N}\) (\(N\geq 2\)) is a bounded domain with the Lipschitz boundary ∂Ω, \(s\in (0,1)\), \(p_{i}(x,\cdot ):\mathbb{R}^{2N}\rightarrow (1,+\infty )\), \(\overline{p}_{i}(x)=p_{i}(x,x)\), \(r_{1}(x)\), \(r_{2}(x)\) are continuous functions, \(\lambda _{1}\), \(\lambda _{2}\) are positive constants, \(A_{1}(x)\), \(A_{2}(x)\) are positive weighted functions, \(g_{i}(x)\geq 0\in L^{1}(\mathbb{R}^{N}\setminus \Omega )\), \(\beta (x)\geq 0\in L^{\infty}(\mathbb{R}^{N}\setminus \Omega )\),
and
where \(P.V\). stands for the Cauchy principal value.
Equations (1) arise from general reaction-diffusion equation
where \(A(\varphi )=|\nabla \varphi |^{p-2}+|\nabla \varphi |^{q-2}\). Problem (2) has applications in biophysics, plasma physics, and chemical reactions. For more details on equation (2), readers are referred to [1, 2]. Combining with a \(\mathbb{Z}_{2}\)-symmetric version of the mountain pass lemma for even functionals and some adequate variational methods, Mihăilescu [3] proved that the equations
have infinitely many weak solutions. In addition, Chung and Toan [4] considered a class of fractional Laplacian problems
using variational techniques and Ekeland’s variational principle. The authors used the variational techniques to discuss the results of the existence of solutions in fractional cases [5–7]. In addition, Heidarkhani et al. [8–10] studied the existence results of variable exponent equations using variational methods and established the critical point theory. Zuo et al. [11] investigated the existence and multiplicity of solutions for the \(p(x,\cdot )\& q(x,\cdot )\) fractional Choquard problems with variable order. On a similar issue, a related study was conducted by Biswas et al. For more details, see [12].
The classical Schrödinger equation is of the following form:
where V, φ denote the potential function and wave function, respectively, and i, h are constants ([13]). Recently, Xiang et al. in [14] and Bu et al. in [15] discussed the fractional Laplace operator Schrödinger equations with variable order and Schrödinger–Kirchhof-type equations, respectively.
The critical local problem involving concave-convex nonlinearities was first studied by Ambrosetti et al. in [16]. Subsequently, variational methods were used [17] to discuss the following equations:
with the variable order concave-convex term. For other similar types of equations, see [18, 19] and the references therein.
The Robin and Neumann boundary problems are interesting topics [20]. Mugnai et al. [21] investigated fractional p-Laplacian problems with nonlocal Neumann boundary conditions. Moreover, Deng [22] considered the following equations:
For double-phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian, Manouni et al. [23] proved the existence of solutions by variational tools, truncation techniques, and comparison methods. In many papers, the Robin and Neumann boundary problems of fractional equations were studied in different ways; e.g., the Morse theory was used in [24, 25], the mountain pass lemma in [26, 27], Ekeland’s variational principle in [28, 29], and the topological degree in [30].
To our knowledge, there is no previous work on the problem (1). This paper is devoted to this topic. We obtain new results by applying the mountain pass lemma, Ekeland’s variational principle, the fountain theorem, and the Krasnoselskii genus theory. Our problem differs from problems (3), (4), and (5) in that we discuss Robin boundary conditions, and it also differs from problem the (6) in that we consider \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations with concave-convex nonlinearities.
Before stating the main results, we introduce the basic assumptions.
-
(P)
\(p_{i}(x,y)\) is a symmetric and continuous function, that is,
$$ p_{i}(x,y)=p_{i}(y,x), \quad \text{for all } (x, y)\in \mathbb{R}^{N} \times \mathbb{R}^{N} $$with
$$ 1< p_{i}^{-}:=\min_{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}p_{i}(x,y) \leq p_{i}(x,y)\leq p_{i}^{+}:=\max _{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}p_{i}(x,y)< +\infty , $$and
$$ 1< p_{1}^{-}\leq p_{1}^{+}\leq p_{2}^{-}\leq p_{2}^{+}< +\infty , $$such that \(sp_{i}^{+}< N\). Let \(0< s<1<p(x,\cdot )\), the fractional critical exponent \(p^{*}_{s}(x)\) be defined as \(p^{*}_{s}(x)=\frac{Np(x,x)}{N-sp(x,x)}\) and \(p(x,\cdot )< p^{*}_{s}(x)\) for all \(x\in \overline{\Omega}\).
-
(G)
\(\int g_{i}(x)\varphi \,dx=-\int g_{i}(x)\varphi \,dx\).
-
(V)
\(V_{i}(x)\) is a continuous function, satisfying \(\inf_{x\in \Omega}V_{i}(x)>V_{i0}>0\), for all \(d_{i}>0\), \(\mathrm{means} (\{x \in \Omega :V_{i}(x)< d_{i}\})<+\infty \).
-
(H)
\(A_{1}(x)\) and \(A_{2}(x)\) are weighted functions in \(C(\overline{\Omega})\) and satisfy \(A_{1}(x)\in L^{s_{1}(x)}(\Omega )\) such that \(1< s_{1}(x) \in C(\overline{\Omega})\) and \(1< s'_{1}(x)r_{1}(x)< p^{*}_{s}(x)\) for all \(x\in \overline{\Omega}\), \(A_{2}(x)\in L^{s_{2}(x)}(\Omega )\) such that \(1< s_{2}(x) \in C(\overline{\Omega})\) and \(1< s'_{2}(x)r_{2}(x)< p^{*}_{s}(x)\) for all \(x\in \overline{\Omega}\). Here, \(s'_{1}(x)\) and \(s'_{2}(x)\) are conjugate exponents of the functions \(s_{1}(x)\) and \(s_{2}(x)\), respectively.
The main results of this paper are as follows:
Theorem 1.1
Assume that assumptions (P), (G), (V), and (H) hold. Equations (1) have two nontrivial weak solutions.
Theorem 1.2
Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) have infinitely many nontrivial weak solutions in X.
Theorem 1.3
Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) possess infinitely many solutions.
In Sect. 2, we state some basic results of the Lebesgue space \(L^{q(x)}(\Omega )\). In Sect. 3, we introduce the workspaces associated with equations (1). In Sect. 4, we verify the \((PS)\) conditions and prove Theorem 1.1 by the mountain pass lemma and Ekeland’s variational principle. In Sect. 5, we prove Theorem 1.2 by applying the fountain theorem. Finally, using the Krasnoselskii genus theory, we give the proof of Theorem 1.3.
2 Preliminaries
In this section, we recall some basic results of the Lebesgue space \(L^{q(x)}(\Omega )\) with a variable exponent. Assume that domain Ω is bounded in \(\mathbb{R}^{N}\) with the Lipschitz boundary ∂Ω. Let
where \(C_{+}(\overline{\Omega})= \{q\in C(\overline{\Omega}): q(x)>1, \text{for all } x\in \overline{\Omega} \}\).
The variable exponent Lebesgue space \(L^{q(x)}(\Omega )\), which is defined by
equipped with the Luxemburg norm
where \((L^{q(x)}(\Omega ), \|\cdot \|_{q(x)})\) is a separable, uniformly convex, and reflexive Banach space [31].
Let \(L^{q'(x)}(\Omega )\) be the conjugate space of \(L^{q(x)}(\Omega )\) and \(1/q(x)+1/q'(x)=1\) (\(p(x)\) and \(q'(x)\) are conjugate indices to each other). For \(\varphi \in L^{q(x)}(\Omega )\) and \(v\in L^{q'(x)}(\Omega )\), the Hölder inequality
holds. If \(q_{i}(x)\in C_{+}(\overline{\Omega})\) (\(i=1,2,\ldots ,\widehat{n}\)) and
for all \(\varphi _{i}(x)\in L^{q_{i}(x)}(\Omega ) \), there exists
Lemma 2.1
([32]) Let \(\rho _{q(x)}\) be the modular of the \(L^{q(x)}(\Omega )\) space, and \(\rho _{q(x)}: L^{q(x)}(\Omega )\rightarrow \mathbb{R}\) defined by \(\rho _{q(x)}(\varphi )=\int _{\Omega}|\varphi (x)|^{q(x)}\,dx\). Then, the following properties hold:
-
(i)
\(\|\varphi \|_{q(x)}<1(=1, >1) \Longleftrightarrow \rho _{q(x)}( \varphi )<1 (=1, >1)\);
-
(ii)
;
-
(iii)
.
Lemma 2.2
([32]) If \(\varphi , \varphi _{n}\in L^{q(x)}(\Omega )\) with \(n\in \mathbb{N}\), then
-
(i)
\(\lim_{n\rightarrow +\infty}\|\varphi _{n}-\varphi \|_{q(x)}=0\);
-
(ii)
\(\lim_{n\rightarrow +\infty}\rho _{q(x)}(\varphi _{n}-\varphi )=0\);
-
(iii)
\(\varphi _{n}(x)\rightarrow \varphi (x)\) a. e. in Ω and \(\lim_{n\rightarrow +\infty}\rho _{q(x)}(\varphi _{n})=\rho _{q(x)}( \varphi ) \).
Lemma 2.3
([33]) Let \(p(x)\), \(q(x)\) be measurable functions such that \(p(x)\in L^{\infty}(\mathbb{R}^{N})\) and \(1< p(x)q(x)<\infty \), for any \(x\in \mathbb{R}^{N}\). Then, there is
with \(\varphi \in L^{q(x)}(\mathbb{R}^{N})\), \(\varphi \neq 0\).
3 The basic properties of functionals and operators
In this section, we state some properties of functionals and operators, and give the definition of weak solutions of equations (1) with Robin boundary conditions. We first introduce the workspaces \((W,\|\cdot \|_{W})\) and \((X,\|\cdot \|_{X})\) associated with equations (1).
The fractional variable Sobolev space \(W:=W^{s,q(x),p(x,\cdot )}(\Omega )\) is given by
Set
as the variable exponent Gagliardo seminorm. W is a Banach space with the norm
We take into account three continuous functions \(p(x,y):\overline{\Omega}\times \overline{\Omega}\rightarrow (1, \infty )\) and \(r_{1}(x), r_{2}(x)\in C_{+}({\overline{\Omega}})\). From condition (P), we know that
Lemma 3.1
([34]) Suppose that \(\Omega \subset \mathbb{R}^{N}\) is a bounded open domain and (8) holds. Then, W is a separable and reflexive space.
Lemma 3.2
([35]) Let smooth bounded domain \(\Omega \subset \mathbb{R}^{N}\), \(sp(x, y)< N\) for \((x, y) \in \overline{\Omega}\times \overline{\Omega}\) with \(s\in (0,1)\), and \(q(x)\geq p(x, x)\) for \(x \in \overline{\Omega}\). Suppose that continuous function \(\widehat{h}(x):\overline{\Omega}\rightarrow (1,\infty )\) satisfies
There exists a positive constant \(C_{0}=C_{0}(N,s,p,\widehat{h},\Omega )\) such that for every \(\varphi \in W\), it holds that
Then, the embedding \(W \hookrightarrow L^{\widehat{h}(x)}\) for all \(\widehat{h}\in (1, p^{*}_{s})\) is compact.
Lemma 3.3
([35, 36]) If \(1< sp^{-}\) and
There exists a positive constant \(C_{1}=C_{1}(N,s,p,\widehat{h},\partial \Omega )\) such that
Then, the embedding \(W^{s,\overline{p}(x),p(x,y)}(\Omega )\hookrightarrow L^{\widehat{h}(x)}( \partial \Omega )\) is compact.
Define nonlinear map \(\mathcal{L}:W\rightarrow W^{*}\)
for all \(\varphi , \psi \in W\), \(\mathcal{L}\) has the following properties.
Lemma 3.4
([28])
-
(i)
\(\mathcal{L}\) is a bounded and strictly monotone operator;
-
(ii)
\(\mathcal{L}\) is a mapping of \((S_{+})\), i.e., if \(\varphi _{n}\rightharpoonup \varphi \) in W and \(\lim_{n\rightarrow \infty}\sup \langle \mathcal{L}(\varphi _{n})- \mathcal{L}(\varphi ), \varphi _{n}-\varphi \rangle \leq 0\), then \(\varphi _{n}\rightarrow \varphi \) in W;
-
(iii)
\(\mathcal{L}:W\rightarrow W^{*}\) is a homeomorphism.
Define function \(S:W\rightarrow \mathbb{R}\)
which is related to (9). The derivative of S is
for all \(\varphi ,\psi \in W\); for more details, see [34].
Let
equipped with the norm
where
and
with \(\Omega ^{c}=\mathbb{R}^{N}\backslash \Omega \).
Lemma 3.5
([28]) Assume that assumptions (P), (G), and (V) hold. Then, \((X_{i},\|\cdot \|_{X_{i}})\) is a reflexive Banach space.
The norm \(\|\cdot \|_{X_{i}}\) on \(X_{i}\) is equivalent to
where the modular \(\rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}:X_{i} \rightarrow \mathbb{R}\) is defined by
Lemma 3.6
Assume that assumptions (P), (G), and (V) hold. The following properties hold:
-
(i)
\(\|\varphi \|_{X_{i}}<1(=1, >1) \Longleftrightarrow \rho _{s,p_{i}(x, \cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}(\varphi )<1 (=1, >1)\);
-
(ii)
;
-
(iii)
;
-
(iv)
\(\rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}( \varphi -v)\rightarrow 0 \Leftrightarrow \|\varphi -v\|_{X_{i}} \rightarrow 0\).
Let \(X=X_{1}\cap X_{2}\) with norm \(\|\varphi \|_{X}=\|\varphi \|_{X_{1}}+\|\varphi \|_{X_{2}}\), which is a separable and reflexive Banach space. The dual space of X is \(X^{*}\). The modular \(\rho _{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}= \rho _{s,p_{1}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}+ \rho _{s,p_{2}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}\). We have the following result.
Lemma 3.7
([28]) Assume that assumptions (P), (G), and (V) hold. Then, from (10), the following properties hold:
-
(i)
The function \(\rho _{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2})}\) is of class \(C^{1}(X,\mathbb{R})\);
-
(ii)
The strictly monotone operator \(\rho '_{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}: X \rightarrow X^{*}\) is coercive, then
$$ \frac{\langle \rho '_{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}},\varphi \rangle _{X}}{ \Vert \varphi \Vert _{X}}\rightarrow +\infty , \qquad \Vert \varphi \Vert _{X} \rightarrow +\infty ; $$ -
(iii)
\(\rho '_{s,p(x, \cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}\) is a mapping of type \((S_{+})\), that is, if \(\varphi _{n}\rightharpoonup \varphi \) in X and \(\lim \sup_{n\rightarrow +\infty} \langle \rho '_{s,p(x,\cdot ), \mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}},\varphi \rangle _{X} \leq 0\), then \(\varphi _{n}\rightarrow \varphi \) in X.
Lemma 3.8
([35, 36]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any \(\widehat{r}\in C_{+}(\overline{\Omega})\) with \(1<\widehat{r}(x)<p_{s}^{*}(x)\) for all \(x\in \Omega \), there is a positive constant \(\varpi ^{*}=\varpi ^{*}(s,p_{i},N,\widehat{r},\Omega )>0\) such that
Moreover, this embedding is compact.
Lemma 3.9
([35]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any \(\widehat{r}\in C_{+}(\mathbb{R}^{N}\backslash \Omega )\) with \(1<\widehat{r}(x)<p_{\partial}^{*}(x)\) for all \(x\in \mathbb{R}^{N}\backslash \Omega \), there is a positive constant \(\widehat{\varpi}^{*}=\widehat{\varpi}^{*}(s,p_{i},N,\widehat{r}, \partial \Omega )>0\) such that
Moreover, this embedding is compact.
More precisely, we now present the divergence theorem and the analogous formula for the partition integral formula in nonlocal case [37].
Lemma 3.10
([29]) Let the hypotheses (P) hold, and let φ be any bounded \(C^{2}\)-function in \(R^{N}\). Then,
Lemma 3.11
Let the hypotheses (P) hold. Suppose that φ and v are bounded \(C^{2}\)-functions in \(\mathbb{R}^{N}\). Then,
for every \(v\in X\).
Proof
According to symmetry, we obtain
□
Lemma 3.12
Assuming that assumption (P) holds and letting φ be a weak solutions of equations (1), we have
Lemma 3.13
Assuming that assumptions (P), (G), (V), and (H) hold, let \(I_{\lambda}: X \rightarrow \mathbb{R}\) be a energy function defined by
for every \(\varphi \in X\). Then, any critical point of \(I_{\lambda}\) is a weak solution of equations (1).
4 Proof of Theorem 1.1
To prove Theorem 1.1, we need a well-known mountain pass lemma.
Theorem 4.1
Let X be a real Banach space and \(I_{\lambda}\in C^{1}(X, \mathbb{R})\) with \(I_{\lambda}(0)=0\). Assume that the following conditions hold:
-
(i)
\(I_{\lambda}\) satisfies \((PS )\) conditions;
-
(ii)
there exist \(\rho , \sigma >0\) such that \(I_{\lambda}(\varphi )\geq \sigma \), for all \(\varphi \in X\), with \(\|\varphi \|_{X}=\rho \);
-
(iii)
there exists \(\upsilon \in X\), satisfying \(\|\upsilon \|_{X}>\rho \) such that \(I_{\lambda}(\upsilon )<0\).
Then, \(I_{\lambda}\) has a critical value \(c>\sigma \), that is,
where \(\Upsilon =\{\gamma \in C^{1}([0,1];X):\gamma (0)=1,\gamma (1)= \upsilon \} \).
Definition 1
Let X be a Banach space, \(I_{\lambda}\in C^{1}(E,\mathbb{R})\). We say that \(I_{\lambda}\) satisfies the \((PS )\) conditions if every sequence \(\{\varphi _{n}\}_{n\in \mathbb{N}}\subset X\) satisfying
has a convergent subsequence in X.
Next, we prove that the \(I_{\lambda}\) defined in Lemma 3.13 satisfies the \((PS)\) conditions.
Lemma 4.1
Assume that assumptions (P), (G), (V), and (H) hold. Then, the sequence \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) is bounded in X.
Proof
According to (H), we get
so from Lemmas 3.8 and 3.9, there exist constants \(M_{1}\) and \(M_{2}\) such that
Let \(\rho >\max \{1,\frac{1}{M_{1}},\frac{1}{M_{2}} \}\) and
Thus, by the Hölder inequality and Lemma 2.3, for all \(\varphi \in X\) with \(\|\varphi \|_{X}=\rho \), we obtain
and
We use the counterfactual method. Suppose \(\|\varphi _{n}\|_{X}\rightarrow \infty \), \(n\rightarrow \infty \). Combining conditions (P), (G), (V), (H), and Lemma 3.8 and letting \(0<\vartheta <\min \{r_{1}^{-}r_{2}^{-} \frac{Z_{A} +Z_{B}-1}{r_{2}^{-}Z_{A}K_{A}+ r_{1}^{-}Z_{B}K_{A}}, r_{1}^{-},r_{2}^{-} \}\), where \(K_{A}=\max \{\|A_{1}\|_{X}, \|A_{2}\|_{X}\}\), \(Z_{A}=4\lambda _{1}M_{1}^{r_{1}^{+}}C_{A_{1}}\), \(Z_{B}=4\lambda _{2}M_{2}^{r_{2}^{+}}C_{A_{2}}\), we have
In addition, we obtain \(c\frac{1}{\|\varphi _{n}\|^{p_{1}^{-}}_{X}}+o(1) \frac{1}{\|\varphi _{n}\|^{p_{1}^{-}}_{X}}\rightarrow 0\) because \(\|\varphi _{n}\|_{X}\rightarrow \infty \), \(n\rightarrow \infty \). Due to
there is a contradiction. Thus, \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) is bounded. □
Inspired by [15], we have the following lemma.
Lemma 4.2
Assume that assumptions (P), (G), (V), and (H) hold. Then, \(I_{\lambda}\) satisfies the \((PS)\) conditions.
Proof
According to Lemma 4.1, \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) is bounded, that is, there is a subsequence \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) and \(\varphi _{0}\) in X such that
Due to \(\varphi _{n}\rightarrow \varphi _{0}\) in \(L^{\widehat{r}(x)}(\mathbb{R}^{N}\setminus \Omega )\), then \(|\varphi _{n}|^{\bar{p}_{i}(x)-2}\varphi _{n}\rightarrow |\varphi _{0}|^{ \bar{p}_{i}(x)-2}\varphi _{0}\). We get
Since \(\lambda _{1} A_{1}(x)|\varphi |^{r_{1}(x)-2}\varphi \) and \(\lambda _{2}A_{2}(x)|\varphi |^{r_{2}(x)-2}\varphi \) in X are sequentially weakly lower semi-continuous, for \(v\in X\) and measurable for all \(\Omega \subset \mathbb{R}^{N}\), we obtain
Hence, \(\{A_{1}(x)(|\varphi |_{n}^{r_{1}(x)-2}\varphi _{n}- |\varphi |_{0}^{r_{1}(x)-2} \varphi _{0})|v|\}_{n\in \mathbb{N}}\) is uniformly integrable in \(\mathbb{R}^{N}\). Then, using the Vitali convergence theorem, we get
Similarly, there is
We need to prove that \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) is strongly convergent,
A discussion similar to Lemma 3.7 gives that \(\varphi _{n}\rightarrow \varphi _{0}\) in X. Combining the Definition 1 and the Lemma 4.1, we complete the proof. □
Lemma 4.3
Assume that assumptions (P), (G), (V), and (H) hold. There exist \(\rho >0\) and \(\sigma >0\) such that, for all \(\varphi \in X\) with \(\|\varphi \| _{X}=\rho \),
holds.
Proof
Combining (12) with (13), for any \(\varphi \in X\) with \(\|\varphi \|_{X}=\rho >1\), we have
Let
where \(r_{1}^{-}< r_{2}^{-}< p_{2}^{+}\). Then, there exists \(\chi >0\) such that \(f(t)=f(\chi )>0\). Choosing \(\|A_{1}\|_{s_{1}(x)}^{{r_{1}^{-}}}<\sigma ^{*}=\frac{r_{1}^{-}}{4\lambda _{1}M_{1}^{r_{1}^{-}}}f(\chi )\), we get
for \(\|\varphi \|_{X}=\chi = \rho \). □
Lemma 4.4
Assume that assumptions (P), (G), (V), and (H) hold. Then, there exists υ, which satisfies \(\|\upsilon \|_{X}>\rho \). Then, there exists \(\upsilon \in X\) such that
Proof
Choosing \(\widehat{\upsilon}\in X \) such that \(\|\widehat{\upsilon}\|_{X}=1\), and for \(t\in (0,1)\) small enough, we obtain
with the fact that \(1< r_{1}^{-}< p_{2}^{+}\). Thus, \(I_{\lambda}(t\widehat{\upsilon})<0\) with \(\|t\widehat{\upsilon}\|_{X}>\rho \). The proof is proved by letting \(\upsilon =t\widehat{\upsilon}\). □
Proof of Theorem 1.1
Combining Lemmas 4.1 and 4.2, it can be inferred that \(I_{\lambda}\) satisfies \((PS)\) conditions. According to Lemmas 4.3 and 4.4, we know that \(I_{\lambda}\) satisfies the mountain pass lemma. Therefore, we have a subsequence \(\{\varphi _{n}\}_{n\in \mathbb{N}}\) and \(\varphi _{0}^{(1)}\in X\) such that \(\varphi _{n}\rightarrow \varphi _{0}^{(1)}\) in X by Lemma 4.1 and \(0<\sigma <c<\infty \). Therefore, \(I_{\lambda}(\varphi _{n})= c>\sigma \), that is, \(\varphi _{0}^{(1)}\) is a solution of problem (1) with positive energy.
Next, we will apply Ekeland’s variational principle to prove that (1) has a solution with negative energy.
By Lemma 4.3, we derive that
where ρ is the positive constant introduced in Lemma 4.3.
From condition (H), there exist \(\epsilon _{1},\epsilon _{2}>0\) and an open set \(\Omega _{0}\subset \subset \Omega \) such that
and we get
Hence,
By Lemma 2.1 and \(g_{i}(x)>0\), we conclude
For sufficiently small \(\tau \in (0,1)\), let \(\eta \in C_{0}^{\infty}(\Omega )\) such that \(\overline{\Omega}_{0}\subset \operatorname{supp}(\eta )\), \(\eta =1\), for all \(x\in \overline{\Omega}_{0}\) and \(0\leq \eta \leq 1\) in Ω. Then, by applying (14), it follows that
Since \({r_{2}}^{-}+\epsilon _{2}< p_{1}^{-}\), we have \(I_{\lambda}(t\eta )<0\).
In addition, combining the Hölder inequality and inequality (11), for any \(\varphi \in B_{\rho} (0)\), we have
This fact gives
Set
By (15), \(I_{\lambda}:\overline{B_{\rho}(0)}\rightarrow \mathbb{R}\) is lower bounded on \(\overline{B_{\rho}(0)}\) and \(I_{\lambda}\in C^{1}(\overline{B_{\rho}(0)},\mathbb{R})\). Using Ekeland’s variational principle, there exists \(\{\varphi _{n}\}_{n\in \mathbb{N}}\in \overline{B_{\rho}(0)}\) such that
Since
we have \(\varphi _{n}\in B_{\rho}(0)\). Define function \(\zeta : \overline{B_{\rho}(0)}\rightarrow \mathbb{R}\) by
which implies \(\zeta (\varphi _{n})<\zeta (\varphi )\) from (16). Then, \(\varphi _{n}\) is a minimum point of ζ, and we have
for small \(t>0\) and any \(v\in B_{1}(0)=\{v\in X: \|v\|_{X}=1\}\). Hence,
Let \(t\rightarrow 0\), then \(\langle I'_{\lambda},v\rangle +\frac{1}{n}\|v\|_{X} \geq 0\). Replace v with −v. Then, we obtain \(\langle -I'_{\lambda},v\rangle +\frac{1}{n}\|v\|_{X} \geq 0\). Thus, \(\|I'_{\lambda}(\varphi _{n})\|_{X^{*}}\leq \frac{1}{n}\). We infer that there exists a sequence \(\{\varphi _{n}\}_{n\in \mathbb{N}} \subset B_{\rho} (0)\) such that
By Lemma 4.2, there is \(\varphi _{n}\rightarrow \varphi _{0}^{(2)}\) in X. Then, we have \(I_{\lambda}'(\varphi _{0}^{(2)})=0 \) and \(I_{\lambda}(\varphi _{0}^{(2)})=\widehat{c}<0\), that is, \(\varphi _{0}^{(2)}\) is another solution of equations (1) with negative energy, which ends the proof. □
Here, we give an example of application of Theorem 1.1.
Example 4.1
Let \(\Omega =\{(x,y)\in \mathbb{R}: x^{2}+y^{2}\leq 1\}\). Consider the problem
By simple calculations, we obtain \(\operatorname{meas}(\partial \Omega )=2\pi \), \(p_{1}^{-}=3\), \(p_{1}^{+}=4\), \(p_{2}^{-}=5\), \(p_{2}^{+}=6\). Conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.1 are fulfilled. Hence, Theorem 1.1 implies that problem (17) admits two nontrivial weak solutions.
5 Proof of Theorem 1.2
To prove Theorem 1.2, we first recall the following lemmas.
Lemma 5.1
([15]) Let X be a reflexive and separable Banach space. Then, there are \(\{e_{n}\}\subset E\) and \(\{e_{n}^{*}\}\subset E^{*}\) such that
and
Denote
Lemma 5.2
([15]) Assume that \(q(x)\in C_{+}(\overline{\Omega})\), \(q(x)< p^{*}(x)\), for any \(x\in \overline{\Omega}\) and denote
then \(\lim_{k\rightarrow \infty}\widetilde{\xi}_{k}=0\).
Now, we recall the fountain theorem.
Theorem 5.1
([15]) Let X be a real Banach space and \(I_{\lambda}\in C^{1}(X, \mathbb{R})\) be a even functional satisfying the \((PS)\) conditions. There exists \(r_{k}>0\) such that \(\widehat{\rho}_{k}> r_{k}>0\) for every \(k\in \mathbb{N}\). Then, the following conditions hold:
-
(i)
\(\alpha _{k}=\max \{I_{\lambda}(\varphi ): \varphi \in X_{k}, \| \varphi \|=\widehat{\rho}_{k}\}\leq 0\);
-
(ii)
\(\beta _{k}=\inf \{I_{\lambda}(\varphi ): \varphi \in Y_{k}, \| \varphi \|= r_{k}\}\rightarrow +\infty \) as \(k\rightarrow \infty \).
Then, \(I_{\lambda}\) possesses a series of critical points \(\varphi _{k}\) such that \(I_{\lambda}(\varphi _{k})\rightarrow +\infty \).
Lemma 5.3
Assume that assumptions (P), (G), (V), and (H) hold. There exists \(\widehat{\rho}_{k}>0\) such that
Proof
Let \(t\in (0,1)\). For \(\|\widehat{\varphi}\|_{X}=\widetilde{\rho}_{k}\geq 1\) and \(\widehat{\rho}_{k}>\widetilde{\rho}_{k}\), there exists φ̂ such that
with \(p_{1}^{-}>r_{1}^{+}>1\). Taking \(\varphi =t\widehat{\varphi}\), for sufficiently small t, it follows that
□
Lemma 5.4
Assume that assumptions (P), (G), (V), and (H) hold. There exists \(r_{k}>0\) such that
Proof
According to Lemma 5.2, for \(\|\varphi \|_{X}=r_{k}>1\), we obtain
Let
and there exists a constant C̃ such that \(r_{\varphi}=\max \{r_{1}^{+}+\widetilde{C}, r_{2}^{+}+\widetilde{C} \}\), where \(1< p_{1}^{-}< r_{\varphi}^{-}< r_{\varphi}\). Therefore,
Choose
Since \(p_{1}^{-}< r_{\varphi}\), we have \(r_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \). By the choice of \(r_{k}\) with \(\|\varphi \|_{X}=r_{k}\) such that \(\widehat{\rho}_{k}>\widetilde{\rho}_{k}>r_{k}>0\), we obtain
□
Proof of Theorem 1.2
Let hypotheses (P), (G), (V), and (H) be satisfied. By Lemma 4.2, \(I_{\lambda} \) satisfies \((PS)\) conditions. Under the definition of \(I_{\lambda} \) in Lemma 3.13, it follows that \(I_{\lambda}(0)=0\) and \(I_{\lambda}\) is an even function. Therefore, from Lemmas 5.3 and 5.4, it can be deduced that \(I_{\lambda}\) satisfies Theorem 5.1. Then, \(I_{\lambda}\) possesses a series of critical points \(\varphi _{k}\) as \(k\rightarrow +\infty \). In conclusion, equations (1) possess infinitely many nontrivial weak solutions. □
Here, we give an example of application of Theorem 1.2.
Example 5.1
Let \(\Omega =\{(x,y)\in \mathbb{R}: x^{2}+y^{2}\leq 1\}\). Consider the problem
By simple calculations, we obtain \(\operatorname{meas}(\partial \Omega )=2\pi \), \(p_{1}^{-}=3\), \(p_{1}^{+}=4\), \(p_{2}^{-}=5\), \(p_{2}^{+}=6\), \(r_{1}^{-}=5/4\), \(r_{1}^{+}=9/4\), \(r_{2}^{-}=7/4\), and \(r_{2}^{+}=11/4\). That is, conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.2 are fulfilled. Hence, Theorem 1.2 implies that problem (18) admits infinitely many nontrivial weak solutions.
6 Proof of Theorem 1.3
We give some results with the aid of the Krasnoselskii genus. Let E be a real Banach space and set
Let \(\mathcal{A}\subset \mathfrak{R}\) and \(E=\mathbb{R}^{k}\). We define genus
If the mapping ψ does not exist for any \(k>0\), and set \(\gamma (\mathcal{A})=\infty \). If \(\mathcal{A}\) is a subset consisting of a finite number of pairs of points, then, \(\gamma (\mathcal{A})=1\). Furthermore, from definition, \({\gamma (\emptyset )=0}\).
Lemma 6.1
([18]) Let \(E=R^{N}\) and ∂Ω be the boundary of an open, symmetric, and bounded subset \(\partial \Omega \subset R^{N}\) with \(0\in \Omega \). Then, \(\gamma (\partial \Omega )=N\).
Corollary 6.1
([18]) \(\gamma (S^{N-1})=N\).
Theorem 6.1
([18]) Let \(I_{\lambda}\in C^{1}(X)\) be a functional satisfying the \((PS)\) conditions and assume that
-
(i)
\(I_{\lambda}\) is bounded from below and even;
-
(ii)
there is a compact set \(K\in \mathbb{R}\) such that \(\gamma (K)=k\) and \(\sup_{x\in K}I_{\lambda}(x)< I_{\lambda}(0)\).
Then, \(I_{\lambda}\) has at least k pairs of distinct critical points whose corresponding critical values are all less than \(I_{\lambda}(0)\).
Proof of Theorem 1.3
Combining (12), (13), and the Hölder inequality (7) for \(\|\varphi \|_{X}>1\), we obtain
Since \(\max \{1, r^{+}_{1}, r^{+}_{2}\}< p_{1}^{-}\), for \(\|\varphi \|_{X}\) large enough, \(I_{\lambda}\) is bounded from below. \(I_{\lambda}\in X\) is an even function by the definition and \(I_{\lambda}(0)=0\). Moreover, \(I_{\lambda}\) is coercive in X and satisfies the \((PS)\) conditions by Lemma 4.2. Let
We obtain
Now we prove that for any \(k \in \mathbb{N}\), there is \(c_{k}<0\). For each k, we take k disjoint open sets \(\widetilde{K}_{i}\) such that \(\bigcup^{k}_{i=1} \widetilde{K}_{i}\subset \Omega \). For \(i = 1,\ldots ,k\), let \(\varphi _{i} \in (X\bigcap C^{\infty}_{0}(\widetilde{K} _{i}) \setminus \{0\}\) with \(\|\varphi _{i}\|_{X} = 1\), and
Since each norm on \(\mathcal{M}_{k}\) is equivalent, there is \(\rho ^{*}_{k} \in (0,1)\) such that \(\varphi \in \mathcal{M}_{k}\) with \(\|\varphi \|_{X} \leq \rho ^{*}_{k}\), which means that \(\|\varphi \|_{\infty}< C_{\rho ^{*}_{k}}<1\). Set
Combining the compactness of \(S^{(k)}_{\rho ^{*}_{k}}\) and \(t\in (0,1)\) for all \(\varphi \in S^{(k)}_{\rho ^{*}_{k}}\),
Sine \(1< r_{1}^{-}< p_{1}^{-}\), there exist \(t_{k}\in (0,1)\) and \(\varepsilon _{k}\) such that
Thus, \(I_{\lambda}(\varphi )<0\) for all \(\varphi \in S^{(k)}_{t_{k}\rho ^{*}_{k}}\). Furthermore, \(\gamma (S^{(k)}_{t_{k}\rho ^{*}_{k}})=k\) such that \(c_{k}<-\varepsilon _{k}<0\) for all k, and the assertion is proved. Each \(c_{k}\) is a critical value by the Krasnoselskii genus theory. Combining Theorem 6.1, \(I_{\lambda}\) has at least k pairs of different critical points. In addition, since k is arbitrary, we obtain an infinite number of critical points of equations (1). □
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Acknowledgements
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 (2023KYQD0044).
Funding
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 (2023KYQD0044).
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Zhenfeng Zhang wrote the main manuscript text and Tianqing An, Weichun Bu, and Shuai Li verified this article. All authors reviewed the manuscript.
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Zhang, Z., An, T., Bu, W. et al. Existence and multiplicity of solutions for fractional \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations with Robin boundary conditions. Bound Value Probl 2024, 37 (2024). https://doi.org/10.1186/s13661-024-01844-4
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DOI: https://doi.org/10.1186/s13661-024-01844-4