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Existence,multiplicity, and nonexistence of solutions for a p-Kirchhoff elliptic equation on \(\mathbb{R}^{N}\)
Boundary Value Problems volume 2017, Article number: 20 (2017)
Abstract
In this paper, we study the multiplicity of solutions for the following nonhomogeneous p-Kirchhoff elliptic equation:
with \(a,\lambda,m>0\) and \(1< p< N\). By variational methods we prove that problem (0.1) admits at least two solutions under appropriate assumptions on \(f(u)\) and \(h(x)\). The main difficulty to overcome is the lack of an a priori bound for Palais-Smale sequence. Motivated by Jeanjean (Proc. R. Soc. Edinb., Sect. A 129:787-809, 1999), we use a cut-off functional to obtain a bounded (PS) sequence. Also, if \(f(u)=|u|^{q-2}u\), \(p< q<\min\{p(m+1), p^{*}=\frac{pN}{N-p}\}\), and \(h(x)=0\), then we prove that problem (0.1) has at least one nontrivial solution for any \(\lambda\in(0, \lambda^{*}]\) and has no nontrivial weak solutions for any \(\lambda\in(\lambda^{*}, +\infty)\).
1 Introduction
In this paper, we are interested in the multiplicity of solutions to the following nonhomogeneous p-Kirchhoff elliptic problem:
where \(\Delta_{p}u=\operatorname {div}(\vert \nabla{u} \vert ^{p-2}\nabla{u})\) is the p-Laplacian operator, and the nontrivial function \(h(x)\) can be seen as a perturbation term. Problem (1.1) is a generalization of the model introduced by Kirchhoff [2]. More precisely, Kirchhoff proposed the model given by the equation
which takes into account the changes in length of string produced by transverse vibration. The parameters in (1.2) have the following meaning: L is the length of the string, h is the area of cross-section, E is the Young modulus of material, ρ is the mass density, and \(P_{0}\) is the initial tension.
The equation
generalizes equation (1.2), where \(M: \mathbb{R}^{+}\to\mathbb{R}\) is a given function, Ω is a domain of \(\mathbb{R}^{N}\). The stationary counterpart of (1.3) is the Kirchhoff-type elliptic equation
Some classical and interesting results on Kirchhoff-type elliptic equations can be found, for example, in [3–9].
Particularly, Li et al. [10] considered the Kirchhoff-type problem
where \(N \geq3\), with constants \(a, b > 0\) and \(\lambda\geq0\) under the following assumptions:
- \((H_{1})\) :
-
\(f\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(\vert f(t)\vert \leq C(1+t^{q-1})\) for all \(t\in\mathbb{R}^{+}=[0,+\infty)\) and some \(q\in(2,2^{*})\), where \(2^{*}=\frac{2N}{N-2}\) for \(N\geq3\);
- \((H_{2})\) :
-
\(\lim_{t\rightarrow0}\frac{f(t)}{t}=0\); \(\lim_{t\rightarrow+\infty}\frac{f(t)}{t}=+\infty\).
It is easy to see that \(f(u)=\vert u\vert ^{q-2}u\), \(2< q<4\), and \(N=3\) satisfy these conditions. They obtained that there exists \(\lambda _{0} > 0\) such that, for any \(\lambda\in[0, \lambda_{0})\), problem (1.5) has at least one positive solution in \(W^{1,2}({\mathbb {R}}^{N})\). The \(\lambda_{0}\) depends on f, a, b, the Sobolev constant, and several test functions in [10]; it is not very clear whether the the existence of solutions for (1.5) still holds for large \(\lambda >0\). Recently, Chen et al. [11] studied the existence of positive solutions to the p-Kirchhoff problem
where \(a,b>0\), \(\tau,\lambda\ge0\), \(\mu\in\mathbb{R}\), and \(1< p< N\). By the Nehari manifold method, they proved that problem (1.6) admits at least a positive ground state solution for any \(\lambda>0\) when \(p(\tau +1)< q< m< p^{*}=\frac{pN}{N-p}\). However, does the existence of solutions for (1.5) still hold for any \(\lambda>0\) when \(p< q< p(\tau+1)\) and \(\mu=0\)? This is a interesting problem. In this paper, we answer positively this question. More interesting results for Kirchhoff-type problems can be found in [1, 2, 5–7, 10–14].
In the present paper, we are ready to extend the analysis to the nonhomogeneous p-Kirchhoff-type equation of (1.1) in \(\mathbb{R}^{N}\) with the nonlinearity \(f(u)\) satisfying the following conditions:
- \((F_{1})\) :
-
\(f\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(\vert f(t)\vert \leq C(t^{p-1}+t^{q-1})\) for all \(t\in\mathbb{R}^{+}\) and some \(q\in(p,p^{*})\), where \(p^{*}={pN}/({N-p})\), \(1< p< N\);
- \((F_{2})\) :
-
\(\lim_{t\rightarrow0^{+}}\frac{f(t)}{t^{p-1}}=0\);
- \((F_{3})\) :
-
\(\lim_{t\rightarrow+\infty}\frac{f(t)}{t^{p-1}}=+\infty\).
In addition, we suppose that the nontrivial and nonnegative function \(h(x)\equiv h(\vert x\vert )\in C^{1}(\mathbb{R}^{N})\cap L^{p'}(\mathbb{R}^{N})\) satisfies
- \((H)\) :
-
there exists \(\xi(x)\in L^{p'}(\mathbb{R}^{N})\cap W^{1,\infty}(\mathbb{R}^{N})\) such that
$$ \bigl\vert \nabla h(x)\cdot x \bigr\vert \leq\xi^{p'}(x), \quad \forall x \in\mathbb{R}^{N}, $$(1.7)
with \(p'=\frac{p}{p-1}\).
We will use the Ekeland variational principle [15] and a version of the mountain pass theorem in [1] to study the existence of multiple solutions of problem (1.1) in \({\mathbb {R}}^{N}\). It is well known that an important technical condition to get a bounded (PS) sequence is the following Ambrosetti-Rabinowitz-type condition (AR): there exists \(\theta>p\) such that \(0<\theta F(s)\le sf(s)\) for \(s>0\). The loss of (AR) condition renders variational techniques more delicate. Inspired by [1, 10], we use a cut-off functional and obtain a bounded (PS) sequence.
In order to state our main result, we introduce some Sobolev spaces and norms. Let \(W^{1,p}(\mathbb{R}^{N})\) be the usual Sobolev space with the norm
We denote by \(\Vert \cdot \Vert _{q}\) the usual \(L^{q}({\mathbb {R}}^{N})\) norm. Then it well known that the embedding \(W^{1,p}(\mathbb {R}^{N})\hookrightarrow L^{q}(\mathbb{R}^{N})\) is continuous for \(q\in (p,p^{*}]\) and there exists a constant \(S_{q}\) such that
Let \(X=W_{r}^{1,p}(\mathbb{R}^{N})\) be the subspace of \(W^{1,p}(\mathbb {R}^{N})\) containing only the radial functional. Then by the Lemma 2.2 in [11] we have that the embedding \(X\hookrightarrow L^{q}(\mathbb {R}^{N})\) is compact for \(q\in(p,p^{*})\).
A function \(u\in X\) is said to be a weak solution of (1.1) if for all \(v \in X\),
Let \(I(u):X\rightarrow\mathbb{R} \) be the energy functional associated with problem (1.1) defined by
where \(F(u)=\int_{0}^{u}f(s)\,ds\). It is easy to see that the functional \(I\in C^{1}(X,\mathbb{R})\) and its Gateaux derivative is given by
Clearly, we see that a weak solution of (1.1) corresponds to a critical point of the functional.
The main result in this paper is as follows.
Theorem 1.1
Let \((F_{1})\)-\((F_{3})\) and \((H)\) hold. Then, there exist \(\lambda_{0}, \widetilde{{m}}_{0}>0\) such that, for any \(\lambda\in[0,\lambda_{0})\), (1.1) has at least two nontrivial solutions in X when \(\Vert h\Vert _{p'}<\widetilde{{m}}_{0}\).
Furthermore, consider \(h(x)=0\) and \(f(x,u)=\vert u\vert ^{q-2}u\), \(p< q<\min\{p(m+1),p^{*}\}\), that is,
We can now state the second main result.
Theorem 1.2
Let \(a>0\) and \(p< q<\min\{p(m+1),p^{*}\}\). Then there exists \(\lambda ^{*}>0\) such that problem (1.13) has at least one nontrivial solution for any \(\lambda\in(0, \lambda^{*}]\) and has no nontrivial weak solutions for any \(\lambda\in(\lambda^{*}, +\infty)\).
Remark 1.3
In [11], Chen and Zhu considered the case \(p< p(m+1)< q<p^{*}\). They proved that problem (1.1) admits at least one positive solution for any \(\lambda>0\).
2 Proof of Theorem 1.1
In this section, we first establish some properties of the functional I and then prove Theorem 1.1. Throughout the paper, we denote by C or \(C_{i}\) s positive constants that may vary from line to line and are not essential to the problem.
Lemma 2.1
If assumptions \((F_{1})\)-\((F_{3})\) hold and \(h(x)\in L^{p'}(\mathbb{R}^{N})\), then there exist \(\rho, \alpha, m_{0}>0\) such that \(I(u)\geq\alpha>0\) with \(\Vert u\Vert =\rho\) and \(\Vert h\Vert _{p'}< m_{0}\).
Proof
It follows from \((F_{1})\)-\((F_{2})\) that
with \(\varepsilon>0\). By the Hölder inequality we have
Thus,
where \(\varepsilon=\epsilon=\frac{a}{4p}\), \(C_{1}\), \(C_{2}\) are some positive constants. Let
We see that there exists \(\rho>0\) such that \(\max_{t\geq0}z(t)=z(\rho)\equiv m_{0}>0\). Then it follows from (2.3) that there exists \(\alpha>0\) such that \(I(u)\geq\alpha\) with \(\Vert u\Vert =\rho\) and \(\Vert h\Vert _{p'}< m_{0}\). This ends the proof of Lemma 2.1. □
We denote by \(B_{r}\) the open ball in X centered at the origin with radius r. By Ekland’s variational principle [15] we get the following lemma, which implies that there exists a function \(u_{0}\) such that \(I'(u_{0})=0\) and \(I(u_{0})<0\) if \(\Vert h\Vert _{p'}\) is small.
Lemma 2.2
Let assumptions \((F_{1})\)-\((F_{3})\) hold, and \(h(x)\in L^{p'}(\mathbb{R}^{N})\), \(h(x)\not\equiv0\), with \(\Vert h\Vert _{p'}< m_{0}\). Then there exists a function \(u_{0}\in X\) such that
and \(u_{0}\) is a nontrivial weak solution of problem (1.1).
Proof
Choose a function \(\phi\in C_{0}^{1}(\mathbb{R}^{N})\) such that \(\int_{\mathbb{R}^{N}}h(x)\phi(x)\,dx>0\). Then
for small \(t>0\) and thus for any open ball \({B}_{\kappa}\subset X\) such that \(-\infty< c_{\kappa}=\inf_{\overline{B}_{\kappa}}I(u)<0\). Thus,
where ρ is given in Lemma 2.1. Let \(\varepsilon_{n}\downarrow0\) be such that
Then, by Ekland’s variational principle [15] there exists \(\{u_{n}\} \subset\overline{B}_{\rho}\) such that
and
Then, it follows from (2.8)-(2.10) that
So \(u_{n}\in B_{\rho}\), and we now consider the function \(F: \overline {B}_{\rho}\to {\mathbb {R}}\) given by
Then (2.10) shows that \(F(u_{n})< F(u)\), \(u\in\overline{{B}}_{\rho}\), \(u_{n}\neq u\), and thus \(u_{n}\) is a strict local minimum of F. Moreover,
Hence,
Passing to the limit as \(t\to0^{+}\), it follows that
Replacing v in (2.15) by −v, we get
so that \(\Vert I'(u_{n})\Vert \leq\varepsilon_{n}\). Therefore, there is a sequence \(\{u_{n}\}\in B\rho\) such that \(I(u_{n})\to c_{\rho}<0\) and \(I'(u_{n})\to0\) in \(X^{*}\) as \(n\to\infty\). In the following, we will prove that \(\{u_{n}\}\) has a convergent subsequence in X. Indeed, since \(\Vert u_{n}\Vert <\rho\), by the reflexivity of X and compact embedding \(X\hookrightarrow L^{q}\) for all \(q\in(p,p^{*})\), passing to a subsequence, we can assume that
By (1.12) we can get
where
It is clear that
By \((F_{1})\) and \((F_{2})\), for any \(\varepsilon>0\), there exists \(C_{\varepsilon}>0\) such that
Hence,
Define the linear function \(g:X\rightarrow\mathbb{R}\) by
Noticing that \(\vert g(\omega)\vert \leq2\Vert u_{0}\Vert ^{p-1}\Vert \omega \Vert \), we can deduce that g is continuous on X. Using \(u_{n}\rightharpoonup u_{0}\) in X, we have
Since \(\Vert u_{n}\Vert <\rho\), we deduce that \(\vert Q_{n}\vert \rightarrow0\) as \(n\rightarrow\infty\).
Combining the above results, we have \(\vert P_{n}\vert \to0\) as \(n\to\infty\), Then, using the standard inequalities in \(\mathbb{R}^{N}\)
where \(\langle\cdot,\cdot\rangle\) denotes the scalar product in \(\mathbb{R}^{N}\), we can show that \(u_{n} \rightarrow u_{0}\) in X. Thus, \(u_{0}\) is a nontrivial weak solution of problem (1.1). The proof is completed. □
Next, we prove that problem (1.1) has a mountain-pass-type solution. To overcome the difficulty of finding a bounded (PS) sequence for the associated functional I, motivated by [1, 10], we use a cut-off function \(\psi\in C_{0}^{1}(\mathbb{R}^{+})\) that satisfies
and study the following modified functional \(I^{T}\) defined by
where \(T>0\) and \(\eta_{T}(u)=\psi(\frac{\Vert u\Vert ^{p}}{T^{p}})\). For \(T>0\) sufficiently large and λ sufficiently small, we will prove that there exists a critical point \(\tilde{u}_{0}\) of \(I_{T}\) such that \(\Vert \tilde{u}_{0}\Vert \leq T\), and so \(\tilde{u}_{0}\) is also a critical point of I. For this purpose, we use the following theorem given in [1].
Lemma 2.3
see[1]
Let X be a Banach space with norm \(\Vert \cdot \Vert _{X}\), and \(K\subset\mathbb{R}^{+}\) be an interval. Consider the family of \(C^{1}\) functionals on X
with B nonnegative and either \(A(u)\rightarrow\infty\) or \(B(u)\rightarrow\infty\) as \(\Vert u\Vert _{X}\rightarrow\infty\) and \(I_{\mu}(0)=0\). For any \(\mu\in K\), we set
If for any \(\mu\in K\), the set \(\Gamma_{\mu}\) is nonempty, and
then, for almost every \(\mu\in K\), there is a sequence \(\{u_{n}\}\subset X\) such that (i) \(\{u_{n}\}\) is bounded; (ii) \(I_{\mu}(u_{n})\rightarrow c_{\mu}\); (iii) \(I'_{\mu}(u_{n})\rightarrow0\) in \(X^{-1}\).
In our case,
So the perturbed functional we study is
and
where \({\widehat{M}}(\Vert u\Vert )=(a + \lambda\eta_{T}(u)\Vert u\Vert ^{pm} + \frac{\lambda}{(m + 1)T^{p}}\eta'_{T}(u)\Vert u\Vert ^{p(m + 1)})\). The following lemmas, Lemma 2.4 and Lemma 2.5, imply that \(I^{T}_{\mu}\) satisfies the conditions of Lemma 2.3.
Lemma 2.4
Let \((F_{1})\)-\((F_{3})\) hold, Then \(\Gamma_{\mu}\neq\emptyset\) for all \(\mu\in[\frac{1}{2},1]\).
Proof
Choose \(\beta(x)\in C_{0}^{1}(\mathbb{R}^{N})\) with \(\beta(x)\geq0\) in \(\mathbb{R}^{N}\), \(\Vert \beta \Vert =1\), and \(\operatorname{supp}(\beta)\subset B_{R}\) for some \(R>0\). By \((F_{3})\) we have that, for any \(C_{3}>0\) with \({C_{3}}/{2}\int_{B_{R}}\beta^{p}\,dx>{a}/{p}\), there exists \(C_{4}>0\) such that
Then, for \(t^{p}>2T^{p}\),
It follows that we can choose \(t>0\) large enough such that \(I^{T}_{\mu}(t\beta)<0\). The proof is completed. □
Lemma 2.5
Let \((F_{1})\)-\((F_{3})\) hold. Then there exists a constant \(c>0\) such that \(c_{\mu}\geq c>0\) for all \(\mu\in[\frac{1}{2},1]\) if \(\Vert h\Vert _{p'}< m_{1}\).
Proof
Similarly as in the proof of Lemma 2.1, we can show that, for every \(\mu\in[\frac{1}{2},1]\), there exists \(c>0\) such that \(I^{T}_{\mu}(u)\geq c\) with \(\Vert u\Vert =\tilde{\rho}\) and \(\Vert h\Vert _{p'}< m_{1}\). Fix \(\mu\in[\frac{1}{2},1]\) and \(\gamma\in\Gamma_{\mu}\). By the definition of \(\Gamma_{\mu}\), \(\Vert \gamma({1})\Vert >\tilde{\rho }\). By the continuity we deduce that there exists \(t_{\gamma}\in(0,1)\) such that \(\Vert \gamma({t_{\gamma}})\Vert _{E}=\tilde{\rho}\). Therefore, for any \(\mu\in[\frac{1}{2},1]\),
which completes the proof. □
Lemma 2.6
For any \(\mu\in[\frac{1}{2},1]\) and \(a>2^{m+1}(\frac{m+3}{m+1})\lambda T^{pm}\), each bounded (PS) sequence of the functional \(I^{T}_{\mu}\) admits a convergent subsequence.
Proof
By Lemmas 2.3-2.5, we obtain that, for a.e. \(\mu\in [1/2,1]\), there is a bounded sequence \(\{u_{n}\}\) in X that satisfies
Since the embedding \(X\hookrightarrow L^{q}(\mathbb{R}^{N})\) is compact for \(q\in(p,p^{*})\), passing to a subsequence, we can assume that
By (2.16) we can get
where
It is clear that
An analogous argument as in (2.22) and (2.25) gives us that
Combining the above results and \(a>2^{m+1}(\frac{m+3}{m+1})\lambda T^{pm}\), we have that \(\vert A_{n}\vert \to0\) as \(n\to\infty\). Then, using a standard equality ([3], Lemma 2.1), we can show that \(u_{n} \rightarrow u\) in X. The proof is completed. □
Lemma 2.7
Assume \((F_{1})\)-\((F_{3})\) and \(a>2^{m+1}(\frac{m+3}{m+1})\lambda T^{pm}\). Then, for almost every \(\mu\in[\frac{1}{2},1]\), there exist \(u^{\mu}\in X\setminus\{0\}\) such that \((I^{T}_{\mu})'(u^{\mu})=0\) and \(I^{T}_{\mu}(u^{\mu})=c_{\mu}\) with \(\Vert h\Vert _{p'}< m_{1}\).
Proof
It follows from Lemmas 2.3-2.5 that, for every \(\mu \in[\frac{1}{2},1]\), there exists a bounded sequence \(\{u_{n}^{\mu}\} \subset X\) such that
By Lemma 2.6 we can suppose that \(u^{\mu}\in X\) and \(u^{\mu}_{n}\rightarrow u^{\mu}\) in X. The proof is completed. □
According to Lemma 2.6, there exists a sequence \(\{\mu_{n}\}\subset[\frac{1}{2},1]\) with \(\mu_{n}\rightarrow1\) and \(\{u_{n}\}\subset X\) as \(n\rightarrow\infty\) such that \(I^{T}_{\mu_{n}}(u_{n})=c_{\mu_{n}}\), \((I^{T}_{\mu_{n}})'(u_{n})=0\), and \(u_{n}\) is a positive solution of
In the following, to obtain \(\Vert u_{n}\Vert < T\), we establish an identity that extends the Kazin-Pohozav identity in ([13], Thm. 29.4) with \(p=2\).
Lemma 2.8
Assume that \(f(x,u):\mathbb{R}^{N}\times\mathbb{R}^{1}\rightarrow\mathbb{R}^{1}\) is a Carethéodary function, \(u\in C^{2}_{\mathrm {loc}}(\mathbb{R}^{N})\) is a solution of
\(\frac{\partial u}{\partial x_{i}}\in L^{p}({\mathbb{R}^{N}})\), \(i=1,2,\ldots\) , and \(F(x,u), F_{1}(x,u)\in L^{1}(\mathbb{R}^{N})\). Then
where \(F(x,u)=\int^{u}_{0}f(x,s)\,ds\) and \(F_{1}(x,u)=\sum^{N}_{i=1}x_{i}\frac{\partial F(x,u)}{\partial x_{i}}\).
Proof
Multiplying equation (2.44) by \(x\cdot\nabla u\) and integrating over the ball \(B_{R}\), we obtain
Then
where \(n_{i}\) are the components of the unit outward normal to \(\partial B_{R}\), and ds is an area element. On the other hand, integrating by parts, we obtain
On \(B_{R}\), we have \(\nabla u=\frac{\partial u}{n}\cdot\vec{n}=\frac {\partial u}{\partial n}\frac{x}{R}\) and
Further, we have
Therefore, we obtain
Since \(F(x,u)\in L^{1}({\mathbb {R}}^{N})\) and \(u\in X\), we claim that
Indeed, otherwise,
Then, there exists \(R_{0}>0\) such that, for \(R\geq R_{0}\),
Let \(R_{n}=R_{0}+n\), \(n=1,2,\dots\). Then \(R_{n}\to\infty\) as \(n\to\infty\). It follows from the integral mean theorem that there is \(\xi_{n}\in (R_{n-1},R_{n})\) and \(\xi_{n}\geq R_{0}\) such that, for \(R\geq R_{0}\),
and thus
This contradicts the fact
Therefore, (2.52) is true. Thus, letting \(R\to\infty\) in (2.51), we have
Then, we finish the proof of Lemma 2.8. □
Lemma 2.9
Let \(a>2^{m+1}(\frac{m+3}{m+1})\lambda T^{pm}\), and let \(u\in X\) be a weak solution of
where \({\widehat{M}}(\Vert u\Vert )=(a + \lambda\eta_{T}(u)\Vert u\Vert ^{pm} + \frac{\lambda}{(m + 1)T^{p}}\eta'_{T}(u)\Vert u\Vert ^{p(m + 1)})\). Then the following identity holds:
Proof
Since \(u\in X\) is a weak solution of (2.59), by standard regularity results, \(u\in C^{2}_{\mathrm {loc}}(\mathbb{R}^{N})\cap W^{1,p}(\mathbb {R}^{N})\). Let
Then \(u\in X\) is also a solution of
By Lemma 2.8,
where \(G(x,u)=\int^{u}_{0}g(x,s)\,ds\) and \(G_{1}(x,u)=\sum^{N}_{i=1}x_{i}\frac {\partial G(x,u)}{\partial x_{i}}\). Then the conclusion holds. □
Lemma 2.10
Assume that \((F_{1})\)-\((F_{3})\) and \((H)\) hold and that \(\Vert h\Vert _{p'}< m_{1}\) for \(m_{1}\) given in Lemma 2.6. Let \(u_{n}\) be a critical point of \(I^{T}_{\mu_{n}}\) at level \(c_{\mu_{n}}\). Then for T sufficiently large, there exists \(\lambda_{0}=\lambda_{0}(T)\) with \(\lambda_{0}< a(\frac{m+1}{m+3})T^{-pm}\) such that, for any \(\lambda\in[0,\lambda_{0})\), subject to a subsequence, \(\Vert u_{n}\Vert < T\) for all \(n\in\mathbb{N}\).
Proof
Since \((I^{T}_{\mu_{n}})'(u_{n})=0\), by Lemma 2.9 \(u_{n}\) satisfies
Using \(I^{T}_{\mu_{n}}(u_{n})=c_{\mu_{n}}\), we have
Therefore, by (2.64), (2.65) and \(a>2^{m+1}(\frac {m+3}{m+1})\lambda T^{pm}\) we deduce that
By the min-max definition of the mountain pass level, Lemma 2.5, and (2.35) we have
Using \((H)\) and the Young equality, we have
We can easily calculate that
Combining the above estimates, we see that
Since \(\xi(x)\in L^{p'}(\mathbb{R}^{N})\cap W^{1,\infty}\), we see that \(\xi^{p'}u_{n}\in X\). It follows from \((I^{T}_{\mu_{n}}(u_{n}))'(\xi^{p'}u_{n})=0\) that
Since \(a>2^{m+1}(\frac{m+3}{m+1})\lambda T^{pm}\), we have \(({3a}/2)\geq \widehat{M}( \Vert \xi^{p'}u_{n}\Vert )\), and it follows from (2.69) and (2.71) that
From (2.70) by the Hölder inequality we deduce that
where C is a constant independent of λ and T.
By \((F_{3})\), for any \(L>0\), there exists \(C(L)>0\) such that
Combining (2.72)-(2.74), we get
For \(L>0\) large enough, we obtain
It follows from (2.70) and (2.76) that
On the other hand,
Suppose that \(\Vert u_{n}\Vert >T\) for \(n\in {\mathbb {N}}\) and T large enough. Then
which is not true if we choose T large and λ small enough. So by setting \(\lambda(T)\) small we obtain that the sequence \(\{u_{n}\}\) is bounded for any \(\lambda\in[0,\lambda_{0})\), and the conclusion holds. □
Lemma 2.11
Let T, \(\lambda_{0}\) be defined by Lemma 2.10, and \(u_{n}\) be the critical point of \(I^{T}_{\mu_{n}}\) at level \(c_{\mu_{n}}\). Then the sequence \(\{u_{n}\}\) is also a (PS) sequence for I.
Proof
From the proof of Lemma 2.10 we may assume that \(\Vert u_{n}\Vert \leq T\). So
Since \(\mu_{n}\rightarrow1\), we can show that \(\{u_{n}\}\) is a (PS) sequence of I. Indeed, the boundedness of \(\{u_{n}\}\) implies that \(\{I^{T}_{\mu_{n}}\}\) is bounded. Also,
Thus, \(I'(u_{n})\rightarrow0\), and \(\{u_{n}\}\) is a bounded (PS) sequence of I. By Lemma 2.5, \(\{u_{n}\}\) has a convergent subsequence. We may assume that \(u_{n}\rightarrow\tilde{u}_{0}\). Consequently, \(I'(\tilde{u}_{0})=0\). According to Lemma 2.4, we have that \(I(\tilde{u}_{0})=\lim_{n\rightarrow\infty} I(u_{n})=\lim_{n\rightarrow\infty}I^{T}_{\mu_{n}}(u_{n})\geq c>0\) and \(\tilde{u}_{0}\) is a solution of problem (1.1). Thus, we completed the proof. □
Proof of Theorem 1.1
By Lemma 2.2 the problem has a solution \(u_{0}\in X\) with \(I(u_{0})<0\). From Lemma 2.9 we know that problem (1.1) possesses a second solution \(\tilde{u}_{0}\in X\) with \(I(\tilde{u}_{0})\geq c>0\). Hence, \(u_{0}\neq\tilde{u}_{0}\), and we complete the proof of Theorem 1.1. □
3 Proof of Theorem 1.2
Let \(I_{\lambda}(u):X\rightarrow\mathbb{R} \) be the energy functional associated with problem (1.13) defined by
where \(F(u)=\int_{0}^{u}f(s)\,ds\). It is easy to see that the functional \(I\in C^{1}(E,\mathbb{R})\) and its Gateaux derivative is given by
Clearly, we see that a weak solution of (1.13) corresponds to a critical point of the functional.
In this part, we first proof the nonexistence for problem (1.13) for large \(\lambda>\lambda^{*}\). which means that if a solution exists, then λ must sufficiently small. Secondly, we obtain that there exists \(\lambda^{**}\) such that problem (1.1) has at least one solution for any \(0<\lambda<\lambda^{**}\). Finally, by the properties of \(\lambda^{*}\) and \(\lambda^{**}\) we deduce that \(\lambda^{*}=\lambda ^{**}\). We will break the proof into six steps.
Proof of Theorem 1.2
Step 1. Nonexistence for large \(\lambda>0\) . It is sufficient to show that if u is a nontrivial solution of problem (1.13), then \(\lambda>0\) must be small. Assume that u is a nontrivial solution of problem (1.1). Then we get \(I'_{\lambda}(u)u=0\), that is,
Since \(p< q<\min\{p(m+1),p^{*}\}\), applying the Young inequality and (1.9), we deduce that
which implies that \(\lambda\leq\lambda_{1}=(S^{q}_{q})^{\frac {pm}{q-p}}a^{-\frac{p(m+1)-q}{q-p}}\). On the other hand, if \(\lambda ^{*}\geq\lambda_{1}\), then we conclude that problem (1.1) has no solution for any \(\lambda\in(\lambda^{*}, +\infty)\).
Step 2. Coercivity of \(I_{\lambda}(u)\) . Indeed, for any \(u\in E\) and all \(\lambda>0\),
Since \(q< p(m+1)\), there exists \(C_{1}=C_{1}(\lambda,q,m,S_{q})\) such that
It follows that
This implies that \(I_{\lambda}(u)\) is coercive.
Step 3. The infimum of \(I_{\lambda}\) is attained. Let \(\{u_{n}\}\) be a minimizing sequence of \(I_{\lambda}\). Then from Step 2 we immediately see that \(\{u_{n}\}\) is bounded in X. Therefore, without loss of generality, we may assume that \(\{u_{n}\}\) is nonnegative and converges weakly and pointwise to some u in X.
Using the compact embedding \(X\hookrightarrow L^{q}(\mathbb{R}^{N})\), we have
by the weak lower semicontinuity of the norm \(\Vert \cdot \Vert \). Thus,
Therefore, u is a global minimum for \(I_{\lambda}\), and hence it is a critical point, namely a weak solution to problem (1.1).
Step 4. The weak solution u is nontrivial if \(\lambda>0\) is sufficiently small. Clearly, \(I_{\lambda}(0)=0\). Therefore, it is sufficient to show that there exists \(\lambda_{0}>0\) such that
Choose \(u_{0}\in C^{\infty}_{0}({\mathbb {R}}^{N})\), \(u_{0}\not\equiv0\), such that \(\Vert u_{0}\Vert _{E}=1\). Denote
where
Then there exist \(\lambda_{0}>0\) and large \(t_{\lambda}>0\) such that \(I_{\lambda}(t_{\lambda}u_{0})<0\) for \(\lambda\in(0,\lambda_{0}]\). Let \(e=t_{\lambda}u_{0}\). Then \(\Vert e\Vert =t_{\lambda}\) and \(I_{\lambda}(e)<0\). This implies that (3.10) is true. So the weak solution u is nontrivial if \(\lambda>0\) is sufficiently small.
Now, we define
Clearly, \(\lambda^{**}\geq\lambda^{*}\). To complete the proof of Theorem 1.2, it suffices to prove the following facts: (a) problem (1.13) has a weak solution for any \(\lambda<\lambda^{**}\); (b) \(\lambda ^{**}=\lambda^{*}\), and problem (1.13) admits a weak solution when \(\lambda=\lambda^{*}\).
Step 5. Problem ( 1.13 ) has a solution for any \(\lambda <\lambda^{**}\) and \(\lambda^{*}=\lambda^{**}\) . Fix \(\lambda<\lambda ^{**}\). By the definition of \(\lambda^{**}\), there exists \(\mu\in (\lambda,\lambda^{**})\) such that \(I_{\lambda}\) has a nontrivial critical point \(u_{\mu}\in E\). Clearly, we have
This implies that \(u_{\mu}\) is a subsolution of problem (1.13). In order to find a supsolution of (1.13) that dominates \(u_{\mu}\), we consider the constrained minimization problem
Arguments similar to those used in Step 3 and Step 4 show that the above minimization has a solution \(u_{\lambda}\geq u_{\mu}\), which is also a weak solution of problem (1.13). Hence, problem (1.13) admits a weak solution for any \(\lambda\in[0,\lambda^{**})\), This means that \(\lambda^{*}\geq\lambda^{**}\) by the definition of \(\lambda^{*}\). But we already know that \(\lambda^{**}\geq\lambda^{*}\), and therefore \(\lambda^{**}=\lambda^{*}\).
Step 6. Problem ( 1.13 ) admits a nontrivial solution when \(\lambda=\lambda^{*}\) . Let \(\{\lambda_{n}\}\) be a increasing sequence converging to \(\lambda ^{*}\), and \(\{u_{n}\}\) be a sequence of solutions of (1.1) corresponding to \(\lambda_{n}\). By Step 2, \(\{u_{n}\}\) is bounded in X, and without loss of generality we may assume that \(u_{n}\rightharpoonup u\) in X, \(u_{n}\rightarrow u\) in \(L^{q}({\mathbb {R}}^{N})\), and \(u_{n}\rightarrow u^{*}\) a.e. in X. It follows from \(I_{\lambda}(u_{n})v=0\) that, for any \(v\in X\),
Then, passing to the limit as \(n\to\infty\), we deduce that \(u^{*}\) satisfies \(I_{\lambda}(u^{*})v=0\) when \(\lambda=\lambda^{*}\). Now, it remains to prove that \(u^{*}\) is a nontrivial critical point for \(I_{\lambda^{*}}\). From \(I'_{\lambda}(u_{n})u_{n}=0\) it is easy to deduce that \(\Vert u_{n}\Vert \geq(\lambda_{n} S_{q}^{-q})^{1/(q-p(m+1))}\), which implies that \(u_{n}\) has a lower bound. Next, since \(\lambda_{n}\nearrow\lambda^{*}\) as \(n\to\infty\), it suffices to show that \(\Vert u_{n}-u^{*}\Vert \to 0\) as \(n\to\infty\).
Since \(u_{n}\) and \(u^{*}\) are the solutions of (1.1) corresponding to \(\lambda_{n}\) and \(\lambda^{*}\), we see that
where
By the Hölder inequality and compact embedding \(u_{n}\to u\) in \(L^{q}(\mathbb{R}^{N},H)\) we have
Next, consider the functional \(j:X\to\mathbb{R}\) defined by
Since \(\vert j(\omega)\vert \leq2\Vert u^{*}\Vert ^{p-1}\Vert \omega \Vert \), j is continuous on X. Using \(u_{n}\rightharpoonup{u^{*}}\) and the boundedness of \(u_{n}\) and \(u^{*}\) in X, we have that
Combining (3.15), (3.16), and (3.18), this forces \(X_{n}\to{0}\) as \(n\to\infty\). Then, using the standard inequality (2.25) in \(\mathbb{R}^{N}\), we have that \(\Vert u_{n}-u^{*}\Vert \to{0}\) as \(n\to\infty\), and thus \(u^{*}\) is a nontrivial weak solution of problem (1.13) corresponding to \(\lambda=\lambda^{*}\). This completes the proof of Theorem 1.2. □
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The authors are highly grateful for the referees’ careful reading and comments on this paper that led to the improvement of the original manuscript. The first author would like to express his gratitude to the second author for her help in the revision of this paper.
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Liu, L., Zhou, C. Existence,multiplicity, and nonexistence of solutions for a p-Kirchhoff elliptic equation on \(\mathbb{R}^{N}\) . Bound Value Probl 2017, 20 (2017). https://doi.org/10.1186/s13661-017-0752-6
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DOI: https://doi.org/10.1186/s13661-017-0752-6