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Multiple positive solutions for quasilinear elliptic problems with combined critical Sobolev–Hardy terms
Boundary Value Problems volume 2019, Article number: 136 (2019)
Abstract
In this paper, we investigate the quasilinear elliptic equations involving multiple critical Sobolev–Hardy terms with Dirichlet boundary conditions on bounded smooth domains \(\varOmega \subset R^{N}\) (\({N \ge 3} \)), and prove the multiplicity of positive solutions by employing Ekeland’s variational principle and the maximum principle.
1 Introduction
In this paper, we consider the following quasilinear elliptic problem:
where \(\varOmega \subset R^{N}\) (\({N \ge 3} \)) is a bounded smooth domain such that the different points 0, \(x_{0}\in \varOmega \), \(- \Delta _{p} u = - \operatorname{div} ( { \vert {\nabla u} \vert ^{p - 2} \nabla u} )\) is the p-Laplacian of u; \(0 \le \mu < \overline{\mu }:=(\frac{N-p}{p})^{p} \), \(1< p < N\) and λ is a positive parameter; \(0\le a\le b< p\), \(1\le q < p\), \(p^{*}(a)= \frac{p(N-a)}{N-p}\), \(p^{*}(b)=\frac{p(N-b)}{N-p}\). Note that \(p^{*}(0)=p^{*}=\frac{Np}{N-p}\), \(p^{*}(p)=p\).
Let \(W_{0}^{1,p}(\varOmega )\) be the completion of \(C_{0}^{\infty }( \varOmega )\) with respect to the norm \((\int _{\varOmega }|\nabla u|^{p}\,dx)^{ \frac{1}{p}}\). The energy functional of problem (1) is defined on \(W_{0}^{1,p}(\varOmega )\) by
Then \(J(u)\in C^{1}(W_{0}^{1,p}(\varOmega ),R)\). Function \(u\in W_{0} ^{1,p}(\varOmega )\backslash \{0\}\) is said to be a nontrivial solution of (1) if \(\langle J^{\prime }(u),v\rangle =0\) for all \(v\in W_{0}^{1,p}(\varOmega )\) and a solution of (1) is a critical point of \(J(u)\). But the appearance of multiple Sobolev–Hardy terms in (1) makes it difficult to investigate the existence of positive solutions for problem (1).
Recall that the functional \(J(u)\) satisfies the \((\mathrm{PS})_{c}\) condition if every \((\mathrm{PS})_{c}\) sequence for \(J(u)\) has a convergent subsequence, and a sequence \(\{u_{n}\} \subset W_{0}^{1,p}(\varOmega )\) is called a \((\mathrm{PS}) _{c}\) sequence for \(J(u)\) if \(J(u_{n}) \to c\) and \(J'(u_{n}) \to 0\).
Elliptic equations with critical growth terms have received wide attention in recent years. In a pioneering work, Pohozaev [18] considered the following elliptic problem:
where Ω is a star-shaped domain with respect to the origin, and obtained that there is no nontrivial solution. However, lower order terms can reverse this situation. Indeed, Brezis and Nirenberg [1] proved the existence of positive solutions for the nonlinear elliptic problem involving the critical Sobolev exponent
Generalizations of this result can be found in [6], and for multiplicity results for elliptic equations with critical exponents see [7].
As for the elliptic problems involving Hardy terms, Jannelli [13] proved the existence of solutions. This problem was also discussed in [2, 3, 8, 9]. The following quasilinear elliptic problems with a singular Hardy term and a critical Sobolev–Hardy term:
have been investigated in recent years, where \(K(x)\) is a continuous nonnegative function and \(g(x,u)\) is a subcritical perturbation. Kang [14] proved the existence of solutions for problem (2) with \(K(x)=1\) and \(g(x,u)=\frac{|u|^{q-2}u}{|x|^{t}}\) where \(p\le t< p^{*}(s)\) by using variational methods and the results crucially depend on the parameters p, q, t, λ, and μ. In [17], Liang consider problem (2) with \(K(x)=1\) and derived the existence of infinitely many small solutions by using the concentration compactness principle and a symmetric mountain pass theorem.
Concerning problems with multiple nonlinearities, there has been little research up to now. Here we mention Gao [5] who studied the elliptic problem with combined critical Sobolev–Hardy terms on smooth bounded domain and obtained some existence results by investigating the limit behavior of the PS sequence for the corresponding energy functional. Li [15] has established the complete asymptotic description for any PS sequence \(\{u_{n}\}\) of the associational energy functional and then proved the existence of nontrivial solutions under different assumptions. As for problems involving multiple critical Sobolev–Hardy terms, we refer to articles [12, 16].
This paper is devoted to the study of the multiplicity of positive solutions for problem (1) when a, b, s, λ, μ satisfy suitable conditions by using variational methods and some ideas from [11, 12].
Problem (1) is related to Sobolev–Hardy inequality
When \(t=p\), \(p^{*}(t)=p\), the well-known Hardy inequality holds:
where \(\overline{\mu }=(\frac{N-p}{p})^{p}\) is the best Hardy constant.
In the space \(W_{0}^{1,p}(\varOmega )\), we employ the following norm if \(\mu <\overline{\mu }\):
Due to Hardy inequality, it is equivalent to the usual norm \((\int _{\varOmega }|\nabla u |^{p}\,dx)^{\frac{1}{p}}\) of the space \(W_{0}^{1,p}(\varOmega )\), and
is also equivalent to the usual norm \((\int _{\varOmega }|\nabla u |^{p}\,dx)^{ \frac{1}{p}}\) of the space \(W_{0}^{1,p}(\varOmega )\) with \(x_{0}\in \varOmega \). Hence we can deduce that
where β is a constant.
Set
Whenever \(A_{\mu ,t}\) is independent of \(\varOmega \subset {R^{N}}\), we will simple denote \(A_{\mu ,t}(\varOmega )=A_{\mu ,t}({R^{N}})=A_{\mu ,t}\). Therefore we conclude that
Let
and
Now we give our main result:
Theorem 1.1
If \(N\ge 3\), \(0\le \mu <\overline{\mu }\), \(0\le a\le b< p\), \(0\le s< p\), \(1 \le q< p\), then we have the following results:
-
(i)
If \(\lambda \in (0, \varLambda _{0})\), then (1) has at least one positive solution in \(W_{0}^{1,p}(\varOmega )\).
-
(ii)
If \(\lambda \in (0, \frac{q}{p}\varLambda _{0})\), then (1) has at least two positive solutions in \(W_{0}^{1,p}( \varOmega )\).
This paper is organized as follows. In Sect. 2, we narrate some useful preliminary knowledge and some properties of Nehari manifolds. In Sect. 3, the multiplicity of positive weak solutions is verified.
Throughout this paper, various positive constants will be denoted by c, and dx in integrals will be omitted for convenience.
2 Preliminary knowledge and main results
Since the functional \(J(u)\) is not bounded from below on \(W_{0}^{1,p}( \varOmega )\), we will work on a Nehari manifold. For \(\lambda >0\), we define
We recall that any nonzero solution of (1) belongs to \(N_{\lambda }\). Moreover, by definition, we have that \(u\in N_{\lambda }\) if and only if
Lemma 2.1
The functional \(J(u) \) is coercive and bounded from below on \(N_{\lambda }\).
Proof
For \(u\in N_{\lambda }\), we have
Set \(R_{0}\) be a positive constant such that \(\varOmega \subset B(0;R _{0})\), where \(B(0;R_{0})=\{x\in {R}^{N}:|x|< R_{0}\}\). Since
where \(\omega _{N}=\frac{2\pi ^{\frac{N}{2}}}{N\varGamma (\frac{N}{2})}\) is the volume of the unit ball in \({R^{N}}\), we have
Thus combining with (4), we get that
Since \(0\le b\), \(s< p\) and \(1\le q< p\), \(J(u)\) is coercive and bounded below on \(N_{\lambda }\). □
Define \(\phi _{\lambda }: W_{0}^{1,p}(\varOmega )\to {R}\) by \(\phi _{ \lambda }(u)=\langle J^{\prime }(u),u\rangle \), that is,
Note that \(\phi _{\lambda }\) is of class \(C^{1}\) with
Furthermore, if \(u\in N_{\lambda }\), then from (3) and (6), we have
and
Now we split \(N_{\lambda }\) into three sets:
The following result shows that minimizers on \(N_{\lambda }\) are the usual critical points for \(J(u)\).
Lemma 2.2
Suppose that \(u_{0}\) is a local minimizer of \(J(u)\) on \(N_{\lambda }\) and \(u_{0}\notin N_{\lambda }^{0}\), then \(J^{\prime }(u_{0})=0\) in \((W_{0}^{1,p}(\varOmega ))^{-1}\).
Proof
It is easy to see that there exists a neighborhood U of \(u_{0}\) in \(W_{0}^{1,p}(\varOmega )\) such that
Furthermore, by the Lagrange Multipliers Theorem, there exists \(\rho \in {R}\) such that \(J^{\prime }(u_{0})=\rho \phi _{\lambda }(u _{0})\). Then, since \(u_{0}\in N_{\lambda }\), we get
Now \(u_{0}\notin N_{\lambda }^{0}\), thus \(\rho =0\), and consequently \(J^{\prime }(u_{0})=0\) in \((W_{0}^{1,p}(\varOmega ))^{-1}\). □
Motivated by the above result, we will get conditions for \(N_{\lambda }^{0}=\emptyset \).
Lemma 2.3
If \(\lambda \in (0,\varLambda _{0})\), then \(N_{\lambda }^{0}=\emptyset \), where \(\varLambda _{0}\) is given in the introduction.
Proof
We argue by contradiction. Suppose that there exists a \(\lambda \in (0,\varLambda _{0})\) such that \(N_{\lambda }^{0}\neq\emptyset \), then from (9),
which implies
Again by using (7), Hölder and Sobolev–Hardy inequalities, we have
Now we distinguish two cases:
Case 1. \(\frac{p^{*}(a)-q}{p-q}\frac{\|u\|_{\mu }^{p^{*}(a)}}{A_{ \mu ,a}^{\frac{p^{*}(a)}{p}}}\le \frac{p^{*}(b)-q}{p-q}\frac{ \beta ^{p^{*}(b)}\|u\|_{\mu }^{p^{*}(b)}}{A_{\mu ,b}^{ \frac{p^{*}(b)}{p}}}\).
It is easy to calculate that
Combining with (11), we conclude that
Case 2. \(\frac{p^{*}(a)-q}{p-q}\frac{\|u\|_{\mu }^{p^{*}(a)}}{A_{ \mu ,a}^{\frac{p^{*}(a)}{p}}}>\frac{p^{*}(b)-q}{p-q}\frac{\beta ^{p ^{*}(b)}\|u\|_{\mu }^{p^{*}(b)}}{A_{\mu ,b}^{\frac{p^{*}(b)}{p}}}\).
As in Case 1, one obtains that
Hence \(\lambda \ge \varLambda _{0}\), which contradicts \(\lambda \in (0, \varLambda _{0})\). Thus \(N_{\lambda }^{0}=\emptyset \). □
Lemma 2.4
If \(\lambda \in (0,\varLambda _{0})\), then for each \(u\in W_{0}^{1,p}( \varOmega )\backslash \{0\}\), the set \(\{\tau u:\tau >0\}\) intersects \(N_{\lambda }\) exactly twice. More specifically, there exist a unique \(\tau ^{-}=\tau ^{-}(u)>0\) such that \(\tau ^{-}u\in N_{\lambda }^{-}\) and a unique \(\tau ^{+}=\tau ^{+}(u)>0\) such that \(\tau ^{+}u\in N_{\lambda }^{+}\). Moreover, \(\tau ^{+}<\tau _{\max }<\tau ^{-}\) and
Proof
The proof is similar to that of Lemma 2.7 in [11], and we omit it here. □
From Lemma 2.3 we obtain that \(N_{\lambda }=N_{\lambda }^{+} \cup N_{\lambda }^{-}\) for all \(\lambda \in (0,\varLambda _{0})\). Furthermore, by Lemma 2.4 it follows that \(N_{\lambda }^{+}\) and \(N_{\lambda }^{-}\) are nonempty and, by Lemma 2.1, we may define
Lemma 2.5
-
(i)
If \(\lambda \in (0,\varLambda _{0})\), then we have \(\alpha _{\lambda } \le \alpha _{\lambda }^{+}<0\).
-
(ii)
If \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then there exists some positive constant \(d_{0}\) such that \(\alpha _{\lambda }^{-}>d_{0}\).
In particular, for each \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), we have that \(\alpha _{\lambda }^{+}<0<\alpha _{\lambda }^{-}\).
Proof
(i) It is enough to prove that there exists \(c>0\) such that \(\alpha _{\lambda }^{+}<-c<0\). Let \(u\in N_{\lambda }^{+}\). Then from (7), we have
Therefore for \(u\in N_{\lambda }^{+}\), we get
where \(q< p< p^{*}(b)\le p^{*}(a)\). Therefore, from the definition of \(\alpha _{\lambda }\) and \(\alpha _{\lambda }^{+}\), we can deduce that \(\alpha _{\lambda }\le \alpha _{\lambda }^{+}<0\).
(ii) Let \(u\in N_{\lambda }^{-}\). By (7),
Thus from the Sobolev–Hardy inequality, we get
Case 1. \(\frac{p^{*}(a)-q}{p-q} A_{\mu ,a}^{-\frac{p^{*}(a)}{p}}\|u\| _{\mu }^{p^{*}(a)}\le \frac{p^{*}(b)-q}{p-q} A_{\mu ,b}^{- \frac{p^{*}(b)}{p}}\beta ^{p^{*}(b)}\|u\|_{\mu }^{p^{*}(b)}\).
It is easy to calculate that for all \(u\in N_{\lambda }^{-}\),
Case 2. \(\frac{p^{*}(a)-q}{p-q} A_{\mu ,a}^{-\frac{p^{*}(a)}{p}}\|u\| _{\mu }^{p^{*}(a)}>\frac{p^{*}(b)-q}{p-q} A_{\mu ,b}^{- \frac{p^{*}(b)}{p}}\|u\|_{\mu }^{p^{*}(b)}\).
It is easy to calculate that for all \(u\in N_{\lambda }^{-}\)
With (5), we deduce that
So if \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then \(J(u)>d_{0}\) for all \(u\in N_{\lambda }^{-}\) for some positive constant \(d_{0}\). □
Remark 1
If \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then by Lemmas 2.4 and 2.5, for each \(u\in W_{0}^{1,p}( \varOmega )\backslash \{0\}\), we can easily deduce that
3 Proof of the main result
Lemma 3.1
-
(i)
If \(\lambda \in (0,\varLambda _{0})\), then \(J(u)\) has a \((PS)_{ \alpha _{\lambda }}\) sequence \(\{u_{n}\}\subset N_{\lambda }\).
-
(ii)
If \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then \(J(u)\) has a \((\mathrm{PS})_{\alpha _{\lambda }^{-}}\) sequence \(\{u_{n}\}\subset N_{\lambda }^{-}\).
Proof
The proof is similar to that of Proposition 3.3 in [11], and we omit it here. □
Now we use the Ekeland’s variational principle [4] to get the following results.
Theorem 3.2
If \(\lambda \in (0,\varLambda _{0})\), then there exists \(u_{\lambda } \in N_{\lambda }^{+}\) such that
-
(i)
\(J(u_{\lambda })=\alpha _{\lambda }=\alpha _{\lambda }^{+}\);
-
(ii)
\(u_{\lambda }\) is a positive solution for problem (1);
-
(iii)
\(\|u_{\lambda }\|_{\mu }\to 0\) as \(\lambda \to 0^{+}\).
Proof
By Lemma 3.1(i), there exists a minimizing sequence \(\{u_{n}\}\subset N_{\lambda }\) such that
Since \(J(u)\) is coercive on \(N_{\lambda }\), we obtain that \(\{u_{n}\}\) is bounded in \(W_{0}^{1,p}(\varOmega )\). Thus, passing to a subsequence if necessary, there exists \(u_{\lambda }\in W_{0}^{1,p}(\varOmega )\) such that as \(n\to \infty \),
From (13) and (14), it is easy to see that \(u_{\lambda }\) is a solution of (1). Furthermore, from \(u_{n}\in N_{\lambda }\) and (4), we deduce that
Let \(n\to \infty \) in (15). Then from (13)–(14) and since \(\alpha _{\lambda }<0\) by Lemma 2.5(i), we get
Thus \(u_{\lambda }\neq 0\). Since \(J^{\prime }(u_{\lambda })=0\), it follows that \(u_{\lambda }\in N_{\lambda }\) and, in particular, \(J(u_{\lambda })\ge \alpha _{\lambda }\).
Next, we will show, up to a subsequence, that \(u_{n} \to u_{\lambda }\) strongly in \(W_{0}^{1,p}(\varOmega )\) and \(J(u_{\lambda })=\alpha _{ \lambda }\). From the fact \(u_{n}\), \(u_{\lambda }\in N_{\lambda }\), (4) and Fatou’s Lemma, it follows that
which implies that \(J(u_{\lambda })=\alpha _{\lambda }\) and \(\lim_{n\to \infty }\|u_{n}\|_{\mu }^{p}=\|u_{\lambda }\|_{\mu } ^{p}\). Standard argument shows that \(u_{n}\to u_{\lambda }\) strongly in \(W_{0}^{1,p}(\varOmega )\). Moreover, \(u_{\lambda }\in N_{\lambda }^{+}\). Otherwise, if \(u_{\lambda }\in N_{\lambda }^{-}\), from Lemma 2.4 there exist unique \(\tau ^{+}_{\lambda }\) and \(\tau ^{-} _{\lambda }\) such that \(\tau ^{+}_{\lambda }u_{\lambda }\in N_{\lambda }^{+}\), \(\tau ^{-}_{\lambda }u_{\lambda }\in N_{\lambda }^{-}\) and \(\tau ^{+}_{\lambda }<\tau ^{-}_{\lambda }=1\). Since
there exists \(\overline{\tau }\in (\tau ^{+}_{\lambda },\tau ^{-}_{ \lambda })\) such that \(J(\tau ^{+}_{\lambda }u_{\lambda })< J(\tau ^{-} _{\lambda }u_{\lambda })\). By Lemma 2.4, we get that
which is a contradiction. Since \(J(u_{\lambda })=J(|u_{\lambda }|)\) and \(|u_{\lambda }|\in N_{\lambda }^{+}\), by Lemma 2.2, we may assume that \(u_{\lambda }\) is a nontrivial nonnegative solution of (1). From the strong maximum principle [19], it follows that \(u_{\lambda }>0\) in Ω. Finally, by (9), Hölder and Sobolev–Hardy inequalities, we obtain
Thus
which implies that \(\|u_{\lambda }\|_{\mu }\to 0\) as \(\lambda \to 0^{+}\). □
Next we will establish the existence of the second positive solution of (1) by proving that \(J(u)\) satisfies the \((\mathrm{PS})_{ \alpha _{\lambda }}\) condition.
Lemma 3.3
Let \(\{u_{n}\}\) be a bounded sequence in \(W_{0}^{1,p}(\varOmega )\). If \(\{u_{n}\}\) is a \((\mathrm{PS})_{c}\) sequence for \(J(u)\) with \(c\in (0,\varLambda _{1})\) where \(\varLambda _{1}\) is defined in the introduction. Then there exists a subsequence of \(\{u_{n}\}\) converging weakly to a nonzero solution solution of (1).
Proof
The proof is similar to that of Corollary 4.3 in [15], and the details are omitted. □
Lemma 3.4
([14])
Assume \(1< p< N\), \(0\le a< p\) and \(0\le \mu <\overline{\mu }\). Then the problem
has radially symmetric ground states
satisfying
where \(U_{p,\mu }(x)=U_{p,\mu }(|x|)\) is the unique radial solution for problem (16) satisfying
and \(D^{1,p}({R}^{N})=\{u\in L^{p^{*}}({R}^{N}):\nabla u\in L^{p}( {R}^{N})\}\). Moreover, \(U_{p,\mu }(x)\) also has the following properties:
where \(c_{1}\) and \(c_{2}\) are positive constants depending on p and N, \(a(\mu )\) and \(b(\mu )\) are the zeros of the function
satisfying
Remark 2
By direct calculation, we deduce that \(\tau _{\min }=\frac{N-p}{p}\) is the only minimum point of \(f(\tau )\). Furthermore, \(f^{\prime }(\tau )<0\) for \(0<\tau < \tau _{\min }\) and \(f^{\prime }(\tau )>0\) for \(\tau >\tau _{ \min }\). Thus, we infer that
Furthermore, by (17) we know that \(b(\mu )> \frac{N}{p}\) implies \(N>p^{2}\).
Lemma 3.5
([7])
Suppose \(1< p< N\), \(0\le b< p \). Then the following holds:
-
(i)
\(A_{0,b}\) is independent of Ω;
-
(ii)
\(A_{0,b}\) is attained when \(\varOmega ={R}^{N}\) by the functions
$$ y_{\varepsilon }(x)= \biggl(\varepsilon (N-b) \biggl(\frac{N-p}{p-1} \biggr)^{p-1} \biggr) ^{\frac{N-p}{p(p-b)}}\bigl(\varepsilon + \vert x-x_{0} \vert ^{\frac{p-b}{p-1}}\bigr)^{ \frac{p-N}{p-b}} $$
for some \(\varepsilon >0\). Moreover, the functions \(y_{\varepsilon }(x)\) solve the equation
and satisfy
Lemma 3.6
If \(0\le \mu <\overline{\mu }\), \(0\le a\), \(b< p\) and \(1\le q< p\), then for any \(\lambda >0\), there exists \(v_{\lambda }\in W_{0}^{1,p}(\varOmega )\) such that
In particular, \(\alpha _{\lambda }^{-}<\varLambda _{1}\) for all \(\lambda \in (0,\varLambda _{0})\), where \(\lambda _{1}\) is defined in the introduction.
Proof
Now we distinguish two cases, that is, \(\frac{p-a}{p(N-a)}A_{\mu , a} ^{\frac{N-a}{p-a}}\le \frac{p-b}{p(N-b)}A_{0, b}^{\frac{N-b}{p-b}}\) and \(\frac{p-a}{p(N-a)}A_{\mu , a}^{\frac{N-a}{p-a}}>\frac{p-b}{p(N-b)}A _{0, b}^{\frac{N-b}{p-b}}\).
Case 1. \(\frac{p-a}{p(N-a)}A_{\mu , a}^{\frac{N-a}{p-a}}\le \frac{p-b}{p(N-b)}A_{0, b}^{\frac{N-b}{p-b}}\).
Assume \(\rho >0\) is small enough such that \(B(0,\rho ) \subset \varOmega \), \(\varphi (x) \in C_{0}^{\infty }(\varOmega )\), \(0\le \varphi (x) \le 1\), \(\varphi (x)=1\) for \(|x| \le \frac{\rho }{2}\), \(\varphi (x)=0\) for \(|x|\ge \rho \). Let
The following estimates are from [10] and [14]:
Now we consider the following functions:
and
Using the definitions of g and \(u_{\varepsilon }\), we get
Combining this with (19) and letting \(\varepsilon \in (0,1)\), there exists \(\tau _{0}\in (0,1)\) independent of ε such that
On the other hand, by the fact that
and from (19) and (20), we obtain that
Hence as \(\lambda >0\), \(1\le q< p\), by (24) we have that
(i) If \(1\le q<\frac{N-s}{b(\mu )}\), then by (21), we obtain that
and since \(b(\mu )>\frac{N-p}{p}\), \(q< p\), we obtain
Combining this with (22) and (25), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that
(ii) If \(\frac{N-s}{b(\mu )}\le q< p\), then by (21) and \(b(\mu )>\frac{N-p}{p}\), we obtain
and \(b(\mu )p+p-N>N-s+(1-\frac{N}{p})q\). Combining this with (22) and (25), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that
From (26) and (27), we obtain the result in Case 1 by taking \(v_{\lambda }=u_{\varepsilon _{\lambda }}\).
Case 2. \(\frac{p-a}{p(N-a)}A_{\mu , a}^{\frac{N-a}{p-a}}> \frac{p-b}{p(N-b)}A_{0, b}^{\frac{N-b}{p-b}}\).
Let
Consider \(\varphi (x)\in C_{0}^{\infty }(\varOmega )\), \(0\le \varphi (x) \le 1\), \(\varphi (x)=1\) for \(|x-x_{0}|\le \frac{R}{2}\), \(\varphi (x)=0\) for \(|x-x_{0}|\ge R\), where \(B(x_{0},R)\subset \varOmega \). Denote
such that
Then we can obtain the following results by the methods used in [7]:
Observing that \(w_{\varepsilon }\) concentrates on \(x=x_{0}\) when \(\varepsilon >0\) is small enough, we can easily estimate
Especially, when \(q=p\), we have
Now we consider the following function:
Since \(\lim_{\tau \to +\infty }h(\tau )=-\infty \) and \(\lim_{\tau \to 0^{+}}h(\tau )<0\), combining this with Remark 1, we get that \(\sup_{\tau \ge 0}h(\tau )\) is attained for some \(0<\tau _{0}<+\infty \). Together with (23) and (28)–(31), we calculate that
(i) If \(p^{2}\ge N\), then we have that \(\frac{N-p}{p-b}> \frac{q(N-b)}{p(p-b)}\). By (32), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that
(ii) If \(p^{2}< N\), then we have that \(\frac{N-p}{p-b}> \frac{p(p-1)}{p-b}\). By (32), for any \(\lambda >0\), we can choose \(\varepsilon _{\lambda }\) small enough such that
From (i) and (ii), we obtain the result in Case 2 by taking \(v_{\lambda }=w_{\varepsilon _{\lambda }}\).
From Lemma 2.4, the definition of \(\alpha _{\lambda }^{-}\) and (18), for any \(\lambda \in (0,\varLambda _{0})\), we obtain that there exists \(\tau _{\lambda }^{-}>0\) such that \(\tau _{\lambda }^{-}v _{\lambda }\in N_{\lambda }^{-}\) and
The proof is thus complete. □
Now we establish the existence of a local minimum of \(J(u)\) on \(N_{\lambda }^{-}\).
Theorem 3.7
Assume that \(N\ge 3\), \(0\le \mu <\overline{\mu }\), \(0\le a\), \(b< p\) and \(1\le q< p\). If \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then there exists \(U_{\lambda }\in N_{\lambda }^{-}\) such that
-
(i)
\(J(U_{\lambda })=\alpha _{\lambda }^{-} \);
-
(ii)
\(U_{\lambda }\) is a positive solution of (1).
Proof
If \(\lambda \in (0,\frac{q}{p}\varLambda _{0})\), then by Lemmas 2.5(ii), 3.1(ii), and 3.6, there exists a \((\mathrm{PS}) _{\alpha _{\lambda }^{-}}\) sequence \(\{u_{n}\}\subset N_{\lambda }^{-}\) in \(W_{0}^{1,p}(\varOmega )\) for \(J(u)\) with \(\alpha _{\lambda }^{-} \in (0,\varLambda _{1})\). Since \(J(u)\) is coercive on \(N_{\lambda }\), we get that \(\{u_{n}\}\) is bounded in \(W_{0}^{1,p}(\varOmega )\). From Lemma 3.3, there exists a subsequence still denoted by \(\{u_{n}\}\) and a nontrivial solution \(U_{\lambda }\in W_{0}^{1,p}(\varOmega )\) of (1) such that \(u_{n}\rightharpoonup U_{\lambda }\) weakly in \(W_{0}^{1,p}(\varOmega )\).
First we prove that \(U_{\lambda }\in N_{\lambda }^{-} \). Arguing by contradiction, we assume \(U_{\lambda }\in N_{\lambda }^{+}\). Since \(N_{\lambda }^{-}\) is closed in \(W_{0}^{1,p}(\varOmega )\), we have \(\|u_{\lambda }\|_{\mu }<\lim \inf_{n\to \infty }\|u_{n}\|_{ \mu }\). Thus by Lemma 2.4, there exists a unique \(\tau _{ \lambda }^{-}\) such that \(\tau _{\lambda }^{-}U_{\lambda }\in N_{ \lambda }^{-}\). From Remark 1, \(u_{n}\in N_{\lambda }^{-}\), \(\|U_{\lambda }\|_{\mu }<\lim \inf_{n\to \infty }\|u_{n}\|_{ \mu }\), and (4), we can deduce that
This is a contradiction. Thus \(U_{\lambda }\in N_{\lambda }^{-}\).
Next, by the same argument as that in Theorem 3.2, we get that \(u_{n}\to U_{\lambda }\) strongly in \(W_{0}^{1,p}(\varOmega )\) and \(J(U_{\lambda })=\alpha _{\lambda }^{-}>0\) for all \(\lambda \in (0, \frac{q}{p}\varLambda _{0})\). Since \(J(U_{\lambda })=J(|U_{\lambda }|) \) and \(|U_{\lambda }|\in N_{\lambda }^{-}\), by Lemma 2.2 we may assume that \(U_{\lambda }\) is a nontrivial nonnegative solution of (1). Finally, by the maximum principle, we obtain that \(U_{\lambda }\) is a positive solution of (1). □
The proof of Theorem 1.1
Now we complete the proof of Theorem 1.1. Part (i) of Theorem 1.1 immediately follows from Theorem 3.2. When \(0<\lambda <\frac{q}{p}\varLambda _{0}<\varLambda _{0}\), by Theorems 3.2 and 3.7, we obtain that (1) has at least two positive solutions \(u_{\lambda }\) and \(U_{\lambda }\) such that \(u_{\lambda }\in N_{\lambda }^{+}\), \(U_{ \lambda }\in N_{\lambda }^{-} \). Since \(N_{\lambda }^{+} \cap N_{ \lambda }^{-}=\emptyset \), this implies that \(u_{\lambda }\) and \(U_{\lambda }\) are distinct. This completes the proof of Theorem 1.1. □
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The authors would like to thank the referees for their valuable comments and suggestions which improved the original manuscript.
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The project was supported by National Natural Science Foundation of China (Grant No. 11871212) and North China University of Water Resources and Electric power (Grant No. 70495).
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Li, Y. Multiple positive solutions for quasilinear elliptic problems with combined critical Sobolev–Hardy terms. Bound Value Probl 2019, 136 (2019). https://doi.org/10.1186/s13661-019-1249-2
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DOI: https://doi.org/10.1186/s13661-019-1249-2
MSC
- 35J20
- 35D30
Keywords
- Quasilinear elliptic equation
- Sobolev–Hardy term
- Positive solution
- Ekeland’s variational principle