Abstract
The paper deals with the following double phase problem
where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \Omega \), \(N\ge 2\), m represents a Kirchhoff coefficient, \(1<p<q<p^*\) with \(p^*=Np/(N-p)\) being the critical Sobolev exponent to p, a bounded weight \(a(\cdot )\ge 0\), \(\lambda >0\) and \(\gamma \in (0,1)\). By the Nehari manifold approach, we establish the existence of at least one weak solution.
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1 Introduction
In this paper, we combine the effects of a nonlocal Kirchhoff coefficient and a double phase operator with a singular term and a critical Sobolev nonlinearity. Precisely, we study the problem
![figure a](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs12215-022-00762-7/MediaObjects/12215_2022_762_Figa_HTML.png)
where along the paper, and without further mentioning, \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \Omega \), dimension \(N\ge 2\), \(\lambda >0\) is a real parameter and exponent \(\gamma \in (0,1)\). The main operator \(\mathcal {L}_{p,q}^{a}\) is the so-called double phase operator given by
with \(W^{1,\mathcal {H}}_0(\Omega )\) being the homogeneous Musielak-Orlicz Sobolev space where we assume that
- (\(\hbox {h}_1\)):
-
\(1<p<N\), \(p<q<p^*\) and \(0 \le a(\cdot )\in L^\infty (\Omega )\) with \(p^*\) being the critical Sobolev exponent to p given by
$$\begin{aligned} p^*=\frac{Np}{N-p}. \end{aligned}$$(1.2)
While the nonlocal term m in (\(P_\lambda \)) denotes a Kirchhoff coefficient satisfying
- (\(\hbox {h}_2\)):
-
\(m:[0,\infty ) \rightarrow [0,\infty )\) is a continuous function defined by
$$\begin{aligned} m(t) = a_0 + b_0 t^{\theta -1} \quad \text {for all } t\ge 0, \end{aligned}$$where \(a_0\ge 0\), \(b_0>0\) with \( \theta \in \left[ 1,p^*/q\right) \).
Problem (\(P_\lambda \)) is said to be of double phase type because of the presence of two different elliptic growths p and q. The study of double phase problems and related functionals originates from the seminal paper by Zhikov [25], where he introduced for the first time in literature the related energy functional to (1.1) defined by
This kind of functional has been used to describe models for strongly anisotropic materials in the context of homogenization and elasticity. Indeed, the modulating coefficient \(a(\cdot )\) dictates the geometry of composites made of two different materials with distinct power hardening exponents p and q. From the mathematical point of view, the behavior of (1.3) is related to the sets on which the weight function \(a(\cdot )\) vanishes or not. In this direction, Zhikov found other mathematical applications for (1.3) in the study of duality theory and of the Lavrentiev gap phenomenon, as shown in [26, 27]. Also, (1.3) belongs to the class of the integral functionals with nonstandard growth condition, according to Marcellini’s terminology [22, 23]. Following this line of research, Mingione et al. provide famous results in the regularity theory of local minimizers of (1.3), see, for example, the works of Baroni-Colombo-Mingione [4, 5] and Colombo-Mingione [9, 10].
Starting from [25], several authors studied existence and multiplicity results for nonlinear problems driven by (1.1) with the help of different variational techniques. In particular, Fiscella-Pinamonti [18] introduced two different double phase problems of Kirchhoff type, with the same variational structure set in \(W^{1,\mathcal {H}}_0(\Omega )\). By the mountain pass and fountain theorems, existence and multiplicity results are provided in [18]. Following this direction, in [17] Fiscella-Marino-Pinamonti-Verzellesi consider some classes of Kirchhoff type problems on a double phase setting but with nonlinear boundary conditions. Combining variational methods, truncation arguments and topological tools, different multiplicity results are established. Recently, the authors [2] were able to study a Kirchhoff problem like (\(P_\lambda \)), but involving a subcritical term. By a suitable Nehari manifold decomposition, the existence of two different solutions are provided in [2]. We also mention the works of Cammaroto-Vilasi [7], Isernia-Repovš [20] and Ambrosio-Isernia [1] for Kirchhoff type problems driven by the \(p(\cdot )\)-Laplacian or the (p, q)-Laplacian.
The main novelty, as well as the main difficulty, of problem (\(P_\lambda \)) is the presence of a critical Sobolev nonlinearity. Indeed, in order to overcome the lack of compactness at critical levels arising from the presence of the critical term in (\(P_\lambda \)), the same fibering analysis used in [2] cannot work. For this, we exploit other variational tools inspired by more recent situations as in [14]. For this, Farkas-Fiscella-Winkert [14] used a suitable convergence analysis of gradients in order to handle the critical Sobolev nonlinearity of problem
Following this direction, we mention [15, 16] concerning existence results for critical double phase problems involving a singular term and defined on Minkowski spaces in terms of Finsler manifolds, that is driven by the Finsler double phase operator
where \(({\mathbb {R}}^N,F)\) stands for a Minkowski space. While, Crespo-Blanco-Papageorgiou-Winkert [12] consider a nonhomogeneous singular Neumann double phase problem with critical growth on the boundary, given by
By the fibering approach introduced by Drábek-Pohozaev [13] along with a Nehari manifold decomposition, the existence of at least two solutions of (1.4) is obtained in [12].
Inspired by the above papers, we solve problem (\(P_\lambda \)) by a variational approach. Indeed, a function \(u \in W^{1,\mathcal {H}}_0(\Omega )\) is said to be a weak solution of problem (\(P_\lambda \)) if \(u^{-\gamma }\varphi \in L^1(\Omega )\), \(u>0\) a.e. in \(\Omega \) and
is satisfied for all \(\varphi \in W^{1,\mathcal {H}}_0(\Omega )\), where \(\langle \cdot ,\cdot \rangle \) denotes the duality pairing between \(W^{1,\mathcal {H}}_0(\Omega )\) and its dual space \(W^{1,\mathcal {H}}_0(\Omega )^*\) In particular, the weak solutions of (\(P_\lambda \)) are the critical points of the energy functional \(J_\lambda :W^{1,\mathcal {H}}_0(\Omega )\rightarrow {\mathbb {R}}\) given by
for any \(u\in W^{1,\mathcal {H}}_0(\Omega )\), where
Hence, the main result reads as follows.
Theorem 1.1
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied. Then there exists \(\lambda ^*>0\) such that for all \(\lambda \in (0,\lambda ^*]\) problem (\(P_\lambda \)) has at least one weak solution \(u_\lambda \) such that \(J_\lambda (u_\lambda )<0\).
The proof of Theorem 1.1 is based on a suitable minimization argument on the Nehari manifold. For this, we extract a minimizing sequence whose energy values converge to a negative number. However, in order to verify that the sequence actually converges to a solution of (\(P_\lambda \)) we need a truncation argument combined with a delicate gradient analysis, inspired by [14].
The paper is organized as follows. In Sect. 2, we recall the main properties of Musielak-Orlicz Sobolev spaces \(W^{1,\mathcal {H}}_0(\Omega )\) and state the main embeddings concerning these spaces. Section 3 gives a detailed analysis of the fibering map, presents the main properties of suitable subsets of the Nehari manifold and finally shows the existence of a weak solution of problem (\(P_\lambda \)).
2 Preliminaries
In this section, we will present the main properties and embedding results for Musielak-Orlicz Sobolev spaces. First, we denote by \(L^{r}(\Omega )=L^r(\Omega ;{\mathbb {R}})\) and \(L^r(\Omega ;{\mathbb {R}}^N)\) the usual Lebesgue spaces with the norm \(\Vert \cdot \Vert _r\) and the corresponding Sobolev space \(W^{1,r}_0(\Omega )\) is equipped with the norm \(\Vert \nabla \cdot \Vert _r\), for \(1\le r\le \infty \).
Suppose hypothesis (\(\hbox {h}_1\)) and consider the nonlinear function \(\mathcal {H}:\Omega \times [0,\infty )\rightarrow [0,\infty )\) defined by
The Musielak-Orlicz Lebesgue space \(L^\mathcal {H}(\Omega )\) is given by
equipped with the Luxemburg norm
where the modular function is given by
Next, we recall the relation between the norm \(\Vert \,\cdot \,\Vert _{\mathcal {H}}\) and the modular function \(\varrho _{\mathcal {H}}\), see Liu-Dai [21, Proposition 2.1] or Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.13].
Proposition 2.1
Let (\(\hbox {h}_1\)) be satisfied, \(u\in L^{\mathcal {H}}(\Omega )\) and \(c>0\). Then the following hold:
-
(i)
If \(u\ne 0\), then \(\Vert u\Vert _{\mathcal {H}}=c\) if and only if \( \varrho _{\mathcal {H}}(\frac{u}{c})=1\);
-
(ii)
\(\Vert u\Vert _{\mathcal {H}}<1\) (resp. \(>1\), \(=1\)) if and only if \( \varrho _{\mathcal {H}}(u)<1\) (resp. \(>1\), \(=1\));
-
(iii)
If \(\Vert u\Vert _{\mathcal {H}}<1\), then \(\Vert u\Vert _{\mathcal {H}}^q\le \varrho _{\mathcal {H}}(u)\le \Vert u\Vert _{\mathcal {H}}^p\);
-
(iv)
If \(\Vert u\Vert _{\mathcal {H}}>1\), then \(\Vert u\Vert _{\mathcal {H}}^p\le \varrho _{\mathcal {H}}(u)\le \Vert u\Vert _{\mathcal {H}}^q\);
-
(v)
\(\Vert u\Vert _{\mathcal {H}}\rightarrow 0\) if and only if \( \varrho _{\mathcal {H}}(u)\rightarrow 0\);
-
(vi)
\(\Vert u\Vert _{\mathcal {H}}\rightarrow \infty \) if and only if \( \varrho _{\mathcal {H}}(u)\rightarrow \infty \).
Moreover, we define the weighted space
endowed with the seminorm
The corresponding Musielak-Orlicz Sobolev space \(W^{1,\mathcal {H}}(\Omega )\) is defined by
equipped with the norm
where \(\Vert \nabla u\Vert _\mathcal {H}=\Vert \,|\nabla u|\,\Vert _{\mathcal {H}}\). In addition, we denote by \(W^{1,\mathcal {H}}_0(\Omega )\) the completion of \(C^\infty _0(\Omega )\) in \(W^{1,\mathcal {H}}(\Omega )\). Thanks to hypothesis (\(\hbox {h}_1\)), we know that
is an equivalent norm in \(W^{1,\mathcal {H}}_0(\Omega )\), see Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.16(ii)]. Furthermore, it is known that \(L^\mathcal {H}(\Omega )\), \(W^{1,\mathcal {H}}(\Omega )\) and \(W^{1,\mathcal {H}}_0(\Omega )\) are uniformly convex and so reflexive Banach spaces, see Colasuonno-Squassina [8, Proposition 2.14] or Harjulehto-Hästö [19, Theorem 6.1.4].
Finally, we recall some useful embedding results for the spaces \(L^\mathcal {H}(\Omega )\) and \(W^{1,\mathcal {H}}_0(\Omega )\), see Colasuonno-Squassina [8, Proposition 2.15] or Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Propositions 2.17 and 2.19].
Proposition 2.2
Let (\(\hbox {h}_1\)) be satisfied and let \(p^*\) be the critical exponent to p given in (1.2). Then the following embeddings hold:
-
(i)
\(L^{\mathcal {H}}(\Omega ) \hookrightarrow L^{r}(\Omega )\) and \(W^{1,\mathcal {H}}_0(\Omega )\hookrightarrow W^{1,r}_0(\Omega )\) are continuous for all \(r\in [1,p]\);
-
(ii)
\(W^{1,\mathcal {H}}_0(\Omega )\hookrightarrow L^{r}(\Omega )\) is continuous for all \(r \in [1,p^*]\) and compact for all \(r \in [1,p^*)\);
-
(iii)
\(L^{\mathcal {H}}(\Omega ) \hookrightarrow L^q_a(\Omega )\) is continuous;
-
(iv)
\(L^{q}(\Omega )\hookrightarrow L^{\mathcal {H}}(\Omega ) \) is continuous.
3 Proof the main result
In order to solve problem (\(P_\lambda \)), we apply a minimization argument for \(J_\lambda \) on a suitable subset of \(W^{1,\mathcal {H}}_0(\Omega )\). For this, we define the fibering function \(\psi _u:[0,\infty ) \rightarrow {\mathbb {R}}\) defined by
which gives
It is easy to see that \(\psi _u \in C^{\infty }((0,\infty ))\). In particular, we have for \(t>0\)
and
Thus, we can introduce the Nehari manifold related to our problem which is defined by
In particular, we have \(u \in \mathcal {N}_\lambda \) if and only if
Also \(tu\in \mathcal {N}_\lambda \) if and only if \(\psi _{tu}'(1)=0\). Observe that \(\mathcal {N}_\lambda \) contains all weak solutions of (\(P_\lambda \)). Moreover, we define the following subsets of \(\mathcal {N}_\lambda \)
In contrast to [2] we are not going to study the set \(\mathcal {N}_\lambda ^{-} = \left\{ u\in \mathcal {N}_\lambda \,:\, \psi _u''(1)< 0\right\} \). The next Lemma can be shown as in [2, Lemmas 3.1 and 3.2] replacing r by \(p^*\).
Lemma 3.1
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied.
-
(i)
The functional \(J_\lambda \big |_{\mathcal {N}_\lambda }\) is coercive and bounded from below for any \(\lambda >0\).
-
(ii)
There exists \(\Lambda _1>0\) such that \(\mathcal {N}_\lambda ^\circ =\emptyset \) for all \(\lambda \in (0,\Lambda _1)\).
Let S be the best Sobolev constant in \(W_0^{1,p}(\Omega )\) defined as
Note that we can write \(\psi _u'(t)\) in the form
where
From this definition we see that \(tu\in \mathcal {N}_\lambda \) if and only if
The next Lemma shows that \(\mathcal {N}_\lambda ^+\) is nonempty whenever \(\lambda \) is sufficiently small.
Lemma 3.2
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied and let \(u\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}\). Then there exist \(\Lambda _2>0\) and unique \(t_1^u<t_{\max }^u<t_2^u\) such that
whenever \(\lambda \in (0,\Lambda _2)\). In particular, \(t_1^u u \in \mathcal {N}_\lambda ^+\) for \(\lambda \in (0,\Lambda _2)\).
Proof
For \(u\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}\) the equation
can be equivalently written as
From \(p^*>q\theta \) and \(\theta \ge 1\) we see that
We denote the left-hand side of (3.4) by
Then, from (3.5) and \(0<\gamma<1<p<q<p^*\), we know that
From the intermediate value theorem along with (i) and (ii) we can find \(t_{\max }^u>0\) such that (3.4) holds. In addition, (iii) implies that \(t_{\max }^u\) is unique due to the injectivity of \(T_u\). Moreover, if we consider \(\sigma '_u(t)>0\), then in place of (3.4) we get
Since \(T_u\) is strictly decreasing, this holds for all \(t<t_{\max }^u\). The same can be said for \(\sigma _u'(t)<0\) and \(t>t_{\max }^u\). Hence, \(\sigma _u\) is injective in \((0,t_{\max }^u)\) and in \((t_{\max }^u,\infty )\). Furthermore,
with the global maximum \(t_{\max }^u>0\) of \(\sigma _u\). Moreover, we have
Applying the estimate \(p \phi _{\mathcal {H}}(\nabla u) \ge \Vert \nabla u\Vert _p^p\) we obtain
which by using Hölder’s inequality and (3.1) results in
Note that \(\sigma _u\) is increasing on \((0,t_{\max }^u)\). Hence from \(p \phi _{\mathcal {H}}(\nabla u) \ge \Vert \nabla u\Vert _p^p\), \(p<q\), Hölder’s inequality, (3.1) and the representation of \(t_0^u\) in (3.7) we have
where \(\Lambda _2\) is given by
From the considerations above we conclude that
whenever \(\lambda \in (0,\Lambda _2)\). Since \(\sigma _u\) is injective in \((0,t_{\max }^u)\) and in \((t_{\max }^u,\infty )\), we can find unique \(t_1^u, t_2^u>0\) such that
Due to (3.3) we have \(t_1^u u\in \mathcal {N}_\lambda \). Then, from the representation in (3.2), we observe that
Finally, since \(\psi _u'(t_1^u)=\psi _u'(t_2^u)=0\) and \(\sigma _u'(t_2^u)<0<\sigma _u'(t_1^u)\) we derive that
This shows, in particular, that \(t_1^u u \in \mathcal {N}_\lambda ^+\) for \(\lambda \in (0,\Lambda _2)\). \(\square \)
Next we show that the modular \(\varrho _\mathcal {H}(\nabla \cdot )\) is upper bounded with respect to the elements of \(\mathcal {N}_\lambda ^+\). The proof is similar to that in [2, Proposition 3.4] and so we omitted it.
Lemma 3.3
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied. Then there exist \(\Lambda _3>0\) and constant \(D_1=D_1(\lambda )>0\) such that
for every \(u\in \mathcal {N}_\lambda ^+\) and for every \(\lambda \in (0,\Lambda _3)\).
By Lemma 3.1(ii), we observe that \(\mathcal N^+_\lambda \) is closed in \(W^{1,\mathcal {H}}_0(\Omega )\) for \(\lambda >0\) small enough. We define
The next proposition shows that \(\Theta _\lambda ^+ <0\). We refer to [2, Proposition 4.1] for its proof.
Proposition 3.4
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied and let \(\lambda \in (0,\min \{\Lambda _1,\Lambda _2\})\), with \(\Lambda _1\), \(\Lambda _2\) given in Lemmas 3.1(ii) and 3.2. Then \(\Theta _\lambda ^+<0\).
Based on the implicit function theorem in its version stated in Berger [6, p. 115] we can proof the following Lemma which proof is similar to the one in [2, Lemma 4.2].
Lemma 3.5
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied and let \(\lambda >0\). Let us consider \(u\in \mathcal {N}_\lambda ^+\). Then there exist \(\varepsilon >0\) and a continuous function \(\zeta :B_\varepsilon (0) \rightarrow (0,\infty )\) such that
where \(B_\varepsilon (0) := \{v\in W^{1,\mathcal {H}}_0(\Omega )\,:\, \Vert v\Vert <\varepsilon \}\).
Now, we set \(\Lambda ^*:=\min \{\Lambda _1,\Lambda _2,\Lambda _3\}\) with \(\Lambda _1\), \(\Lambda _2\) and \(\Lambda _3>0\) given in Lemmas 3.1(ii), 3.2 and 3.3 . Let \(\lambda \in (0,\Lambda ^*)\). Applying Ekeland’s variational principle, we obtain a sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) satisfying
for any \(u\in {\mathcal {N}}_\lambda ^+\). By Lemma 3.1(i), we know that \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(W^{1,\mathcal {H}}_0(\Omega )\). Hence, by Proposition 2.2(ii) along with the reflexivity of \(W^{1,\mathcal {H}}_0(\Omega )\), there exist a subsequence, still denoted by \(\{u_n\}_{n\in {\mathbb {N}}}\), and an element \(u_\lambda \in W^{1,\mathcal {H}}_0(\Omega )\) such that
for any \(s\in [1,p^*)\). By the coercivity given in Lemma 3.1(i), we can assume that there exist \(E_1,\;E_2\ge 0\) such that
We get the following technical results.
Lemma 3.6
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied, let \(\lambda \in (0,\Lambda ^*)\) and let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) be a sequence satisfying (3.8)–(3.9). Then \(u_\lambda \ne 0\).
Proof
Let us assume by contradiction that \(u_\lambda =0\). Then \(\psi '_{u_n}(1)=0\) implies
Using (3.10), (3.11) and letting \(n\rightarrow \infty \), we get
where we set
Moreover by (3.8) we have
which implies that
Recall that \(E_1\), \(E_2\ge 0\). Then, taking the value of \(d^{p^*}\) from (3.12) into (3.13), we derive that
This implies
and so
which is a contradiction because of \(p<q\le q\theta <p^*\). \(\square \)
Lemma 3.7
Let hypotheses (\(\hbox {h}_1\))–(\(\hbox {h}_2\)) be satisfied, let \(\lambda \in (0,\Lambda ^*)\) and let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) be a sequence satisfying (3.8)–(3.9). Then \(\liminf \limits _{n\rightarrow \infty }\psi ''_{u_n}(1)>0\), that is,
Proof
Since \(\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {N}}_\lambda ^+\), we have \(\psi _{u_n}'(1) =0\) and \(\psi _{u_n}''(1)>0\), that is,
and
Thus, in order to prove the lemma, it is enough to show that
By contradicting (3.14), let us assume that
By Lebesgue dominated convergence theorem, we obtain
Using (3.16) in (3.15), we get
which yields, by applying (3.11),
From this, due to \(p<q< p^*\), we have
Considering \(\psi _{u_n}'(1)=0\) and (3.16), we have
From this and (3.17), we obtain
For any fixed \(w\in W^{1,\mathcal {H}}_0(\Omega )\setminus \{0\}\), we know that there exists a unique \(t_{\max }>0\) such that \(\sigma '_w(t_{\max })=0\). From this and (3.6), we conclude that
It is easy to verify that \(t_{\max }\ge t_{00}\ge t^w_0 \) as defined in (3.7) and from the proof of Lemma 3.2, we know that \(\Lambda _2>0\) is chosen in such a way that
We define
with \(t_{00}\) given in (3.20). Taking \(w= u_n\) in (3.21) and then passing to the limit as \(n\rightarrow \infty \) we get
On the other hand, by Lemma 3.6 and (3.11), we have that at least one of \(E_1\) and \(E_2\) is not zero. Let us assume, without any loss of generality, that \(E_1>0\), \(E_2 \ge 0\). Then by (3.18), (3.19), (3.20) along with \(q\theta < p^*\) and \(\lambda \in (0, \Lambda _2)\), we obtain
This proves the assertion of the lemma. \(\square \)
Let \(h\in W^{1,\mathcal {H}}_0(\Omega )\) be nonnegative. From Lemma 3.5 there exists a sequence of maps \(\{\zeta _n\}_{n\in {\mathbb {N}}}\) such that \(\zeta _n(0) =1\) and \(\zeta _n(th)(u_n+th)\in {\mathcal {N}}_\lambda ^+\) for sufficiently small \(t>0\) and for each \(n\in {\mathbb {N}}\). From this and \(u_n\in {\mathcal {N}}_\lambda \), we have the equations
and
where \(w_n = u_n+th\).
Lemma 3.8
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied, let \(\lambda \in (0,\Lambda ^*)\) and let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) be a sequence satisfying (3.8)–(3.9). For any nonnegative function \(h\in W^{1,\mathcal {H}}_0(\Omega )\), the sequence \(\{\langle \zeta '_n(0),h \rangle \}_{n\in {\mathbb {N}}}\) is uniformly bounded.
Proof
Subtracting (3.22) from (3.23), we get
For notational convenience, we set
We have the following limits
Taking into account
since h is nonnegative, dividing both sides of (3.24) by \(t>0\) and then passing the limit as \(t\rightarrow 0^+\), we obtain
This implies
Therefore, using the fact that \(u_n \in \mathcal {N}_\lambda \), we have
Now using Lemma 3.7 and taking into account the boundedness of \(\{u_n\}_{n\in {\mathbb {N}}}\) in \(W^{1,\mathcal {H}}_0(\Omega )\), we infer that \(\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}\) is bounded below for any nonnegative \(h\in W^{1,\mathcal {H}}_0(\Omega )\).
It remains to show that \(\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}\) is bounded above for any nonnegative \(h\in W^{1,\mathcal {H}}_0(\Omega )\). Assume by contradiction that \(\limsup _{n\rightarrow \infty } \langle \zeta _n'(0),h \rangle =\infty \). Thus, without loss of generality, we can consider \(\zeta _n(th)>\zeta _n(0)=1\) for \(n\in {\mathbb {N}}\) large enough . It is easy to see that
Applying this in (3.9) with \(u= \zeta _n(th)w_n\), we get
Using (3.22) and (3.23) in the inequality above, we obtain
Now dividing the above inequality by \(t>0\), passing to the limit as \(t\rightarrow 0^+\) and using (3.25), we have
which gives a contradiction if we take the limits \(n\rightarrow \infty \) on both sides, considering \(\limsup _{n\rightarrow \infty }\langle \zeta _n'(0),h \rangle =\infty \), since by Lemma 3.7 and the boundedness of \(\{u_n\}_{n\in {\mathbb {N}}}\), there exists some \(M_1>0\) such that
for \(n\in {\mathbb {N}}\) large enough. Thus \(\{\langle \zeta _n'(0),h \rangle \}_{n\in {\mathbb {N}}}\) must be bounded above. \(\square \)
Since \(J_\lambda (u_n)=J_\lambda (|u_n|)\), without loss of generality, we may assume that \(u_n\ge 0\) a. e. in \(\Omega \) and so, \(u_\lambda \ge 0\) a. e. in \(\Omega \). With this assumption, we state our next result.
Lemma 3.9
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied, let \(\lambda \in (0,\Lambda ^*)\) and let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) be a sequence satisfying (3.8)–(3.9). For any \(h\in W^{1,\mathcal {H}}_0(\Omega )\) and \(n\in {\mathbb {N}}\), \(u_n^{-\gamma }h \in L^1(\Omega )\) and as \(n\rightarrow \infty \)
Proof
Let \(h\in W^{1,\mathcal {H}}_0(\Omega )\) be nonnegative and recall the following estimate from the proof of Lemma 3.8
Dividing the above equation with \(t>0\) and then passing to limit as \(t\rightarrow 0^+\), we get
where we used \(u_n\in {\mathcal {N}}_\lambda \) that is \(\psi '_{u_n}(1)=0\). This implies
Observe that \(|u_n+th|^{1-\gamma }-|u_n|^{1-\gamma }\ge 0\), so we can use Fatou’s lemma in (3.27) to obtain
Recall that \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(W^{1,\mathcal {H}}_0(\Omega )\). Then, passing to the limit as \(n\rightarrow \infty \) in the above estimate, we obtain
for each nonnegative \(h\in W^{1,\mathcal {H}}_0(\Omega )\), where we used the uniform boundedness from Lemma 3.8.
We aim to establish that (3.28) holds true for any arbitrary \(h\in W^{1,\mathcal {H}}_0(\Omega )\). For this, we replace h in (3.28) by \((u_n+\varepsilon h)^+\) with \(\varepsilon >0\) and \(h\in W^{1,\mathcal {H}}_0(\Omega )\). Renaming as \(h_{\varepsilon } = u_n+\varepsilon h\) and using (3.28), we get
We define \(\Omega _\varepsilon = \{x\in \Omega \,:\, u_n+\varepsilon h \le 0\}\). Using \(u_n\in {\mathcal {N}}_\lambda \) and \(\int _\Omega u_n^{-\gamma } h_\varepsilon ^-\,\mathrm {d}x\ge 0\) in the above estimate, we get
Note that
and similarly,
Moreover, applying Hölder’s inequality and \(u_n\le -\varepsilon h\) in \(\Omega _\varepsilon \), we have
Putting all these in (3.29), we infer that
Since \(|\Omega _\varepsilon |\rightarrow 0\) as \(\varepsilon \rightarrow 0^+\) and by the boundedness of \(\{u_n\}_{n\in {\mathbb {N}}}\) in \(W^{1,\mathcal {H}}_0(\Omega )\), if we divide (3.30) by \(\varepsilon >0\) and then pass to the limit as \(\varepsilon \rightarrow 0^+\), we obtain
as \(n\rightarrow \infty \). By the arbitrariness of \(h\in W^{1,\mathcal {H}}_0(\Omega )\), (3.31) actually implies (3.26) which completes the proof. \(\square \)
Now, we prove the compactness property of the energy functional \(J_\lambda \) in a suitable range of \(\lambda \). For this purpose, we set for any \(\lambda >0\)
where
and
Also, for any \(k \in {\mathbb {N}}\), let \(T_k\) be the truncation defined by
Proposition 3.10
Let hypotheses (\(\hbox {h}_1\))-(\(\hbox {h}_2\)) be satisfied, let \(\lambda \in (0,\Lambda ^*)\) and let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \mathcal {N}_\lambda ^+\) be a sequence satisfying (3.8)–(3.9) and
Then \(\{u_n\}_{n\in {\mathbb {N}}}\) possesses a strongly convergent subsequence in \(W^{1,\mathcal {H}}_0(\Omega )\).
Proof
Fixing \(k \in {\mathbb {N}}\) and taking \(h= T_k(u_n-u_\lambda ) \in W^{1,\mathcal {H}}_0(\Omega )\) as a test function in (3.26), we get
Using Young’s inequality, Propositions 2.1(iii)–(iv), 2.2(ii) and boundedness of the sequences \(\{u_n\}_{n \in {\mathbb {N}}}\), \(\{T_k(u_n-u_\lambda )\}_{n \in {\mathbb {N}}}\) in \(W^{1,\mathcal {H}}_0(\Omega )\), we obtain
with a constant C independent of n and k, that is, the sequence \(\{u_n^{-\gamma } \ T_k(u_n-u_\lambda )\}_{n \in {\mathbb {N}}}\) is uniformly integrable. Then, using (3.10) and Vitali’s convergence theorem, we get
By Hölder’s inequality, we observe that
is a bounded linear functional. From (3.10), we see that \(\nabla T_k(u_n -u_\lambda ) \rightharpoonup 0\) in \([L^{\mathcal {H}}(\Omega )]^N\), so we can get
Let \(\phi _{\mathcal {H}}(\nabla u_n) \rightarrow \beta := \frac{E_1}{p} + \frac{E_2}{q}\) as \(n\rightarrow \infty \), where \(E_1\) and \(E_2\) are defined in (3.11). Thus, by using (3.36)–(3.37) in (3.35) and the fact that \(a_0 \ge 0\), \(b_0>0, \beta >0\), we get
By Simon’s inequalities, see [24, formula (2.2)], we rewrite the above estimate as
Set
Note that \(s_n(x) \ge 0\) a. e. in \(\Omega \). We divide the set \(\Omega \) by
where \(k, n \in {\mathbb {N}}\) are fixed. Let \(\eta \in (0,1)\). Then, from the definition of \(T_k\), Hölder’s inequality, (3.38) and the fact that \(\lim _{n \rightarrow \infty } |F_n^k| =0\), we get
Letting \(k \rightarrow 0^+\), we obtain that \(s_n^\eta \rightarrow 0\) in \(L^1(\Omega )\). Thus, we may assume that \(s_n \rightarrow 0\) a. e. in \(\Omega \) (up to a subsequence) which along with Simon’s inequalities [24, formula (2.2)] gives that
Let M be the nodal set of the weight function \(a(\cdot )\) given by
Since, the sequences \(\{|\nabla u_n|^{p-2} \nabla u_n\}_{n \in {\mathbb {N}}}\) and \(\{|\nabla u_n|^{q-2} \nabla u_n\}_{n \in {\mathbb {N}}}\) are bounded in \(L^{p'}(\Omega )\) and \(L^{q'}(\Omega \setminus M, a(x) \,\mathrm {d}x)\), respectively, then by using (3.39) and [3, Proposition A.8], we conclude that
and
Furthermore, using (3.10), (3.39) and the Brezis-Lieb Lemma, we obtain
as \(n \rightarrow \infty \). Let \(\Vert u_n -u_\lambda \Vert _{p^*} \rightarrow \ell \) for some \(\ell \ge 0.\) Now, by taking \(u_n-u_\lambda \) as a test function in (3.26), we get
as \(n \rightarrow \infty .\) Hence, by (3.10) and (3.40) it follows that
which further gives
Now, we claim that \(\ell =0\). Assume by contradiction that \(\ell >0.\) By (3.1) and (3.42), we have
Note that (3.42) implies that
Using (3.43) in (3.44), we get
From (3.45) and (3.1), we obtain
This gives
and so we have
Combining (3.45) and (3.46), we obtain
For any \(n \in {\mathbb {N}}\), we have
From this, as \(n \rightarrow \infty \), by (3.47), (3.40), Hölder’s and Young’s inequality, we derive
where \(\alpha _0\), \(\alpha _1\) are defined in (3.32). The above estimates gives a contradiction to (3.34). Hence \(\ell =0\) and using (3.41) and Proposition 2.1(v), we conclude the proof. \(\square \)
Remark 3.11
By taking \(\lambda \in (0, \Lambda _*)\) with \(\Lambda _*:= \left( \alpha _2 \alpha _1^{-1}\right) ^{\frac{p^* -1+\gamma }{p^*}}\) and \(\alpha _1\), \(\alpha _2\) are defined in (3.32) and (3.33) respectively, we have \(c_\lambda >0.\)
Proof
(Proof of Theorem 1.1) Fix \(\lambda < \lambda ^*:= \min \{\Lambda ^*, \Lambda _*\}\). From Lemma 3.1(ii) and Ekeland’s variational principle there exists a minimizing sequence \(\{u_n\}_{n \in {\mathbb {N}}} \in \mathcal {N}_\lambda ^+ \setminus \{0\}\) verifying (3.8), (3.9), (3.10) and (3.34) with \(c= \Theta _\lambda ^+\). Hence, by combining Propositions 3.4 and 3.10 , we obtain \(u_n \rightarrow u_\lambda \) strongly in \(W^{1,\mathcal {H}}_0(\Omega )\) (up to a subsequence). This further implies that \(u_\lambda \in \mathcal {N}_\lambda \) and by Lemma 3.7, we get \(u_\lambda \in \mathcal {N}_\lambda ^+\) with \(u_\lambda \) achieving \(\Theta _\lambda ^+\) since \(J_\lambda \) is continuous on \(W^{1,\mathcal {H}}_0(\Omega ).\) Since \(0 \not \in \mathcal {N}_\lambda ^+\) and \(u_n \ge 0\) we have \(u_\lambda \not \equiv 0\) and \(u_\lambda \ge 0.\) Letting \(n \rightarrow \infty \) in (3.26), we obtain that u satisfies \(u_\lambda ^{-\gamma }\varphi \in L^1(\Omega )\) and
for all \(\varphi \in W^{1,\mathcal {H}}_0(\Omega )\). Finally, by using Proposition 3.4, Lemma 3.5 and by repeating the proof of [2, Proposition 4.3 and Proposition 4.4, Step 1], we obtain \(u_\lambda > 0\) a. e. in \(\Omega .\) \(\square \)
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Acknowledgements
R. Arora acknowledges the support of the Research Grant from Czech Science Foundation, project Project GA22-17403S. A. Fiscella is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi" (INdAM). A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled "Equazioni alle derivate parziali: problemi e modelli" (Prot_20191219-143223-545) and of the FAPESP Thematic Project titled "Systems and partial differential equations" (2019/02512-5).
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Arora, R., Fiscella, A., Mukherjee, T. et al. On critical double phase Kirchhoff problems with singular nonlinearity. Rend. Circ. Mat. Palermo, II. Ser 71, 1079–1106 (2022). https://doi.org/10.1007/s12215-022-00762-7
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DOI: https://doi.org/10.1007/s12215-022-00762-7
Keywords
- Critical growth
- Double phase operator
- Fibering method
- Nehari manifold
- Nonlocal Kirchhoff term
- Singular problem