Abstract
We focus on two-phase problems with singular and superlinear parametric terms on the right-hand side. Using fibering maps and the Nehari manifold method, we prove that there are at least two non-trivial positive solutions in a geometric setting that is locally similar to Euclidean spaces but has different global properties for all except the smallest values of parameter \(\mu > 0.\) Singularities may appear at discrete locations in the manifold, which is a challenge for the work due to the unpredictable behavior of the solution. The findings presented here generalize some known results.
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1 Introduction
Given \( \left( {\mathcal {E}}, g \right) \) a smooth complete compact Riemannian N-manifold. Consider the following singular two-phase problem:
where \({\mathcal {L}}_{p \left( z \right) , q \left( z \right) }^{\omega \left( z \right) } \left( \textrm{u} \right) = - \, \text{ div } \left( \vert D \textrm{u} \left( z \right) \vert ^{p \left( z \right) - 2} D \textrm{u} \left( z \right) + \omega \left( z \right) \vert D \textrm{u} \left( z \right) \vert ^{q \left( z \right) - 2} D \textrm{u} \left( z \right) \right) \). We make the following hypotheses:
-
(i)
The functions \(p \left( . \right) , q \left( . \right) , r \left( . \right) \in C \left( \overline{{\mathcal {E}}} \right) \) verify the following assumptions:
$$\begin{aligned} 1< q^{-} \le q^{+}< p^{-} \le p^{+}< r^{-} \le r^{+} < + \infty , \end{aligned}$$(1.1)where \( q^{+} = \displaystyle \sup _{z \in \overline{{\mathcal {E}}}} q \left( z \right) , \ q^{-} = \inf _{z \in \overline{{\mathcal {E}}}} q \left( z \right) , \ p^{+} = \sup _{z \in \overline{{\mathcal {E}}}} p \left( z \right) , \ p^{-} = \inf _{z \in \overline{{\mathcal {E}}}} p \left( z \right) , \ r^{+} = \sup _{z \in \overline{{\mathcal {E}}}} r \left( z \right) , \ \text{ and } \ r^{-} = \inf _{z \in \overline{{\mathcal {E}}}} r \left( z \right) .\)
$$\begin{aligned} \frac{p^{-}}{q^{+}} < 1 + \frac{1}{N}. \end{aligned}$$(1.2) -
(ii)
\(\omega : \overline{{\mathcal {E}}} \rightarrow \left[ 1, + \infty \right) \) is Lipschitz continuous.
-
(iii)
\( g \left( z \right) \in L^{\infty } \left( {\mathcal {E}} \right) \) and \( g \left( z \right) \ge 0\) for a.a \( z \in {\mathcal {E}} \) with \( g \not \equiv 0\).
-
(iv)
\(0< \gamma \left( z \right) \in C \left( \overline{{\mathcal {E}}} \right) , \ 0< \gamma ^{-} \le \gamma ^{+} < 1.\)
-
(v)
\( \mu \) is a positive parameter.
Of late, the study of nonlinear elliptic equations and variational problems with growth conditions with variable exponents has gained significant popularity due to its wide range of applications. These applications include but are not limited to studying fluid filtration in porous media, elastoplasticity, constrained heating, optimal control, and more. Additional information on these applications can be found in [10, 18, 30] and the cited references. Originally, while studying the behavior of strongly anisotropic materials, Zhikov discovered that their hardening properties changed radically with the point, which is called the Lavrentiev phenomenon, see for example [35, 36]. To explain this phenomenon, he introduced the functional
when the integrand toggles between two distinct elliptic behaviors.
Numerous intriguing studies have been conducted on the problem of two phases subject to a Dirichlet boundary condition. For example, Liu et al. [22] established the existence and multiplicity of solutions to the following problem
and showed a ground-state solution with sign change by variational method. For the Kirchhoff version, the authors in [4] showed an existing result using minimax techniques and Trudinger–Moser inequalities. Based on the Nehari manifold method, Gasiński et al. in [15] have proposed an alternative treatment. Following this direction and employing the Nehari method, the authors in [26] showed that when the sum of the \(p-\)Laplace operator and a weighted \(q-\)Laplacian \(\left( \ \text {with} \ q < p \right) \) drives the two-phase problem, along with a reaction term and a weight function that is unconstrained away from zero, the resulting system is \(\left( p - 1 \right) -\)superlinear. Moreover, the system possesses a ground-state solution of the constant sign as well as a nodal solution that changes the sign. In particular, using the theory of pseudomonotone operators, Gasiński, and Winkert in [16] have demonstrated that a weak solution to quasilinear elliptic equations with two-phase phenomena and a reaction term that depends on the gradient exists and is unique, provided that certain general conditions are met for the convection term and some linear conditions are imposed on the gradient variable. For a more comprehensive understanding, we recommend referring to [2, 5, 7,8,9, 11, 23, 27] and the cited sources.
Next, we want to highlight a few research papers that focus on the two-phase issues of singularity. In their work, Papageorgiou, Repovš, and Vetro [25] examined the presence of positive solutions to a class of two-phase Dirichlet equations that involve interconnected singular and parametric superlinear effects. For insights and methods that ensure the existence of at least two meaningful positive solutions to two-phase problems related to singularity, readers can refer to [21, 25] also to [29] where the fractional singular \(\left( p, q \right) -\)Laplacian elliptic problems are posed in a smooth bounded domain in \(\mathbb {R}^{N}.\) The strategy of proof is based on the theory of the fractional Sobolev space via the variational method and critical point theory.
For two-phase problems with variable exponents, which few authors consider, we refer to the papers of Ragusa and Tachikawa [28], Tachikawa [31], Aberqi, Bennouna, Benslimane, and Ragusa [3], Vetro and Winkert [33], and Zeng, Radulescu, and Winkert [34]. The readers may refer to the work due to [1, 6, 7, 14, 17, 19, 24] and the references given there.
The current article has two objectives. In the first one, we expand upon the findings in [25] and [21] by moving from the case of Musielak–Orlicz Sobolev spaces with constant exponents p and q to Sobolev–Orlicz spaces with variable exponents on a complete manifold. In the second one, we show the existence of at least two non-trivial solutions to the problem \(\left( {\mathcal {P}} \right) \), which contains a singular term and a parametric superlinear term in the setting of Sobolev spaces with variable exponents in complete manifolds. These results are entirely novel and represent a significant advance in this area of research. However, the challenges presented in the present work are twofold—first, the non-Euclidean setting of \(\left( {\mathcal {P}} \right) \). Second, the more complex nonlinearities associated with the \(p \left( z \right) -\)Laplacian and \( q \left( z \right) -\)Laplacian operators, which are not homogeneous like the \(p-\)Laplace and \(q-\)Laplace operators. Moreover, unlike the \(p-\)Laplace operator, we cannot use the Lagrange Multiplier Theorem in many problems involving this operator, making our task even more difficult.
Now, we define the problem’s solution \(\left( {\mathcal {P}} \right) \) in a weaker sense.
Definition 1.1
A function \(\textrm{u} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) is said to be a weak solution if \( \textrm{u} > 0\) for a.a \( z \in {\mathcal {E}} \) and
is satisfied for every \( \varphi \in {\mathcal {D}} \left( {\mathcal {E}} \right) .\) Let us denote by \({\mathcal {D}} \left( {\mathcal {E}} \right) \) the space of \(C^{\infty }\) functions with compact support in \({\mathcal {E}}.\)
Remark 1.2
For the singular term, the weak solution is by definition a function \( \textrm{u} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) such that \(\textrm{u} \left( z \right) ^{- \gamma \left( z \right) } \, \varphi \left( z \right) \in L^{1} \left( {\mathcal {E}} \right) \) for every \( \varphi \in {\mathcal {D}} \left( {\mathcal {E}} \right) \). Hence, the definition 1.1 is well defined.
Remark 1.3
When \( q \left( z \right) < r \left( z \right) ,\) the coercivity qualities of the functional energy associated with \(\left( {\mathcal {P}} \right) \) (see Lemma 3.1 below) ensure compactness for sequences with a uniformly bounded energy. Unfortunately, when \(q \left( z \right) = r \left( z \right) \) would fail our coercivity properties.
The main result of our paper can be stated as follows.
Theorem 1.4
Assuming that the complete N-manifold \(\left( {\mathcal {E}}, g \right) \) has property \(B_{vol} \left( \mu , v \right) \). Then, under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) ,\) here exists \(\tilde{\mu }^{*}_{0} > 0\) such as for every \( \mu \in \left( 0, \tilde{\mu }^{*}_{0} \right] \), \(\left( {\mathcal {P}} \right) \) has at least two positive solutions \(\tilde{\textrm{u}}^{*}, \, \tilde{v}^{*} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) such as \( J_{\mu } \left( \tilde{\textrm{u}}^{*} \right) < 0 \le J_{\mu } \left( \tilde{v}^{*} \right) .\)
This article is structured as follows: Sect. 2 provides an overview of Sobolev spaces with variable exponents on complete manifolds, referencing papers [6,7,8, 19, 32] for further information. In Sect. 3, the Nehari manifold associated with \(\left( {\mathcal {P}} \right) \) is introduced and analyzed, specifically focusing on local minima, local maxima, and inflection points. The article also demonstrates that two non-trivial positive solutions exist when the parameter \( \mu > 0\) is small enough.
2 Preliminaries
In what follows, we suppose that \( \left( {\mathcal {E}}, g \right) \) is a smooth Riemannain N-manifolds. We give some facts on space \( W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \), which is called the Sobolev spaces with variable exponents in complete manifolds setting. For more results, the reader can consult [6, 7, 12, 14, 17, 19] and references therein.
Definition 2.1
[14] We define the \( L_{k}^{q \left( \cdot \right) } \left( {\mathcal {E}} \right) \) spaces as the completion of \( C^{q \left( \cdot \right) }_{k} \left( {\mathcal {E}} \right) \) with respect to the norm \( \Vert \textrm{u} \Vert _{L^{q \left( \cdot \right) }_{k}}\), with
the symbol \( D^{k} \textrm{u} \) represents the \(k-\)th covariant derivative of the function \(\textrm{u} \), and the norm of \( D^{k} \textrm{u} \) is denoted by \( \vert D^{k} \textrm{u} \vert \), which is defined in local coordinates as
and
\(\bullet \) If \(\Omega \subset {\mathcal {E}}\), then \(L^{q \left( \cdot \right) }_{k, 0} \left( \Omega \right) \) is the completion of \(C^{q \left( \cdot \right) }_{k} \left( {\mathcal {E}} \right) \cap C_{0} \left( {\mathcal {E}} \right) \) with respect to \( \Vert \cdot \Vert _{L^{q \left( \cdot \right) }_{k}},\) where \(C_{0} \left( \Omega \right) \) refers to the the vector space of continuous functions that have a compact support contained in \(\Omega .\)
Definition 2.2
[17] The Sobolev space \( W^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) consists of functions \( \textrm{u}\) belonging to \( L^{q \left( z \right) } \left( {\mathcal {E}} \right) \) such that \( D^{j} \textrm{u} \in L^{q \left( z \right) } \left( {\mathcal {E}} \right) \) for \(j = 1, 2, \cdots , n.\) The norm is given as
\(\bullet \) We define the space \( W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) as the closure of \( C^{\infty } \left( {\mathcal {E}} \right) \) in \( W^{1, q \left( z \right) } \left( {\mathcal {E}} \right) .\)
Definition 2.3
[19] Let \( \xi : \ \left[ \alpha , \beta \right] \longrightarrow {\mathcal {E}} \) a curve of class \( C^{1} \). The length of \( \xi \) is
and to describe the distance between two points z and y in the set \({\mathcal {E}}\), we define the function \( d_{g} \left( z, y \right) \) as
Definition 2.4
[19] A function \( t: {\mathcal {E}} \longrightarrow \mathbb {R} \) is considered to be log-Hölder continuous if for every pair of points \(\left\{ z, y\right\} \) in \( {\mathcal {E}}\), there exists a constant c such that
The set of variable exponents which are log-Hölder continuous is denoted \( {\mathcal {P}}^{log} \left( {\mathcal {E}} \right) \). The link with \( {\mathcal {P}}^{log} \left( {\mathcal {E}} \right) \) and \( {\mathcal {P}}^{log} \left( \mathbb {R}^{N} \right) \) is given by the following proposition:
Proposition 2.5
[6, 14] Let \( q \in {\mathcal {P}}^{log} \left( {\mathcal {E}} \right) \), and let \( \left( B_{\frac{R}{3}} \left( q \right) , \phi \right) \) be a chart such that
can be expressed as bilinear forms, in which \( \delta _{i j} \) represents the delta Kronecker symbol. Then \( q \circ \phi ^{-1} \in {\mathcal {P}}^{log} \left( \phi \left( B_{\frac{R}{3}} \left( q \right) \right) \right) .\)
Definition 2.6
[19] The N-dimensional manifold \( \left( {\mathcal {E}}, g \right) \) possesses the characteristic \( B_{vol} \left( \mu , v \right) ,\) if its geometry is confined in the following way:
-
The Ricci tensor of g denoted by Rc(g) satisfies \(Rc \left( g \right) \ge \mu \left( N - 1 \right) \ g \) for a certain constant \(\mu \), where N represents the dimension of \({\mathcal {E}}.\)
-
For every point z in \({\mathcal {E}},\) there is a positive value v such that \( \vert B_{1} \left( z \right) \vert _{g} \ge v,\) where \(B_{1} \left( z \right) \) are the balls of radius 1 centered at some point z in terms of the volume of smaller concentric balls.
Proposition 2.7
[6, 19] If \({\mathcal {E}}\) is complete, and the embedding \( L^{1}_{1} \left( {\mathcal {E}} \right) \hookrightarrow L^{\frac{n}{N - 1}} \left( {\mathcal {E}} \right) \) holds, then whenever the real numbers q and p satisfy
and
the embedding \( L^{q}_{1} \left( {\mathcal {E}} \right) \hookrightarrow L^{p} \left( {\mathcal {E}} \right) \) also holds.
Proposition 2.8
[17] Given \( \textrm{u} \in L^{q \left( z \right) } \left( {\mathcal {E}} \right) , \ \left\{ \textrm{u} _{k} \right\} \subset L^{q \left( z \right) } \left( {\mathcal {E}} \right) , \ k \in \mathbb {N},\) then
-
(1)
\(\Vert \textrm{u} \Vert _{q \left( z \right) } < 1 \ ( \text{ resp }. = 1, \)>\( 1 ) \iff \rho _{q \left( z \right) } \left( \textrm{u} \right) < 1 \,\,( \text{ resp }. = 1, \)>1),
-
(2)
For \( \textrm{u} \in L^{q \left( z \right) } \left( {\mathcal {E}} \right) \backslash \left\{ 0 \right\} , \ \Vert \textrm{u} \Vert _{q \left( z \right) } = \lambda \Longleftrightarrow \rho _{q \left( z \right) } \left( \frac{\textrm{u} }{\lambda } \right) = 1.\)
-
(3)
\( \Vert \textrm{u} \Vert _{q \left( z \right) } < 1 \Rightarrow \Vert \textrm{u} \Vert _{q \left( z \right) }^{q^{+}} \le \rho _{q \left( z \right) } \left( \textrm{u} \right) \le \Vert \textrm{u} \Vert _{q \left( z \right) }^{q^{-}},\)
-
(4)
\( \Vert \textrm{u} \Vert _{q \left( z \right) } > 1 \Rightarrow \Vert \textrm{u} \Vert _{q \left( z \right) }^{q^{-}} \le \rho _{q \left( z \right) } \left( \textrm{u} \right) \le \Vert \textrm{u} \Vert _{q \left( z \right) }^{q^{+}},\)
-
(5)
\(\displaystyle \lim _{k \rightarrow + \infty } \Vert \textrm{u} _{k} - \textrm{u} \Vert _{q \left( z \right) } = 0 \iff \lim _{k \rightarrow + \infty } \rho _{q \left( z \right) } \left( \textrm{u} _{k} - \textrm{u} \right) = 0. \)
Theorem 2.9
[3] Let \({\mathcal {E}}\) is compact and \( q \left( z \right) , p \left( z \right) \in C \left( \overline{{\mathcal {E}}} \right) \cap L^{\infty } \left( {\mathcal {E}} \right) .\) Assume that
Then,
and
are compact and continuous embeddings.
Proposition 2.10
[6] If \(\left( {\mathcal {E}}, g \right) \) is complete, then \(W^{1, q \left( z \right) } \left( {\mathcal {E}} \right) = W^{1, q \left( z \right) }_{0} \left( {\mathcal {E}} \right) .\)
The weighted variable exponent Lebesgue space \(L_{\omega \left( z \right) }^{q \left( z \right) } \left( {\mathcal {E}} \right) \) is defined by
endowed with
Moreover, the weighted modular on \(L_{\omega \left( z \right) }^{q \left( z \right) } \left( {\mathcal {E}} \right) \) is the mapping \( \rho _{q \left( \cdot \right) , \omega \left( \cdot \right) } : L_{\omega \left( z \right) }^{q \left( z \right) } \left( {\mathcal {E}} \right) \rightarrow \mathbb {R}\) defined by
Example 2.11
We may take \( \omega \left( z \right) = \left( 1 + \vert z \vert \right) ^{\varepsilon \left( z \right) } \) with \(\varepsilon \left( \cdot \right) \in C_{+} \left( \overline{{\mathcal {E}}} \right) \) as an example,
where \( C_{+} \left( \overline{{\mathcal {E}}} \right) = \left\{ q \ \vert \ q \in C \left( \overline{{\mathcal {E}}} \right) \ \text{ with } \ q \left( z \right) > 1 \ \text{ for } \ z \in \overline{{\mathcal {E}}} \right\} .\)
3 Existence results
We consider the functional \(\textrm{ J}_{\mu } : W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \rightarrow \mathbb {R}\) given by:
Due to the singular term \(g \left( z \right) \ \vert \textrm{u} \left( z \right) \vert ^{1 - \gamma \left( z \right) },\) \(\textrm{ J}_{\mu }\) is not \(C^{1}.\) For that, we consider the Nehari manifold that corresponds to the functional \(\textrm{ J}_{\mu } \), which is defined as follows:
It is easy to see that \({\mathcal {N}}_{\mu }\) is smaller than \( W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) and it contains the non-trivial solutions of \(\left( {\mathcal {P}} \right) \). The functional \(\textrm{ J}_{\mu } \bigg \vert _{{\mathcal {N}}_{\mu }}\) can have nice properties which fail to be true globally.
For further considerations, we decompose the set \({\mathcal {N}}_{\mu }\) into three disjoint parts:
Note that, since \( \omega \left( \cdot \right) : \overline{{\mathcal {E}}} \longrightarrow \left[ 1, + \infty \right) ,\) then, there exists \(\omega _{0} > 0,\) and for all \( z \in {\mathcal {E}},\) we have that \( \omega \left( z \right) > \omega _{0}.\)
Lemma 3.1
If hypotheses \(\left( \textit{i} \right) - \left( \textit{v} \right) \) hold, then \(\textrm{ J}_{\mu } \bigg \vert _{{\mathcal {N}}_{\mu }}\) is coercive.
Proof
Let \( \textrm{u} \in {\mathcal {N}}_{\mu } \) with \(\Vert \textrm{u} \Vert > 1,\) where \(\Vert \cdot \Vert \) is the induced norm of \(W^{1, q \left( z \right) }_{0} \left( {\mathcal {E}} \right) \backslash \left\{ 0 \right\} .\) From the definition of \({\mathcal {N}}_{\mu }\), we have
Hence, according to (1.1), Propositions 2.7, 2.8, Theorem 2.9, and based on inequality 2.3 in [3] we obtain
for some \(c_{1}, c_{2} > 0 \) ( since \( q^{+}< p^{+} < r^{-} \) ), and c is the Poincaré constant.
From that, since \( 0< \gamma ^{+} < 1\) and \( 1 - \gamma ^{+}< 1 < p^{+},\) we conclude that \(\textrm{ J}_{\mu } \bigg \vert _{{\mathcal {N}}_{\mu }}\) is coercive. \(\square \)
Let \( \sigma _{\mu }^{+} = \inf _{{\mathcal {N}}_{\mu }} \textrm{ J}_{\mu }.\)
Lemma 3.2
If the assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) \) are satisfied and \( {\mathcal {N}}_{\mu }^{+} \ne 0,\) then \(\sigma _{\mu }^{+} < 0.\)
Proof
Assume that \( \textrm{u} \in {\mathcal {N}}_{\mu }^{+},\) hence by the definition of \( {\mathcal {N}}_{\mu }^{+}\), we get
Since \( {\mathcal {N}}_{\mu }^{+} \subset {\mathcal {N}}_{\mu }, \) we get
which implies that
Hence,
\(\square \)
Lemma 3.3
Under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) ,\) there exists \(\mu ^{*} > 0\) such as \( {\mathcal {N}}_{\mu }^{0} = \emptyset \) for all \( \mu \in \left( 0, \mu ^{*} \right) .\)
Proof
Suppose otherwise, that is \( {\mathcal {N}}_{\mu }^{0} \ne \emptyset \) for all \( \mu > 0,\) we can find \( \textrm{u} \in {\mathcal {N}}_{\mu }^{0}\) such as
since \( \textrm{u} \in {\mathcal {N}}_{\mu },\) we also have
Subtracting (3.1) from (3.2) yields
According to Theorems 13.17 of Hewitt–Stromberg [20, p. 196], Theorem 2.9, and Propositions 2.7, 2.8, we deduce from (3.3) that
Thus,
From (3.1) and Theorems 2.9, we have
which implies that
Hence, the norm of function \(\Vert \textrm{u} \Vert \longrightarrow + \infty \) when \( q^{+}< p^{+} < r^{-}\) and the value of \( \mu \rightarrow 0^{+}, \) which is contradictory to (3.4). \(\square \)
Lemma 3.4
Under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) ,\) there exists \(\tilde{\mu }^{*} \in \left( 0, \mu ^{*} \right] \) such that \( {\mathcal {N}}_{\mu }^{\pm } \ne \emptyset \) for all \(\mu \in \left( 0, \tilde{\mu }^{*} \right) .\) In addition, for all \( \mu \in \left( 0, \tilde{\mu }^{*} \right) ,\) there exists \( \tilde{\textrm{u}}^{*} \in {\mathcal {N}}_{\mu }^{+} \) such that \( \textrm{ J}_{\mu } \left( \tilde{\textrm{u}}^{*} \right) = \sigma _{\mu }^{+} < 0 \) and \( \tilde{\textrm{u}}^{*} \left( z \right) \ge 0 \) for a.a \( z \in {\mathcal {E}}.\)
Proof
Suppose that \( \textrm{u} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \backslash \left\{ 0 \right\} \) and \(\zeta _{\textrm{u}} : \left( 0, + \infty \right) \rightarrow \mathbb {R} \) a function given by
Since \( r^{-} - p^{-} < r^{-} + \gamma ^{-} - 1 \) there exists \( \tilde{s}_{0} > 0\) such as
Then, we have
Furthermore, we have
If G represents the best Sobolev constant, then by Proposition 2.8, we have for \(\Vert u \Vert < 1\) that
Moreover, we have
According to (3.5), (3.6), and (3.7), we have
Hence, there exists \( \tilde{\mu }^{*} \in \left( 0, \mu ^{*} \right] \) independent of \(\textrm{u}\) such as
Now, for \( \textrm{u} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) we consider the function \( \tilde{\zeta }_{z} : \left( 0, + \infty \right) \rightarrow \mathbb {R} \) defined by
According to (1.1), we get that
Thus, there exists \( s_{0} > 0\) such as
Obviously, we have \( \tilde{\zeta }_{\textrm{u}} \ge \zeta _{\textrm{u}} \) and by (3.8), we find the existence of \( \tilde{\mu }^{*} \in \left( 0, \mu ^{*} \right] \) independent of u such as
Therefore, there exists \( s_{1}< s_{0} < s_{2}\) such that
and
where
We will now examine the fibering function \(\psi _{\textrm{u}} : \left[ 0, + \infty \right) \rightarrow \mathbb {R}\), which is defined as \( \psi _{\textrm{u}} \left( s \right) = \textrm{ J}_{\mu } \left( s \textrm{u} \right) \) for all \( s \ge 0.\)
Since \( \psi _{\textrm{u}} \left( s \right) \in C^{2} \left( \left( 0, \infty \right) \right) ,\) we deduce that
and
According to (3.9) and (3.10), we get
which implies by multiplying by \( \gamma ^{-} s_{1}^{r^{-} - 2} \) and \( - \left( r^{-} - 1 \right) \ s_{1}^{r^{-} - 2},\) respectively, that
and
Applying (3.13) in (3.12), we obtain
On the other hand, using the same technique, we apply (3.14) in (3.12), we deduce that
From (3.15) and (3.16), we conclude that
Thus,
Hence,
We use the same method for the point \(s_{2}\) and \(\Vert u \Vert > 1\). Then, according to (3.9) and (3.10), we can see that \({\mathcal {N}}_{\mu }^{-} \ne \emptyset .\)
This demonstrates the proposition’s first statement. Now, consider a minimizer sequence \(\left\{ \textrm{u}_{m} \right\} _{m \in \mathbb {N}} \subset {\mathcal {N}}_{\mu }^{+}\) such as
According to the fact that \( {\mathcal {N}}_{\mu }^{+} \subseteq {\mathcal {N}}_{\mu }\) and Lemma 3.1, we have that
Therefore, we may assume that
From (3.17) and (3.18), we know that
Hence,
Arguing by contradiction, suppose that \( \textrm{u}_{m} \not \rightarrow \tilde{\textrm{u}}^{*}\) in \(W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) .\) Then, we will have
Thus, using (3.19), we have
According to (3.9) and (3.10), we obtain that
which implies that there exists \(m_{0} \in \mathbb {N} \) such as
Since \( \textrm{u}_{m} \in {\mathcal {N}}_{\mu }^{+} \subseteq {\mathcal {N}}_{\mu }\) and \( \psi ^{\prime }_{\textrm{u}_{m}} = s^{r^{-} - 1} \left[ \zeta _{\textrm{u}_{m}} - \mu \displaystyle \int _{{\mathcal {E}}} \vert \textrm{u}_{m} \left( z \right) \vert ^{r \left( z \right) } \ dv_{g} \left( z \right) \right] ,\) we have
Then , by (3.21), we have \(s_{1} > 0.\)
Then, function \( \psi ^{\prime }_{\tilde{\textrm{u}}^{*}} \left( \cdot \right) \) is decreasing on \(\left( 0, 1 \right) .\) Hence, from (3.21), we have
However, \(s_{1} \tilde{\textrm{u}}^{*} \in {\mathcal {N}}_{\mu }^{+}.\) Hence by (3.22), we get
which is a contradiction. Hence, \( \textrm{u}_{m} \rightarrow \tilde{\textrm{u}}^{*} \) in \(W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) holds, and we have
which implies that
Since \( \textrm{u}_{m} \in {\mathcal {N}}_{\mu }^{+} \) for all \( m \in \mathbb {N},\) we have
Letting \( m \rightarrow + \infty ,\) we have
Remind that \( \mu \in \left( 0, \tilde{\mu }^{*} \right) \) and \( \tilde{\mu }^{*} \le \mu ^{*}.\) Then, by Lemma 3.2, we find that equality in (3.23) cannot hold. Therefore, we conclude that \(\tilde{\textrm{u}}^{*} \left( z \right) > 0\) for a.a \( z \in {\mathcal {E}}\) with \( \tilde{\textrm{u}}^{*} \ne 0.\) This completes the proof of Lemma 3.4. \(\square \)
Lemma 3.5
Suppose that hypotheses \(\left( \textit{i} \right) - \left( \textit{v} \right) \) hold, suppose \( w \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) and let \( \mu \in \left( 0, \tilde{\mu }^{*} \right] .\) Then, there exists \( \alpha > 0 \) such as for all \( s \in \left[ 0, \alpha \right] ,\) we have
Proof
Let \(\eta _{w} : \left[ 0, + \infty \right] \longrightarrow \mathbb {R}\) given as
Since \( \tilde{\textrm{u}}^{*} \in {\mathcal {N}}_{\mu }^{+},\) we have \(\eta _{w} \left( 0 \right) > 0.\) According to the continuity of \(\eta _{w} \left( \cdot \right) \) we find \( \alpha > 0\) such as
Thus, \( \tilde{\textrm{u}}^{*} + s w \in {\mathcal {N}}_{\mu }^{+}\) for all \( s \in \left[ 0, \alpha \right] .\) Hence, by Lemma 3.4, we deduce that
This ends the demonstration. \(\square \)
Proposition 3.6
Under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) ,\) let \( \mu \in \left( 0, \tilde{\mu }^{*} \right] .\) Then, \(\tilde{\textrm{u}}^{*}\) is a weak solution of \(\left( {\mathcal {P}} \right) .\)
Proof
Let \( w \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) ,\) by Lemma 3.5 we have for all \( s \in \left[ 0, \alpha \right] \) that
Then,
Dividing the above inequality by s, then \( s \rightarrow 0^{+}.\) We deduce that
Since, \( w \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) is arbitrary. Then that equality must hold, and so \(\tilde{\textrm{u}}^{*} \) is a weak solutions of \( \left( {\mathcal {P}} \right) \) for all \( \mu \in \left( 0, \tilde{\mu }^{*} \right] .\) \(\square \)
Now, using the manifold \( {\mathcal {N}}_{\mu }^{-},\) we will achieve a second weak solution when the parameter \(\mu > 0\) is sufficiently small.
Lemma 3.7
If hypotheses \(\left( \textit{i} \right) - \left( \textit{v} \right) \) are satisfied. Then, there exists \(\tilde{\mu }^{*}_{0} \in \left( 0, \tilde{\mu }^{*} \right] \) such as for all \( \mu \in \left( 0, \tilde{\mu }^{*}_{0} \right] \) we have \(\textrm{ J}_{\mu } \bigg \vert _{{\mathcal {N}}_{\mu }^{-}} \ge 0.\)
Proof
Let \( \textrm{u} \in {\mathcal {N}}_{\mu }^{-}.\) According to Lemma 3.4, Theorem 2.9 and the definition of \({\mathcal {N}}_{\mu }^{-}\), we obtain
Then,
where c being the Poincaré constant. Thus, by Theorem 2.9, we have that
where \(c_{8} \) is the embedding Theorem 2.9 constant. Hence,
If we suppose that the Lemma is invalid. Then by contradiction, we can find \( \textrm{u} \in {\mathcal {N}}_{\mu }^{-}\) such as \(\textrm{ J}_{\mu } \left( \textrm{u} \right) < 0,\) that is
Since \( \textrm{u} \in {\mathcal {N}}_{\mu }^{-} \subseteq {\mathcal {N}}_{\mu },\) we know that
from (3.27) and (3.28), we have
which implies that
Thus, since \(q^{-}< p^{-} < r^{+} \), we have
Hence,
Applying (3.29) in (3.26), we get
Hence,
which is a contradiction, since \(1< q^{+} < r^{-} \) and \(\gamma \left( \cdot \right) \in \left( 0, 1 \right) .\)
Thus, we conclude that we can find \( \tilde{\mu }^{*}_{0} \in \left( 0, \tilde{\mu }^{*} \right] \) such as for all \(\mu \in \left( 0, \tilde{\mu }^{*}_{0} \right] \), we have
\(\square \)
Lemma 3.8
Under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) ,\) let \(\mu \in \left( 0, \tilde{\mu }^{*}_{0} \right] .\) Then, there exists \(\tilde{v}^{*} \in {\mathcal {N}}_{\mu }^{-}\) with \( \tilde{v}^{*} \ge 0\) such as
Proof
Using the same method as Lemma 3.4. If \(\left\{ v_{m} \right\} _{m \in \mathbb {N}} \subseteq {\mathcal {N}}_{\mu }^{-}\) is a minimizing sequence, then, by Lemma 3.1, we have that \(\left\{ \textrm{u}_{m} \right\} _{m \in \mathbb {N}} \subseteq W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) \) is bounded. Then, we suppose that
From (3.9) and (3.10), we can find \(0 < s_{2} \) such as
We contend as in the proof of Lemma 3.4 and using (3.30), we obtain that \( \tilde{v}^{*} \in {\mathcal {N}}_{\mu }^{-}, \ \tilde{v}^{*} \ge 0, \quad \sigma _{\mu }^{-} = \textrm{ J}_{\mu } \left( \tilde{v}^{*} \right) .\) \(\square \)
Lemma 3.9
Under assumptions \(\left( \textit{i} \right) - \left( \textit{v} \right) \) and \(\mu \in \left( 0, \tilde{\mu }^{*} \right] , \ \tilde{v}^{*} \) is a weak solution of \(\left( {\mathcal {P}} \right) \).
Proof
To prove this Lemma, we use the same reasoning as in the proofs of Lemma 3.5 and Proposition 3.6. \(\square \)
According to the above Lemmas, our problem \(\left( {\mathcal {P}} \right) \) has at least two positive solutions \(\tilde{\textrm{u}}^{*}, \ \tilde{v}^{*} \in W_{0}^{1, q \left( z \right) } \left( {\mathcal {E}} \right) ,\) such as \(\textrm{ J}_{\mu } \left( \tilde{\textrm{u}}^{*} \right) < 0 \le \textrm{ J}_{\mu } \left( \tilde{v}^{*} \right) \) for all \( \mu \in \left( 0, \ \tilde{\mu }^{*}_{0} \right] \), where \(\tilde{\mu }^{*}_{0} > 0.\)
Data Availability
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Benslimane, O., Aberqi, A. Singular two-phase problem on a complete manifold: analysis and insights. Arab. J. Math. 13, 45–62 (2024). https://doi.org/10.1007/s40065-023-00443-y
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DOI: https://doi.org/10.1007/s40065-023-00443-y