Abstract
In this paper, p-biharmonic equations involving Hardy potential and negative exponents with a parameter λ are considered. By means of the structure and properties of Nehari manifold, we give uniform lower bounds for \(\varLambda >0\), which is the supremum of the set of λ. When \(\lambda \in (0, \varLambda )\), the above problems admit at least two positive solutions.
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1 Introduction and preliminaries
In this paper, we consider a p-biharmonic equation with Hardy potential and negative exponents:
where \(0\in \varOmega \subset \mathbb{R}^{N}\) is a bounded smooth domain with \(1< p<\frac{N}{2}\), \(\Delta ^{2}_{p}u=\Delta (\vert \Delta u\vert ^{p-2} \Delta u)\) is the p-biharmonic operator. \(\lambda >0\) is a parameter, \(0<\mu <\mu _{N,p}=(\frac{(p-1)N(N-2p)}{p^{2}})^{p}\), \(0< q<1\) and \(p-1<\gamma <p^{*}-1\), where \(p^{*}=\frac{Np}{N-2p}\) is called the critical Sobolev exponent. \(f(x)\geq 0\), \(f(x)\not \equiv 0\), \(g(x)\) satisfies the requirement that the set \(\{x\in \varOmega : g(x)>0 \}\) has positive measures, \(\operatorname{supp}f \cap \{x\in \varOmega : g(x)>0 \} \neq \emptyset \) and \(f, g\in C(\overline{\varOmega })\). Biharmonic equations describe the sport of a rigid body and the deformations of an elastic beam. For example, this type of equation provides a model for considering traveling wave in suspension bridges [5, 16, 27, 30, 36]. Various methods and tools have been adopted to deal with singular problems, such that fixed point theorems [14], topological methods [37], Fourier and Laurent transformation [18, 19], monotone iterative methods [21], global bifurcation theory [12], and degree theory [22, 31].
In recent years, there was much attention focused on the existence, multiplicity and qualitative properties of solutions for p-biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions with Hardy terms [4, 15, 17, 32, 34]. Xie and Wang [32] studied the following p-biharmonic equation with Dirichlet boundary conditions:
where \(\frac{\partial }{\partial n}\) is the outer normal derivative. By using the variational method, the existence of infinitely many solutions with positive energy levels for (1.2) was established. Huang and Liu [15] considered the following p-biharmonic equation with Navier boundary conditions:
where \(1< p<\frac{N}{2}\). By using invariant sets of gradient flows, the authors proved that (1.3) possesses a sign-changing solution. Furthermore, Yang, Zhang and Liu [34] showed that (1.3) has a positive solution, a negative solution and a sequence of sign-changing solutions when f satisfies appropriate conditions. Bhakta [4] established the qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with Hardy terms.
On the other hand, nonlinear biharmonic equations with negative exponents have been studied expensively [1, 6, 8, 13, 20]. Guerra [13] gave a complete description of entire radially symmetric solutions for the following biharmonic equation:
where \(q>1\). Moreover, Cowan et al. [8] dealt with the regularity of the extremal solution of the following fourth order boundary value problems:
Very recently, Ansari, Vaezpour and Hesaaraki [1] considered fourth order elliptic problem with the combinations of Hardy term and negative exponents,
where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\geq 1\)) is a bounded \(C^{4}\)-domain, λ and μ are positive parameters and \(0<\alpha <1\), \(0<\gamma <1\) are constants. Here M, h and k are given continuous functions satisfying suitable hypotheses. By using the Galerkin method and the sharp angle lemma, the authors proved that problem (1.4) has a positive solution for \(0<\mu < (\frac{N(N-4)}{4} ) ^{2}\).
We say that \(u\in W:=W^{2,p}(\varOmega )\cap W_{0}^{1,p}(\varOmega )\) is a weak solution of (1.1), if for every \(\varphi \in W\), there holds
The following Rellich inequality will be used in this paper:
and it is not achieved [9, 24]. For any \(u\in W\), and \(0<\mu <\mu _{N,p}\). The energy functional corresponding to (1.1) is defined by
For \(\mu \in [0,\mu _{N,p})\), W is equipped with the following norm:
Negative exponent term \(u^{-q}\) implies that \(I_{\lambda ,\mu }\) is not differential on W, therefore, critical point theory cannot be applied to the problem (1.1) directly. We consider the following manifold:
and make the following splitting for \(\mathcal{M}\):
In this paper, we will study the dependence of problem (1.1) on q, γ, f, g and Ω and evaluate the extremal value of λ related to multiplicity of positive solutions for problem (1.1). Our idea comes from [7, 28, 29]. Our results improve and complement previous ones obtained in [23, 25]. Denote \(\Vert u\Vert _{t}^{t}= \int _{\varOmega }\vert u\vert ^{t}\,dx\) and \(D^{2, p}(\mathbb{R}^{N})\) be the closure of \(C_{0}^{\infty }(\mathbb{R}^{N})\) with respect to the norm \((\int _{\mathbb{R}^{N}}\vert \Delta u\vert ^{p} \,dx )^{\frac{1}{p}}\).
\(\lambda _{1}\) denotes the smallest eigenvalue for
and \(\varphi _{1}\) denotes the corresponding eigenfunction with \(\varphi _{1}>0\) in Ω [3, 10, 26, 33, 35]. The following minimization problem will be useful in the following discussions:
and \(S_{\mu }\) is achieved by a family of functions [4, 11]. Thus, for every \(u\in W\setminus \{0\}\), \(\Vert u\Vert _{p^{*}}\leq \frac{ \Vert u\Vert _{\mu }}{\sqrt[p]{S_{\mu }}}\). Therefore, combining with the Hölder inequality, we deduce that
and
Our main results are stated in the following theorems.
Theorem 1.1
Assume that \(\lambda \in (0,\varLambda )\), where
Then problem (1.1) admits at least two solutions \(u_{0}\in \mathcal{M}^{+}\), \(U_{0}\in \mathcal{M}^{-}\), with \(\Vert U_{0}\Vert _{\mu }> \Vert u_{0}\Vert _{\mu }\).
Corollary 1.2
Let \(U_{\lambda , \mu ,\varepsilon } \in \mathcal{M}^{-}\) be the solution of problem (1.1) with \(\gamma = \varepsilon +p-1\), where \(\lambda \in (0,T_{\mu })\). Then
with
Theorem 1.3
There exists \(\lambda ^{*} =\lambda ^{*} (N, \varOmega , \mu , q, \gamma )>0\) such that problem (1.1) with \(f=g=1\) admits at least a positive solution for every \(0<\lambda <\lambda ^{*}\) and has no solution for every \(\lambda >\lambda ^{*}\).
2 Some lemmas
Lemma 2.1
Assume that \(\lambda \in (0,T_{\mu })\), where \(T_{\mu }\) is defined in (1.15). Then \(\mathcal{M}^{\pm }\neq \emptyset \) and \(\mathcal{M}^{0}=\{0\}\).
Proof
(i) We can choose \(u^{*}\in \mathcal{M}\setminus \{0 \}\) such that \(\int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx>0\) and \(\int _{\varOmega }g(x) \vert u^{*}\vert ^{\gamma +1}\,dx>0\) from the conditions imposed on f and g. Denote
Note that \(\varphi '_{\mu }(t)=(p-1-\gamma )t^{p-2-\gamma }\Vert u^{*}\Vert _{\mu }^{p}+(q+\gamma ) t^{-1-q-\gamma } \int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx\). Let \(\varphi '_{\mu }(t)=0\), we have
It is easy to check that \(\varphi _{\mu }(t)\rightarrow -\infty \) as \(t\rightarrow 0^{+}\) and \(\varphi _{\mu }(t)\rightarrow -\lambda \int _{\varOmega } g(x)\vert u^{*}\vert ^{\gamma +1}\,dx<0\) as \(t\rightarrow \infty \). Furthermore, \(\varphi _{\mu }(t)\) attains its maximum at \(t_{\max }\). By (1.12) and (1.13), we obtain
When \(A(\mu ,\lambda )=0\), we get
where we use the following two equalities:
and
In turn, this is also true. Hence \(A(\mu ,\lambda )=0\) if and only if \(\lambda =T_{\mu }\). Thus for \(\lambda \in (0,T_{\mu })\), we have \(A(\mu ,\lambda )>0\). Moreover, by (2.2), we derive that \(\varphi _{ \mu }(t_{\max })>0\). Consequently, there exist two numbers \(t_{\mu } ^{-}\) and \(t_{\mu }^{+}\) such that \(0< t_{\mu }^{-}< t_{\max }< t_{ \mu }^{+}\), and
It follows that \(t_{\mu }^{-}u^{*}\in \mathcal{M}^{+}\), and \(t_{\mu }^{+}u^{*}\in \mathcal{M}^{-}\). In fact, if \(\varphi _{\mu }(t)=0\), then
namely
Hence \(tu\in \mathcal{M}\). Furthermore, if \(\varphi '_{\mu }(t)>0\), then
That is
i.e.,
Note that \(tu\in \mathcal{M}\), we have
Thus \(tu\in \mathcal{M}^{+}\). By a similar argument, if \(\varphi _{ \mu }(t)=0\) and \(\varphi '_{\mu }(t)<0\), then \(tu\in \mathcal{M}^{-}\). Therefore, both \(\mathcal{M}^{+}\) and \(\mathcal{M}^{-}\) are non-empty sets for every \(\lambda \in (0,T_{\mu })\).
(ii) We claim that \(\mathcal{M}^{0}=\{0\}\). Otherwise, we suppose that there exists \(u_{*}\in \mathcal{M}^{0}\) and \(u_{*}\neq 0\). Since \(u_{*}\in \mathcal{M}^{0}\), we have
moreover
For \(\lambda \in (0,T_{\mu })\) and \(u_{*}\neq 0\), combining with (2.2), we deduce that
which is a contradiction, Thus \(u_{*}=0\). That is, \(\mathcal{M}^{0}= \{0\}\). □
The gap structure in \(\mathcal{M}\) is embodied in the following lemma.
Lemma 2.2
Assume that \(\lambda \in (0,T_{\mu })\), then
where
Proof
If \(u\in \mathcal{M}^{+}\subset \mathcal{M}\), then
We obtain from (1.13) that
which leads to
which implies that
If \(U\in \mathcal{M}^{-}\subset \mathcal{M}\), combining with (1.12), we derive that
which leads to
Furthermore
which means that
Therefore
where we have used the following facts:
and
Similarly
where we have applied the following equalities:
and
Consequently, \(M_{\mu }(\lambda )=M_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) and \(N_{\mu }(\lambda )=N_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) respectively. This completes the proof of Lemma 2.2. □
Lemma 2.3
Assume that \(\lambda \in (0,T_{\mu })\). Then \(\mathcal{M}^{-}\) is a closed set in W-topology.
Proof
We choose a sequence \(\{U_{n}\}\) such that \(\{U_{n}\} \subset \mathcal{M}^{-}\) and \(U_{n}\rightarrow U_{0}\) with \(U_{0} \in W\). Then
and
Hence \(U_{0} \in \mathcal{M}^{-} \cup \mathcal{M}^{0}\). By Lemma 2.2, we have
that is, \(U_{0}\neq 0\). Combining with Lemma 2.1, we obtain \(U_{0}\notin \mathcal{M}^{0}\). Thus \(U_{0}\in \mathcal{M}^{-}\). Therefore \(\mathcal{M}^{-}\) is a closed set in W-topology for every \(\lambda \in (0,T_{\mu })\). □
Lemma 2.4
For \(u\in \mathcal{M}^{\pm }\), there exist a number \(\varepsilon >0\) and a continuous function \(\widetilde{g}(h)>0\) with \(h\in W\) and \(\Vert h\Vert <\varepsilon \) such that
Proof
We only prove the case that \(\mathcal{M}^{+}\). Define a function \(\widetilde{F}: W\times \mathbb{R}^{+}\rightarrow \mathbb{R}\) by:
Note that \(u\in \mathcal{M}^{+}\), we obtain
and
At \((0,1)\), using the implicit function theorem, we know that there exists \(\overline{\varepsilon }>0\) such that for \(h\in W\) and \(\Vert h\Vert <\overline{\varepsilon }\), the equation \(\widetilde{F}(h,s)=0\) has a unique continuous solution \(s=\widetilde{g}(h)>0\). Hence \(\widetilde{g}(0)=1\) and
i.e.,
and
together with (2.3), these imply that we can choose \(\varepsilon >0\) small enough (\(\varepsilon <\overline{\varepsilon }\)) such that for every \(h\in W\) and \(\Vert h\Vert <\varepsilon \)
that is,
This completes the proof of Lemma 2.3. □
3 Proof of Theorem 1.1
For every \(u\in \mathcal{M}\), by (1.13), we have
Let
We have
Since \(\mathcal{K}''(\Vert u\Vert _{\mu })>0\) for all \(\Vert u\Vert _{\mu }>0\) with \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow 0\) as \(\Vert u\Vert _{\mu }\rightarrow 0\) and \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow \infty \) as \(\Vert u\Vert _{ \mu }\rightarrow \infty \). Therefore \(\mathcal{K}(u)\) attains its minimum at \((\Vert u\Vert _{\mu })_{\min }\), and
By (3.1), we deduce that
namely, \(I_{\lambda ,\mu }(u)\) is coercive on \(\mathcal{M}\). Combining with (3.1), we have
Thus \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). According to Lemma 2.3, if \(\lambda \in (0,T_{\mu })\), then \(\mathcal{M}^{+} \cup \mathcal{M}^{0}\) and \(\mathcal{M}^{-}\) are two closed sets in \(\mathcal{M}\). Therefore, we apply the Ekeland variational principle [2] to derive a minimizing sequence \(\{u_{n}\}\subset \mathcal{M}^{+}\cup \mathcal{M}^{0}\) satisfying:
Assume that \(u_{n}\geq 0\) on \(\varOmega \setminus \{0\}\). Note that \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). By (3.2), we get
for n large enough and a positive constant \(C_{1}\). Hence \(\{u_{n}\}\) is bounded in \(\mathcal{M}\). Let us, for a subsequence, suppose that
For every \(u\in \mathcal{M}^{+}\), we deduce from \(p>1\) that
which implies that \(\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). For \(\lambda \in (0,T_{\mu })\), it follows from Lemma 2.1 that \(\mathcal{M}^{0}=\{0 \}\). Thus \(u_{n}\in \mathcal{M}^{+}\) for n large enough and \(\inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u) =\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). Therefore
i.e., \(u_{0}\geq 0\) and \(u_{0}\neq 0\).
In the following, we prove that, when \(\lambda \in (0,T_{\mu })\),
For \(\{u_{n}\}\subset \mathcal{M}^{+}\), we have
We suppose that
It follows from \(u_{n}\in \mathcal{M}\), the weak lower semi-continuity of the norm and (3.5) that
Hence, for every \(\lambda \in (0,T_{\mu })\) and \(u_{0}\neq 0\), combining with (2.2), we obtain
which is a contradiction. In view of (3.4), we get
for n large enough and some positive constant \(C_{2}\). Since \(u_{n}\in \mathcal{M}\), we have
Set \(\phi \in \mathcal{M}\) with \(\phi \geq 0\). Using Lemma 2.4, there exists \(\widetilde{g}_{n}(t)\) such that \(\widetilde{g}_{n}(0)=1\) and \(\widetilde{g}_{n}(t)(u_{n}+t\phi )\in \mathcal{M}^{+}\). Thus
and
Therefore
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we have
where \(\widetilde{g}'_{n}(0)\) denotes the right derivative of \(\widetilde{g}_{n}(t)\) at zero. If it does not exist, \(\widetilde{g}'_{n}(0)\) should be replaced by \(\lim_{k\rightarrow \infty }\frac{\widetilde{g}_{n}(t_{k})- \widetilde{g}_{n}(0)}{t_{k}}\) for some sequence \(\{t_{k}\}_{k=1}^{ \infty }\) with \(\lim_{k\rightarrow \infty }t_{k} =0\) and \(t_{k}>0\).
Combining with (3.7) and (3.8), we have \(\widetilde{g}'_{n}(0)\neq - \infty \). Now we prove that \(\widetilde{g}'_{n}(0)\neq +\infty \). Otherwise, we suppose that \(\widetilde{g}'_{n}(0)=+\infty \). Note that \(\widetilde{g}_{n}(t)>\widetilde{g}_{n}(0)=1\) for n large enough, and
Using condition (ii) with \(u=\widetilde{g}_{n}(t)(u_{n}+t\phi ) \in \mathcal{M}^{+}\), we deduce that
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we obtain
that is,
which is not true since \(\widetilde{g}'_{n}(0)=+\infty \) and
It follows from (3.7), (3.8) and (3.10) that
for n sufficiently large and a suitable positive constant \(C_{4}\).
In the following, we prove that \(u_{0}\in \mathcal{M}^{+}\) is a solution of problem (1.1). By (3.9) and condition (ii) again, we have
Dividing by \(t>0\) and letting \(t\rightarrow 0^{+}\), we derive that
Noting \(f(x) [(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} ]\geq 0\), for every \(x \in \varOmega \) and \(t>0\), together with the Fatou lemma, we find that
is integrable, and
Applying the Fatou lemma again, we have
Since \(\int _{\varOmega }u_{0}^{-q}\varphi _{1}\,dx<\infty \), we have \(u_{0}>0\) a.e. in Ω. For every \(\phi \in \mathcal{M}\) and \(\phi \geq 0\), we have
Set \(\phi =u_{0}\) in (3.11), we derive that
Furthermore
Hence
Therefore \(u_{n}\rightarrow u_{0}\) in \(\mathcal{M}\) and \(u_{0}\in \mathcal{M}\). By (3.4), we have
i.e., \(u_{0}\in \mathcal{M}^{+}\).
Next, we only need to show that \(u_{0}\) is a positive weak solution of problem (1.1). Define
Substituting Φ into (3.11), combining with (3.12), we deduce that
where \(\varOmega _{1}=\{x\vert u_{0}(x)+\varepsilon \phi (x)>0, x\in \varOmega \}\) and \(\varOmega _{2}=\{x\vert u_{0}(x)+\varepsilon \phi (x)\leq 0, x\in \varOmega \}\). Since the measure of \(\varOmega _{2}\) tends to zero as \(\varepsilon \rightarrow 0\), we have \(\int _{\varOmega _{2}} \vert \Delta u_{0}\vert ^{p-2}\Delta u_{0} \Delta \phi \,dx \rightarrow 0\) as \(\varepsilon \rightarrow 0\). By the same arguments, we have \(\lambda \varepsilon ^{\gamma }\Vert g\Vert _{\infty } \int _{\varOmega _{2}}\vert \phi \vert ^{\gamma +1}\,dx \longrightarrow 0\) and \(\lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx\longrightarrow 0\) as \(\varepsilon \rightarrow 0\). Dividing by ε and taking the limit for \(\varepsilon \rightarrow 0\), we deduce that
Therefore \(u_{0}\) is a positive weak solution of problem (1.1).
We adopt the Ekeland variational principle again to derive a minimizing sequence \(U_{n}\subset \mathcal{M}^{-}\) for the minimization problem \(\inf_{\mathcal{M}^{-}} I_{\lambda ,\mu }\) such that for \(U_{n}\in \mathcal{M}\), \(U_{n} \rightharpoonup U_{0}\) weakly in \(\mathcal{M}\) and pointwise a.e. in Ω. By similar arguments to those in (3.4) and (3.6), for \(\lambda \in (0,T_{\mu })\), we have
which leads to
for n large enough and a positive constant \(C_{5}\). Therefore \(U_{0}>0\) is the positive weak solution of problem (1.1). Furthermore \(U_{0}\in \mathcal{M}\). By (3.14), we obtain
i.e., \(U_{0}\in \mathcal{M}^{-}\). According to Lemma 2.2, we know that problem (1.1) has at least two positive weak solutions \(u_{0}\in \mathcal{M}^{+}\) and \(U_{0}\in \mathcal{M}^{-}\) with \(\Vert U_{0}\Vert _{ \mu }>\Vert u_{0}\Vert _{\mu }\) for every \(\lambda \in (0,T_{\mu })\). This completes the proof of Theorem 1.1.
4 Proof of Corollary 1.2
For every \(U\in \mathcal{M}^{-}\), by Lemma 2.2, we deduce that
Combining with the definition of \(T_{\mu }\), we have
where we adopted the following facts:
Let \(U_{\lambda , \mu ,\varepsilon }\in \mathcal{M}^{-}\) be the solution of problem (1.1) with \(\gamma =\varepsilon +p-1\), where \(\lambda \in (0,T_{\mu })\). Then
where \(C_{\mu , \varepsilon }\) is given in (1.16). This completes the proof of Corollary 1.2.
5 Proof of Theorem 1.3
For simplicity, we consider problem (1.1) with \(f=g=1\),
Let us define
Using Theorem 1.1, we provide uniform estimates for \(\lambda ^{*}(N, \varOmega ,\mu ,q,\gamma )\).
Lemma 5.1
For \(1< p<\frac{N}{2}\), \(0<\mu <\mu _{N,p}\), \(0< q<1<\gamma <p^{*}-1\) and \(\varOmega \in \mathbb{U}\), where \(\mathbb{U}=\{\varOmega \in \mathbb{R}^{N}: \varOmega \textit{ is an open and bounded domain}\}\), we have
where
and
Proof
(1) Assume that \(\lambda \in (0,\lambda ^{-})\), then problem (5.1) has at least two solutions. By the definition of \(\lambda ^{*}\), we have \(\lambda ^{*}\geq \lambda ^{-}>0\).
(2) Assume that (5.1) has a positive solution u. Integrating over Ω by multiplying (5.1) by \(\varphi _{1}\), we obtain
We claim that there exists \(\lambda ^{+}>0\) such that
In fact, letting
We have
i.e.,
Then \(G_{\lambda }(t)\) attains minimum at \(t_{\min }\), and
We may choose \(\lambda =\lambda _{1}^{\frac{\gamma +q}{q-1+p}} (\frac{ \gamma -p+1}{\gamma +q} )^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+\frac{1}{2}=\lambda ^{+} >0\) such that
Therefore
Using (5.3) with \(t=u\), we have
Combining with (5.2) and (5.5), we obtain \(\lambda \leq \lambda ^{+}\). Since λ is arbitrary, we have \(\lambda ^{*}\leq \lambda ^{+}< \infty \). □
Proof of Theorem 1.3
We only prove the case that \(0<\lambda <\lambda ^{*}\). By the definition of \(\lambda ^{*}\), there exists \(\overline{\lambda }\in (\lambda , \lambda ^{*})\) such that the problem
has a positive solution, denoted by \(u_{\overline{\lambda }}\). It follows that
Hence \(u_{\overline{\lambda }}\) is an upper solution of (5.1). Note that \(\lim_{t\rightarrow 0^{+}}G_{\lambda }(t)=\infty \), we can take \(\varepsilon >0\) small enough with \(\varepsilon \varphi _{1}< u_{\overline{ \lambda }}\) and \(G_{\lambda }(\varepsilon \varphi _{1})\geq 0\). Thus
i.e.,
Combining with (1.10) and (5.6), we obtain
namely, \(\varepsilon \varphi _{1}\) is a lower solution of (5.1). Note that \(\Delta _{p}^{2}-\frac{\mu }{\vert x\vert ^{2p}}\) is monotone, then problem (5.1) has a positive solution \(u_{\lambda }\) with \(\varepsilon \varphi _{1}\leq u_{\lambda }\leq u_{\overline{\lambda }}\). □
6 Conclusions
In this paper, we study a class of p-biharmonic equations with Hardy potential and negative exponents. We establish the dependence of the above problem on q, γ, f, g and Ω and evaluate the extremal value of λ related to the multiplicity of positive solutions for this problem.
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This project is supported by the Natural Science Foundation of Shanxi Province (201601D011003), and the Natural Science Foundation of Shandong Province of China (ZR2017MA036).
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Sang, Y., Guo, S. An exact estimate result for p-biharmonic equations with Hardy potential and negative exponents. J Inequal Appl 2019, 26 (2019). https://doi.org/10.1186/s13660-019-1977-y
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DOI: https://doi.org/10.1186/s13660-019-1977-y