Open Access

Existence,multiplicity, and nonexistence of solutions for a p-Kirchhoff elliptic equation on \(\mathbb{R}^{N}\)

Boundary Value Problems20172017:20

https://doi.org/10.1186/s13661-017-0752-6

Received: 4 October 2016

Accepted: 20 January 2017

Published: 2 February 2017

Abstract

In this paper, we study the multiplicity of solutions for the following nonhomogeneous p-Kirchhoff elliptic equation:
$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(|\nabla{u}|^{p}+|u|^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(-\Delta_{p}u+|u|^{p-2}u \bigr) =f(u)+h(x),\quad x\in\mathbb{R}^{N}, $$
(0.1)
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 \(\lambda\in(\lambda^{*}, +\infty)\).

Keywords

p-Kirchhoff elliptic equationbounded potentialvariational methodsmountain pass lemma

1 Introduction

In this paper, we are interested in the multiplicity of solutions to the following nonhomogeneous p-Kirchhoff elliptic problem:
$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{p}+\vert u \vert ^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(- \Delta_{p}u+\vert u\vert ^{p-2}u \bigr) =f(u)+h(x),\quad x\in \mathbb{R}^{N}, $$
(1.1)
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
$$ \rho_{tt}- \biggl(\frac{P_{0}}{h}+\frac{E}{2L} \int^{L}_{0}u^{2}_{x}\,dx \biggr)u_{xx}=0,\quad 0< x< L, t>0, $$
(1.2)
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
$$ \rho_{tt}-M \bigl(\Vert \nabla u\Vert ^{2}_{2} \bigr)\Delta u=f(x,u), \quad x\in\Omega, t>0, $$
(1.3)
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
$$ -M \bigl(\Vert \nabla u\Vert ^{2}_{2} \bigr) \Delta u=f(x,u), \quad x\in\Omega, t>0. $$
(1.4)
Some classical and interesting results on Kirchhoff-type elliptic equations can be found, for example, in [39].
Particularly, Li et al. [10] considered the Kirchhoff-type problem
$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{2}+b \vert u\vert ^{2} \bigr)\,dx \biggr) \biggr) (-\Delta u+bu) =f(u),\quad x \in \mathbb{R}^{N}, $$
(1.5)
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
$$ \textstyle\begin{cases} (a+\lambda( \int_{\mathbb{R}^{N}} (\vert \nabla u\vert ^{p}+b \vert u\vert ^{p} )\,dx )^{\tau} ) (- \Delta_{p}u+b\vert u\vert ^{p-2}u ) \\ \quad =\vert u\vert ^{m-2}u+\mu \vert u\vert ^{q-2}u, \quad x\in \mathbb{R}^{N}, \\ u(x)>0, \quad x\in\mathbb{R}^{N},\quad \quad u(x)\in W^{1,p} ( \mathbb{R}^{N} ), \end{cases} $$
(1.6)
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, 57, 1014].
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
$$ \Vert u\Vert = \biggl( \int_{\mathbb{R}^{N}}\vert \nabla u\vert ^{p}+\vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}}, \quad 1< p< \infty. $$
(1.8)
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
$$ \Vert u\Vert _{q}\leq S_{q}\Vert u \Vert , \quad \forall u\in W^{1,p} \bigl(\mathbb{R}^{N} \bigr). $$
(1.9)
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 \(v \in X\),
$$ \bigl(a+\lambda \Vert u\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u\vert ^{p-2}\nabla u \nabla v+\vert u\vert ^{p-2}uv \bigr) \,dx= \int_{\mathbb{R}^{N}} \bigl(f(u)+h \bigr)v \,dx. $$
(1.10)
Let \(I(u):X\rightarrow\mathbb{R} \) be the energy functional associated with problem (1.1) defined by
$$ I(u)=\frac{a}{p}\Vert u\Vert ^{p}+\frac{\lambda}{p(m+1)}\Vert u \Vert ^{p(m+1)}- \int_{\mathbb{R}^{N}} \bigl(F(u)+hu \bigr)\,dx, $$
(1.11)
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
$$ \begin{aligned}[b] I'(u)v&= \bigl(a+\lambda \Vert u\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u\vert ^{p-2} \nabla u \nabla v+\vert u\vert ^{p-2}uv \bigr)\,dx \\ &\quad{} - \int_{\mathbb{R}^{N}} \bigl(f(u)+h \bigr)v\,dx, \quad \forall v\in X. \end{aligned} $$
(1.12)
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,
$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{p}+\vert u \vert ^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(- \Delta_{p}u+\vert u\vert ^{p-2}u \bigr) =\vert u\vert ^{q-2}u, \quad x\in\mathbb{R}^{N}. $$
(1.13)
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
$$ F(s)\leq\varepsilon \vert s\vert ^{p}+C_{\varepsilon} \vert s \vert ^{q}, \quad \forall s\in\mathbb{R}, $$
(2.1)
with \(\varepsilon>0\). By the Hölder inequality we have
$$ \biggl\vert \int_{\mathbb{R}^{N}}hu\,dx \biggr\vert \leq S_{q}^{-1} \Vert h\Vert _{p'}\Vert u\Vert \leq\epsilon \Vert u\Vert ^{p}+C_{\epsilon} \Vert h\Vert _{p'}^{p'}. $$
(2.2)
Thus,
$$ \begin{aligned}[b] I(u)&\geq\frac{a}{p}\Vert u\Vert ^{p}- \varepsilon \Vert u\Vert ^{p}-C_{\varepsilon} \Vert u\Vert ^{q}-\epsilon \Vert u\Vert ^{p}-C_{\epsilon} \Vert h \Vert _{p'}^{p'} \\ &\geq\frac{a}{2p}\Vert u\Vert ^{p}-C_{1}\Vert u\Vert ^{q}-C_{2} \Vert h\Vert ^{p'}_{p'}, \end{aligned} $$
(2.3)
where \(\varepsilon=\epsilon=\frac{a}{4p}\), \(C_{1}\), \(C_{2}\) are some positive constants. Let
$$ z(t)=\frac{a}{2p}t^{p}-C_{1}t^{q}, \quad t\geq0. $$
(2.4)
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
$$ I(u_{0})=\inf \bigl\{ I(u):u\in{\overline{B}_{\rho}} \bigr\} < 0, $$
(2.5)
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
$$ I(t\phi)\leq\frac{a}{p}t^{p}\Vert \phi \Vert ^{p}+ \frac{\lambda}{p(m+1)}t^{p(m+1)}\Vert \phi \Vert ^{p(m+1)}-t \int_{\mathbb{R}^{N}}h(x)\phi \,dx< 0 $$
(2.6)
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,
$$ c_{\rho}=\inf_{u\in\overline{B}_{\rho}}I(u)< 0 \quad \mbox{and}\quad \inf _{u\in\partial B_{\rho}}I(u)>0, $$
(2.7)
where ρ is given in Lemma 2.1. Let \(\varepsilon_{n}\downarrow0\) be such that
$$ 0< \varepsilon_{n}< \inf_{u\in\partial B_{\rho}}I(u)- \inf_{u\in B_{\rho}}I(u). $$
(2.8)
Then, by Ekland’s variational principle [15] there exists \(\{u_{n}\} \subset\overline{B}_{\rho}\) such that
$$ c_{\rho}\leq I(u_{n})< c_{\rho}+ \varepsilon_{n} $$
(2.9)
and
$$ I(u_{n})< I(u)+\varepsilon_{n}\Vert u_{n}-u\Vert \qquad\mbox{for all } u\in\overline{B_{\rho}}, u_{n}\neq u. $$
(2.10)
Then, it follows from (2.8)-(2.10) that
$$ I(u_{n})< c_{\rho}+\varepsilon_{n}\leq\inf _{u\in B_{\rho}}I(u)+ \varepsilon_{n}< \inf_{u\in\partial B_{\rho}}I(u). $$
(2.11)
So \(u_{n}\in B_{\rho}\), and we now consider the function \(F: \overline {B}_{\rho}\to {\mathbb {R}}\) given by
$$ F(u)=I(u)+\varepsilon_{n}\Vert u_{n}-u\Vert , \quad u\in \overline{B}_{\rho}. $$
(2.12)
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,
$$ t^{-1} \bigl(F(u_{n}+tv)-F(u_{n}) \bigr)\geq0 \quad\mbox{for small } t>0, \forall v\in B_{1}. $$
(2.13)
Hence,
$$ t^{-1} \bigl(I(u_{n}+tv)-I(u_{n}) \bigr)+ \varepsilon_{n}\Vert v\Vert \geq0. $$
(2.14)
Passing to the limit as \(t\to0^{+}\), it follows that
$$ I'(u_{n})v+\varepsilon_{n} \Vert v\Vert \geq0,\quad\forall v\in B_{1}. $$
(2.15)
Replacing v in (2.15) by −v, we get
$$ -I'(u_{n})v+\varepsilon_{n}\Vert v\Vert \geq0,\quad\forall v\in B_{1}, $$
(2.16)
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
$$ u_{n}\rightharpoonup u_{0}, \quad \text{in } X; \quad\quad u_{n} \rightarrow u_{0},\quad L^{q} \bigl( \mathbb{R}^{N} \bigr); \quad\quad u_{n}\rightarrow u_{0}, \quad \mbox{a.e. in } \mathbb{R}^{N}. $$
(2.17)
By (1.12) we can get
$$ \bigl(I(u_{n})-I(u_{0}) \bigr)'(u_{n}-u_{0})=P_{n}+Q_{n}+K_{n}, $$
(2.18)
where
$$ \begin{gathered} P_{n} = \bigl(a+\lambda \Vert u_{n}\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}-\vert \nabla u\vert ^{p-2} \nabla u \bigr)\nabla(u_{n}-u_{0}) \\ \hphantom{P_{n} =}{} + \bigl(\vert u_{n}\vert ^{p-2}u_{n}-u_{0}^{p-2}u_{0} \bigr) (u_{n}-u_{0})\,dx, \\ Q_{n}= \lambda \bigl( \bigl(\Vert u_{n}\Vert ^{pm}-\Vert u_{0}\Vert ^{pm} \bigr) \bigr) \int_{\mathbb{R}^{N}}\vert \nabla u_{0}\vert ^{p-2} \nabla{u_{0}}\nabla(u_{n}-u_{0}) \\ \hphantom{Q_{n}=}{}+\vert u_{0}\vert ^{p-2}u_{0}(u_{n}-u_{0})\,dx, \\ K_{n}= \int_{\mathbb{R}^{N}} \bigl(f(u_{n})-f(u_{0}) \bigr) (u_{n}-u_{0})\,dx. \end{gathered} $$
(2.19)
It is clear that
$$ \bigl(I(u_{n})-I(u_{0}) \bigr)'(u_{n}-u_{0}) \rightarrow0 \quad\text{as } n\rightarrow\infty. $$
(2.20)
By \((F_{1})\) and \((F_{2})\), for any \(\varepsilon>0\), there exists \(C_{\varepsilon}>0\) such that
$$ \bigl\vert f(t) \bigr\vert \leq\varepsilon \vert t\vert ^{p-1}+C_{\varepsilon} \vert t\vert ^{q-1},\quad t\in \mathbb{R}. $$
(2.21)
Hence,
$$\begin{aligned} \begin{aligned}[b] \vert K_{n}\vert &= \biggl\vert \int_{\mathbb{R}^{N}} \bigl(f(u_{n})-f(u_{0}) \bigr) (u_{n}-u_{0})\,dx \biggr\vert \\ &\leq\varepsilon \bigl(\Vert u_{n}\Vert ^{p-1}+\Vert u_{0}\Vert ^{p-1} \bigr)\Vert u_{n}-u_{0} \Vert +C_{\varepsilon}\bigl(\Vert u_{n}\Vert ^{q-1}_{q}+\Vert u_{0}\Vert ^{q-1}_{q} \bigr)\Vert u_{n}-u_{0} \Vert _{q} \\ &\to0 \quad \mbox{as } n\to\infty. \end{aligned} \end{aligned}$$
(2.22)
Define the linear function \(g:X\rightarrow\mathbb{R}\) by
$$ g(\omega)= \int_{\mathbb{R}^{N}}\vert \nabla u_{0}\vert ^{p-2} \nabla u_{0}\nabla\omega+\vert u_{0}\vert ^{p-2}u_{0}\omega \,dx. $$
(2.23)
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
$$ \begin{aligned}[b] g(u_{n}-u_{0})&= \int_{\mathbb{R}^{N}}\vert \nabla u_{0}\vert ^{p-2} \nabla u_{0}\nabla(u_{n}-u_{0})+\vert u_{0}\vert ^{p-2}u_{0}(u_{n}-u_{0})\,dx \\ & \rightarrow0 \quad \mbox{as } n\rightarrow\infty. \end{aligned} $$
(2.24)
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 \(\mathbb{R}^{N}\)
$$\begin{aligned} \begin{aligned} &\bigl\langle \vert x\vert ^{p-2}x- \vert y\vert ^{p-2}y,x-y \bigr\rangle \geq{C_{p}}\vert x-y\vert ^{p}, \quad p\geq2, \\ &\bigl\langle \vert x\vert ^{p-2}x-\vert y\vert ^{p-2}y,x-y \bigr\rangle \geq\frac{C_{p}\vert x-y\vert ^{p}}{{\vert x\vert +\vert y\vert }^{2-p}},\quad 2>p>1, \end{aligned} \end{aligned}$$
(2.25)
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
$$ \begin{aligned} &\psi(t)=1, \quad \forall t\in[0,1]; \quad\quad 0\leq\psi\leq1, \quad \forall t\in(1,2); \\& \psi(t)\equiv0,\quad \forall t \in[2,+\infty); \quad\quad \bigl\Vert \psi' \bigr\Vert _{\infty}\leq2, \end{aligned} $$
(2.26)
and study the following modified functional \(I^{T}\) defined by
$$ I^{T}(u)=\frac{a}{p}\Vert u\Vert ^{p}+ \frac{\lambda}{p(m+1)}\eta_{T}(u)\Vert u\Vert ^{p(m+1)}- \int_{\mathbb{R}^{N}} \bigl(F(u)+hu \bigr)\,dx, \quad u\in X, $$
(2.27)
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
$$ I_{\mu}(u)=A(u)-\mu B(u),\quad \mu\in K, $$
(2.28)
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
$$ \Gamma_{\mu}= \bigl\{ \gamma\in \bigl( C[0,1], X \bigr): \gamma(0)=0,I_{\mu}\bigl(\gamma(1) \bigr)< 0 \bigr\} . $$
(2.29)
If for any \(\mu\in K\), the set \(\Gamma_{\mu}\) is nonempty, and
$$ c_{\mu}=\inf_{\gamma\in \Gamma_{\mu}}\max_{t\in[0,1]}I_{\mu}\bigl(\gamma(t) \bigr)>0, $$
(2.30)
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,
$$ A(u)=\frac{a}{p}\Vert u\Vert ^{p}+\frac{\lambda}{p(m+1)} \eta_{T}(u)\Vert u\Vert ^{p(m+1)}, \quad\quad B(u)= \int_{\mathbb{R}^{N}} \bigl(F(u)+hu \bigr)\,dx. $$
(2.31)
So the perturbed functional we study is
$$ I^{T}_{\mu}(u)=\frac{a}{p}\Vert u\Vert ^{p}+\frac{\lambda}{p(m+1)}\eta_{T}(u)\Vert u\Vert ^{p(m+1)}-\mu \int_{\mathbb{R}^{N}} \bigl(F(u)+hu \bigr)\,dx, $$
(2.32)
and
$$ \bigl(I^{T}_{\mu}(u) \bigr)'v={\widehat{M}} \bigl(\Vert u\Vert \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{p - 2} \nabla{u}\nabla{v} + \vert u\vert ^{p - 2}uv \bigr)\,dx -\mu \int_{\mathbb{R}^{N}} \bigl(f(u)+h \bigr)v\,dx, $$
(2.33)
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
$$ F(t)\geq C_{3}\vert t\vert ^{p}-C_{4}, \quad t\in \mathbb{R}^{+}. $$
(2.34)
Then, for \(t^{p}>2T^{p}\),
$$ \begin{aligned}[b] I^{T}_{\mu}(t\beta)&= \frac{a}{p}\Vert t\beta \Vert ^{p}+\frac{\lambda}{p(m+1)}\psi \biggl( \frac{\Vert t\beta \Vert ^{p}}{T^{p}} \biggr)\Vert t\beta \Vert ^{p(m+1)}-\mu \int_{\mathbb{R}^{N}} \bigl(F(t\beta)+ht\beta \bigr)\,dx \\ &=\frac{a}{p}\Vert t\beta \Vert ^{p}-\mu \int_{\mathbb{R}^{N}} \bigl(F(t\beta)+ht\beta \bigr)\,dx\leq \biggl( \frac{a}{p} -\frac{C_{3}}{2} \int_{B_{R}}\beta^{p}\,dx \biggr)t^{p}+C_{5}. \end{aligned} $$
(2.35)
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 \(\mu\in[\frac{1}{2},1]\),
$$ c_{\mu}=\inf_{\gamma\in\Gamma_{\mu}}\max_{t\in[0,1]}I^{T}_{\mu}\bigl( \gamma(t) \bigr)\geq\inf_{\gamma\in\Gamma_{\mu}} I^{T}_{\mu}\bigl( \gamma(t_{\gamma}) \bigr)\geq c>0, $$
(2.36)
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
$$ I^{T}_{\mu}(u_{n})\rightarrow c_{\mu},\quad\quad \bigl(I^{T}_{\mu}(u_{n}) \bigr)' \rightarrow0 \quad \text{in } X^{*},\quad \text{and} \quad\sup_{n} \Vert u_{n}\Vert < T. $$
(2.37)
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
$$ u_{n}\rightharpoonup u, \quad\text{in } X; \quad\quad u_{n} \rightarrow u, \quad L^{q} \bigl(\mathbb{R}^{N} \bigr);\quad\quad u_{n} \rightarrow u, \quad \text{a.e. in } \mathbb{R}^{N}. $$
(2.38)
By (2.16) we can get
$$ \bigl(I^{T}_{\mu}(u_{n})-I^{T}_{\mu}(u) \bigr)'(u_{n}-u)=A_{n}+B_{n}+\mu C_{n}, $$
(2.39)
where
$$ \begin{gathered} A_{n}= \widehat{M}(u_{n}) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n} -\vert \nabla u\vert ^{p-2} \nabla u \bigr)\nabla(u_{n}-u) \\ \hphantom{A_{n}=}{}+ \bigl(\vert u_{n}\vert ^{p-2}u_{n}-u^{p-2}u \bigr) (u_{n}-u)\,dx, \\ B_{n}= \bigl(\widehat{M}(u_{n})-\widehat{M}(u) \bigr) \int_{\mathbb{R}^{N}}\vert \nabla u\vert ^{p-2}\nabla{u} \nabla(u_{n}-u)+\vert u\vert ^{p-2}u(u_{n}-u)\,dx, \\ C_{n}= \int_{\mathbb{R}^{N}} \bigl(f(u_{n})-f(u) \bigr) (u_{n}-u)\,dx. \end{gathered} $$
(2.40)
It is clear that
$$ \bigl(I^{T}_{\mu}(u_{n})-I^{T}_{\mu}(u) \bigr)'(u_{n}-u)\rightarrow0 \quad\text{as } n \rightarrow \infty. $$
(2.41)
An analogous argument as in (2.22) and (2.25) gives us that
$$ B_{n}\to0 \quad \mbox{and}\quad C_{n}\to0 \quad\text{as } n\to\infty. $$
(2.42)
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
$$ I^{T}_{\mu}\bigl(u^{\mu}_{n} \bigr)\rightarrow c_{\mu}\quad \mbox{and} \quad \bigl(I^{T}_{\mu}\bigr)' \bigl(u^{\mu}_{n} \bigr)\rightarrow0\quad \mbox{as } n\rightarrow\infty. $$
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
$$\begin{aligned} {\widehat{M}} \bigl(\Vert u\Vert \bigr) \bigl(-\Delta_{p} u+ \vert u\vert ^{p-2}u \bigr)=\mu_{n} \bigl(f(u)+h(x) \bigr). \end{aligned}$$
(2.43)

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
$$ \textstyle\begin{cases} -\Delta_{p} u+f(x,u)=0\quad \textit{in } \mathbb{R}^{N}, \\ u(x)\rightarrow0 \quad\textit{as } \rightarrow0, \end{cases} $$
(2.44)
\(\frac{\partial u}{\partial x_{i}}\in L^{p}({\mathbb{R}^{N}})\), \(i=1,2,\ldots\) , and \(F(x,u), F_{1}(x,u)\in L^{1}(\mathbb{R}^{N})\). Then
$$ \frac{N-p}{p} \int_{\mathbb{R}^{N}}\vert \nabla u\vert ^{p}\,dx+ \int_{\mathbb{R}^{N}} \bigl(NF(x,u)+F_{1}(x,u) \bigr)\,dx=0, $$
(2.45)
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
$$\begin{aligned} \int_{B_{R}}f(x,u)x\cdot\nabla u\,dx= \int_{B_{R}}\operatorname {div}\bigl(\vert \nabla u\vert ^{p-2} \nabla u \bigr)x\cdot\nabla u\,dx. \end{aligned}$$
(2.46)
Then
$$ \begin{aligned}[b] \int_{B_{R}}f(x,u)x\cdot\nabla u\,dx&=\sum ^{N}_{i=1} \int_{B_{R}}x_{i} f(x,u)\frac{\partial u}{\partial x_{i}}\,dx \\ &=\sum^{N}_{i=1} \int_{B_{R}} \biggl(\frac{\partial}{\partial x_{i}} \bigl(x_{i}F(x,u) \bigr)- \biggl(F(x,u)+x_{i}\frac{\partial F(x,u)}{\partial x_{i}} \biggr) \biggr)\,dx \\ &=\sum^{N}_{i=1} \int_{\partial B_{R}}F(x,u) x_{i}n_{i}\,ds- \int_{B_{R}} \bigl(NF(x,u)+F_{1}(x,u) \bigr)\,dx \\ &=R \int_{\partial B_{R}}F(x,u)\,ds- \int_{B_{R}} \bigl(NF(x,u)+F_{1}(x,u) \bigr)\,dx, \end{aligned} $$
(2.47)
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
$$ \begin{aligned}[b] & \int_{B_{R}}\operatorname {div}\bigl(\vert \nabla u\vert ^{p-2} \nabla u \bigr)x\cdot\nabla u\,dx \\ &\quad =\sum^{N}_{j=1} \int_{B_{R}}\frac{\partial}{\partial x_{j}} \biggl(\vert \nabla u\vert ^{p-2}\frac{\partial u}{\partial x_{j}} \biggr)\sum^{N}_{i=1}x_{i} \frac{\partial u}{\partial x_{i}}\,dx \\ &\quad =\sum^{N}_{j=1} \int_{B_{R}} \Biggl(\frac{\partial}{\partial x_{j}} \Biggl(\vert \nabla u \vert ^{p-2}\frac{\partial u}{\partial x_{j}}\sum^{N}_{i=1}x_{i} \frac{\partial u}{\partial x_{i}} \Biggr)-\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}} \Biggl(\frac{\partial}{\partial x_{j}}\sum^{N}_{i=1}x_{i} \frac{\partial u}{\partial x_{i}} \Biggr) \Biggr)\,dx \\ &\quad = \int_{\partial B_{R}} \vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial n}x\cdot\nabla u\,ds- \int_{B_{R}}\vert \nabla u\vert ^{p}\,dx \\ &\quad\quad{} - \int_{B_{R}}\sum^{N}_{j=1} \vert \nabla u\vert ^{p-2} \Biggl(\sum^{N}_{i=1}x_{i} \frac{\partial^{2} u}{\partial x_{i}\,\partial x_{j}}\frac{\partial u}{\partial x_{j}} \Biggr)\,dx. \end{aligned} $$
(2.48)
On \(B_{R}\), we have \(\nabla u=\frac{\partial u}{n}\cdot\vec{n}=\frac {\partial u}{\partial n}\frac{x}{R}\) and
$$ \int_{\partial B_{R}}\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial n}x\cdot\nabla u\,dx=R \int_{\partial B_{R}}\vert \nabla u\vert ^{p}\,ds. $$
(2.49)
Further, we have
$$ \begin{aligned}[b] & \int_{B_{R}}\sum^{N}_{j=1} \vert \nabla u\vert ^{p-2} \Biggl(\sum^{N}_{i=1}x_{i} \frac{\partial^{2} u}{\partial x_{i}\,\partial x_{j}}\frac{\partial u}{\partial x_{j}} \Biggr)\,dx \\ &\quad =\frac{1}{p}\sum^{N}_{i=1} \int_{B_{R}} \biggl(\frac{\partial}{\partial x_{i}} \bigl(x_{i} \vert \nabla u\vert ^{p} \bigr)-\vert \nabla u\vert ^{p} \biggr) \,dx \\ &\quad =\frac{R}{p} \int_{\partial B_{R}}\vert \nabla u\vert ^{p}\,ds- \frac{N}{p} \int_{B_{R}}\vert \nabla u\vert ^{p}\,dx. \end{aligned} $$
(2.50)
Therefore, we obtain
$$ R \int_{\partial B_{R}} \biggl(F- \biggl(1-\frac{1}{p} \biggr)\vert \nabla u\vert ^{p} \biggr)\,ds+ \biggl(1-\frac{N}{p} \biggr) \int_{B_{R}}\vert \nabla u\vert ^{p}\,dx- \int_{B_{R}}(NF+F_{1})\,dx=0. $$
(2.51)
Since \(F(x,u)\in L^{1}({\mathbb {R}}^{N})\) and \(u\in X\), we claim that
$$ \liminf_{n\to\infty}R \int_{\partial B_{R}} \bigl( \bigl\vert F(x,u) \bigr\vert +\vert \nabla u\vert ^{p} \bigr)\,dS=0. $$
(2.52)
Indeed, otherwise,
$$ \liminf_{n\to\infty}R \int_{\partial B_{R}} \bigl( \bigl\vert F(x,u) \bigr\vert +\vert \nabla u\vert ^{p} \bigr)\,dS=a_{0}>0. $$
(2.53)
Then, there exists \(R_{0}>0\) such that, for \(R\geq R_{0}\),
$$ R \int_{\partial B_{R}} \bigl( \bigl\vert F(x,u) \bigr\vert +\vert \nabla u\vert ^{p} \bigr)\,dS\geq\frac{a_{0}}{2}. $$
(2.54)
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 \(R\geq R_{0}\),
$$ \int^{R_{n}}_{R_{n-1}} \int_{\partial B_{R}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,ds\,dR=\xi_{n} \int_{\partial B_{\xi_{n}}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,ds\geq\frac{a_{0}}{2}, $$
(2.55)
and thus
$$ \int^{\infty}_{R_{0}} \int_{\partial B_{R}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,ds\,dR\geq\sum^{\infty}_{n=2} \int^{R_{n}}_{R_{n-1}} \int_{\partial B_{R}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,ds\,dR=\infty. $$
(2.56)
This contradicts the fact
$$ \int_{{\mathbb {R}}^{N}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,dx= \int^{\infty}_{0} \int_{\partial B_{R}} \bigl(\vert F\vert +\vert \nabla u\vert ^{p} \bigr)\,ds\,dR< \infty. $$
(2.57)
Therefore, (2.52) is true. Thus, letting \(R\to\infty\) in (2.51), we have
$$ \frac{N-p}{p} \int_{\mathbb{R}^{N}}\vert \nabla u\vert ^{p}\,dx+ \int_{\mathbb{R}^{N}} \bigl(NF(x,u)+F_{1}(x,u) \bigr)\,dx=0. $$
(2.58)
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
$$ {\widehat{M}} \bigl(\Vert u\Vert \bigr) \bigl(- \Delta_{p} u+\vert u\vert ^{p-2}u \bigr)=\mu \bigl(f(u)+h(x) \bigr), $$
(2.59)
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:
$$ \begin{aligned}[b] &{\widehat{M}} \bigl(\Vert u\Vert \bigr) \biggl( \frac{N-p}{p} \int_{\mathbb{R}^{N}} \vert \nabla u\vert ^{p}\,dx + \frac{N}{p} \int_{\mathbb{R}^{N}}\vert u\vert ^{p}\,dx \biggr) \\ &\quad = N\mu \int_{\mathbb{R}^{N}} \bigl(F(u)+hu \bigr)\,dx + \mu \int_{\mathbb{R}^{N}}\nabla h\cdot xu\,dx. \end{aligned} $$
(2.60)

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
$$ g(x,u)=\frac{\mu(f(u)+h(x))}{{\widehat{M}}(\Vert u\Vert )}-\vert u\vert ^{p-2}u. $$
(2.61)
Then \(u\in X\) is also a solution of
$$ -\Delta_{p} u=g(x,u). $$
(2.62)
By Lemma 2.8,
$$ \frac{N-p}{p} \int_{\mathbb{R}^{N}}\vert \nabla u\vert ^{p}\,dx= \int_{\mathbb{R}^{N}} \bigl(NG(u)+ G_{1}(x, u) \bigr)\,dx, $$
(2.63)
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
$$ \begin{aligned}[b] & {\widehat{M}} \bigl( \Vert u\Vert \bigr) \biggl( \frac{N}{p}\Vert u\Vert ^{p}+ \int_{{\mathbb {R}}^{N}}\vert \nabla u\vert ^{p}\,dx \biggr) \\ &= N \mu_{n} \int_{\mathbb{R}^{N}} \bigl(F(u_{n}) + hu_{n} \bigr)\,dx + \mu_{n} \int_{\mathbb{R}^{N}} \nabla h\cdot xu_{n}\,dx. \end{aligned} $$
(2.64)
Using \(I^{T}_{\mu_{n}}(u_{n})=c_{\mu_{n}}\), we have
$$ \frac{aN}{p}\Vert u_{n}\Vert ^{p}+ \frac{\lambda N}{p(m+1)}\eta_{T}(u_{n})\Vert u_{n} \Vert ^{p(m+1)}=N\mu_{n} \int_{\mathbb{R}^{N}} \bigl(F(u_{n})+h u_{n} \bigr)\,dx+Nc_{\mu_{n}}. $$
(2.65)
Therefore, by (2.64), (2.65) and \(a>2^{m+1}(\frac {m+3}{m+1})\lambda T^{pm}\) we deduce that
$$\begin{aligned}& \frac{a}{2} \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p}\,dx \\& \quad \leq{\widehat{M}} \bigl( \Vert u_{n}\Vert \bigr) \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p}\,dx \\& \quad =N c_{\mu_{n}}+N \biggl(\widehat{M} \bigl( \Vert u_{n} \Vert \bigr)-\frac{a}{p} \biggr)\Vert u\Vert ^{p}- \frac{\lambda N}{p(m+1)}\eta_{T}(u_{n})\Vert u_{n} \Vert ^{p(m+1)} - \mu_{n} \int_{\mathbb{R}^{N}}\nabla hx\cdot u_{n}\,dx \\& \quad \leq N c_{\mu_{n}} + \frac{\lambda Nm}{p(m+1)}\eta_{T}(u_{n}) \Vert u_{n}\Vert ^{p(m+1)} + \frac{\lambda N}{p(m+1)T^{p}} \eta'_{T}(u_{n})\Vert u_{n}\Vert ^{p(m+2)} \\& \quad{} - \mu_{n} \int_{\mathbb{R}^{N}}\nabla hx\cdot u_{n}\,dx. \end{aligned}$$
(2.66)
By the min-max definition of the mountain pass level, Lemma 2.5, and (2.35) we have
$$ \begin{aligned}[b] c_{\mu_{n}}&\leq\max_{t} I^{T}_{\mu_{n}}(t \beta) \\ &\leq\max_{t} \biggl\{ \biggl(\frac{a}{p}- \frac{C_{3}}{2} \int_{B_{R}}\vert \beta \vert ^{p}\,dx \biggr)t^{p}+C_{5} \biggr\} +\max_{t} \frac{\lambda}{p(m+1)} \psi \biggl(\frac{t^{p}}{T^{p}} \biggr)t^{p(m+1)} \\ &\leq\frac{\lambda2^{m+1}}{p(m+1)}T^{p(m+1)}+C_{5}. \end{aligned} $$
(2.67)
Using \((H)\) and the Young equality, we have
$$ \begin{aligned}[b] \int_{\mathbb{R}^{N}}\nabla h\cdot xu_{n} \,dx &\leq\frac{ 1}{p'} \int_{\mathbb{R}^{N}}\vert \xi \vert ^{p'}\,dx+ \frac{1}{p} \int_{\mathbb{R}^{N}}\vert \xi \vert ^{p'}\vert u_{n}\vert ^{p}\,dx \\ & \leq\frac{1}{p} \int_{\mathbb{R}^{N}}\vert \xi \vert ^{p'}\vert u_{n}\vert ^{p}\,dx+C_{6}. \end{aligned} $$
(2.68)
We can easily calculate that
$$ \eta_{T}(u_{n})\Vert u_{n} \Vert ^{p(m+1)}\leq2^{m+1}T^{p(m+1)}, \quad\quad \eta'(u_{n}) \Vert u_{n}\Vert ^{p(m+2)}\leq2^{m+2}T^{p(m+2)}. $$
(2.69)
Combining the above estimates, we see that
$$ \frac{a}{2} \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p}\,dx \leq\frac{\lambda N(m+5)}{p(m+1)}2^{m+1}T^{p(m+1)}+ \frac{1}{p} \int_{\mathbb{R}^{N}}\vert \xi \vert ^{p'}\vert u_{n}\vert ^{p}\,dx+C_{7}. $$
(2.70)
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
$$ \begin{aligned}[b] & \widehat{M} \bigl( \bigl\Vert \xi^{p'}u_{n} \bigr\Vert \bigr) \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla \bigl(\xi^{p'}u_{n} \bigr) + \vert u_{n}\vert ^{p-2}u \bigl(\xi^{p'}u_{n} \bigr)\,dx \\ &\quad = \mu_{n} \int_{\mathbb{R}^{N}} \bigl(f(u_{n})+h \bigr) \xi^{p'}u_{n}\,dx. \end{aligned} $$
(2.71)
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
$$ ({3a}/2) \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla \bigl(\xi^{p'}u_{n} \bigr) +\vert u_{n}\vert ^{p}\xi^{p'}\,dx\geq(1/ {2}) \int_{\mathbb{R}^{N}}f(u_{n})u_{n} \xi^{p'}\,dx. $$
(2.72)
From (2.70) by the Hölder inequality we deduce that
$$ \begin{aligned}[b] &{3a} \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla \bigl(\xi^{p'}u_{n} \bigr)\,dx \\ &\quad \leq{3a} \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p-2} \nabla u_{n} \bigl(p'\xi^{p'-1}u_{n} \nabla\xi+\xi^{p'}\nabla u_{n} \bigr)\,dx \\ &\quad \leq3 \bigl(\Vert \xi \Vert ^{\infty}_{p'}+\Vert \nabla \xi \Vert ^{\infty}_{p'} \bigr) \biggl({a} \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p}\,dx \biggr)+{3a} {(p-1)^{-1}} \int_{\mathbb{R}^{N}}\xi^{p'}\vert u_{n}\vert ^{p}\,dx \\ &\quad \leq C\lambda T^{p(m+1)}+C \int_{\mathbb{R}^{N}}\xi^{p'}\vert u_{n}\vert ^{p}\,dx+C, \end{aligned} $$
(2.73)
where C is a constant independent of λ and T.
By \((F_{3})\), for any \(L>0\), there exists \(C(L)>0\) such that
$$ f(s)s\geq Ls^{p}-C(L) \quad\text{for all } s>0. $$
(2.74)
Combining (2.72)-(2.74), we get
$$ \biggl(\frac{1}{2}L-C \biggr) \int_{\mathbb{R}^{N}}\xi^{p'}\vert u_{n}\vert ^{p}\,dx\leq C\lambda T^{p(m+1)}+C. $$
(2.75)
For \(L>0\) large enough, we obtain
$$ \int_{\mathbb{R}^{N}}\xi^{p'}\vert u_{n}\vert ^{p}\,dx\leq C\lambda T^{p(m+1)}+C. $$
(2.76)
It follows from (2.70) and (2.76) that
$$ \int_{\mathbb{R}^{N}}\vert \nabla u_{n}\vert ^{p}\,dx \leq C\lambda T^{p(m+1)}+C. $$
(2.77)
On the other hand,
$$ \begin{aligned}[b] &a\Vert u_{n}\Vert ^{p} + \eta_{T}(u_{n})\Vert u_{n} \Vert ^{p(m+1)} + \frac{\lambda}{m+1}\eta'_{T}(u_{n}) \Vert u_{n}\Vert ^{p(m+2)} \\ &\quad = \mu_{n} \int_{\mathbb{R}^{N}} \bigl(f(u_{n})u_{n}+hu_{n} \bigr)\,dx \\ &\quad \leq\varepsilon \Vert u_{n}\Vert ^{p} + C_{\varepsilon} \Vert u_{n}\Vert _{p^{*}}^{p^{*}} + \frac{1}{p'} \Vert h\Vert ^{p'}_{p'} + \frac{1}{p}\Vert u \Vert ^{p}. \end{aligned} $$
(2.78)
By (2.77) and (2.78) we have
$$ \begin{aligned}[b] (a-\varepsilon-{1}/{p})\Vert u_{n}\Vert ^{p}&\leq C_{\varepsilon} \Vert u_{n}\Vert ^{p*}_{p*}-{\lambda}/ \bigl({(m+1)T^{p}} \bigr) \eta'_{T}(u_{n})\Vert u_{n}\Vert ^{p(m+2)}+C \\ &\leq C\Vert \nabla u_{n}\Vert ^{p*}_{p}+{ \lambda2^{m+2}} {(m+1)}^{-1}T^{p(m+1)}+C \\ &\leq C\lambda T^{p^{*}(m+1)}+C\lambda T^{p(m+1)}+C. \end{aligned} $$
(2.79)
Suppose that \(\Vert u_{n}\Vert >T\) for \(n\in {\mathbb {N}}\) and T large enough. Then
$$ T^{p}< \Vert u_{n}\Vert ^{p}\leq C\lambda T^{p^{*}(m+1)}+C\lambda T^{p(m+1)}+C, $$
(2.80)
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
$$ I(u_{n})=I^{T}_{\mu_{n}}(u_{n})+( \mu_{n}-1) \int_{\mathbb{R}^{N}} \bigl(F(u_{n})+hu_{n} \bigr)\,dx. $$
(2.81)
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,
$$ I'(u_{n})v= \bigl(I^{T}_{\mu_{n}} \bigr)'(u_{n},v)+(\mu_{n}-1) \int_{\mathbb{R}^{N}} \bigl(f(u_{n})+h(u_{n}) \bigr)v\,dx, \quad v\in X. $$
(2.82)
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
$$ I_{\lambda}(u)=\frac{a}{p}\Vert u\Vert ^{p}+ \frac{\lambda}{p(m+1)}\Vert u\Vert ^{p(m+1)}-\frac{1}{q}\Vert u \Vert ^{q}_{q}, $$
(3.1)
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
$$ \begin{aligned}[b] I_{\lambda}'(u)v&= \bigl(a+\lambda \Vert u\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u\vert ^{p-2} \nabla u \nabla v+\vert u\vert ^{p-2}uv \bigr)\,dx \\ &\quad{} - \int_{\mathbb{R}^{N}} \vert u\vert ^{q-2}uv\,dx,\quad \forall v \in E. \end{aligned} $$
(3.2)
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,
$$ a\Vert u\Vert ^{p}+\lambda \Vert u\Vert ^{p(m+1)}=\Vert u\Vert _{q}^{q}. $$
(3.3)
Since \(p< q<\min\{p(m+1),p^{*}\}\), applying the Young inequality and (1.9), we deduce that
$$ a\Vert u\Vert ^{p}+\lambda \Vert u\Vert ^{p(m+1)}=\Vert u\Vert _{q}^{q}\leq S_{q}^{q}\Vert u\Vert ^{q}_{E}\leq a \Vert u\Vert ^{p}_{E}+\lambda_{1}\Vert u \Vert ^{p(m+1)}_{E}, $$
(3.4)
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 \(\lambda>0\),
$$ \begin{aligned}[b] I_{\lambda}(u)&=\frac{a}{p}\Vert u \Vert ^{p}+\frac{\lambda}{p(m+1)}\Vert u\Vert ^{p(m+1)}- \frac{1}{q}\Vert u\Vert ^{q}_{q} \\ &\geq\frac{a}{p}\Vert u\Vert ^{p}+\frac{\lambda}{2p(m+1)} \Vert u\Vert ^{p(m+1)}+\frac{\lambda}{2p(m+1)}\Vert u\Vert ^{p(m+1)}- \frac{S^{q}_{q}}{q}\Vert u\Vert ^{q}. \end{aligned} $$
(3.5)
Since \(q< p(m+1)\), there exists \(C_{1}=C_{1}(\lambda,q,m,S_{q})\) such that
$$ \frac{S^{q}_{q}}{q}\Vert u\Vert ^{q}\leq\frac{\lambda}{2p(m+1)}\Vert u \Vert ^{p(m+1)}+C_{1}. $$
(3.6)
It follows that
$$ I_{\lambda}(u)\geq\frac{a}{p}\Vert u\Vert ^{p}+ \frac{\lambda}{2p(m+1)}\Vert u\Vert ^{p(m+1)}-C_{1}. $$
(3.7)
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
$$ \Vert u\Vert _{q}=\lim_{n\to\infty} \Vert u_{n}\Vert _{q} \quad\text{and} \quad \Vert u\Vert \leq \liminf_{n\to\infty} \Vert u_{n}\Vert $$
(3.8)
by the weak lower semicontinuity of the norm \(\Vert \cdot \Vert \). Thus,
$$\begin{aligned} \begin{aligned}[b] I_{\lambda}(u)&=\frac{a}{p}\Vert u \Vert ^{p}+\frac{\lambda}{p(m+1)}\Vert u\Vert ^{p(m+1)}- \frac{1}{q}\Vert u\Vert ^{q}_{q} \\ &\leq\liminf_{n\to\infty} \biggl(\frac{a}{p}\Vert u_{n}\Vert ^{p}+\frac{\lambda}{p(m+1)}\Vert u_{n} \Vert ^{p(m+1)} \biggr)-\frac{1}{q}\lim_{n\to\infty} \Vert u_{n}\Vert ^{q}_{q} \\ &\leq\liminf_{n\to\infty} \biggl(\frac{a}{p}\Vert u_{n}\Vert ^{p}+\frac{\lambda}{p(m+1)}\Vert u_{n} \Vert ^{p(m+1)}- \frac{1}{q}\Vert u_{n} \Vert ^{q}_{q} \biggr)=\liminf_{n\to\infty}I_{\lambda}(u_{n}). \end{aligned} \end{aligned}$$
(3.9)
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
$$ \inf_{u\in E}I_{\lambda}(u)< 0, \quad \mbox{for any } \lambda\in(0, \lambda_{0}). $$
(3.10)
Choose \(u_{0}\in C^{\infty}_{0}({\mathbb {R}}^{N})\), \(u_{0}\not\equiv0\), such that \(\Vert u_{0}\Vert _{E}=1\). Denote
$$ I_{\lambda}(tu_{0})=t^{p}s(t), \quad\quad s(t)=B_{1}+ \lambda B_{2}t^{pm}-B_{3}t^{q-p}, \quad t\geq0, $$
(3.11)
where
$$B_{1}=\frac{a}{p}, \quad\quad B_{2}=\frac{1}{p(m+1)}>0,\quad\quad B_{3}=\frac{1}{q} \int_{{\mathbb {R}}^{N}}\vert u_{0}\vert ^{q}\,dx>0. $$
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
$$\begin{aligned}& \lambda^{**}=\sup \bigl\{ \lambda>0, \mbox{problem (1.13) admits a nontrival weak solution} \bigr\} , \\& \lambda^{*}=\inf \bigl\{ \lambda>0, \mbox{problem (1.13) does not admit any nontrival weak solution} \bigr\} . \end{aligned}$$
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
$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u_{m}u}\vert ^{p}+\vert u_{\mu} \vert ^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(-\Delta_{p}u_{\mu}+\vert u_{\mu} \vert ^{p-2}u_{\mu}\bigr)\leq \vert u_{\mu} \vert ^{q-2}u_{\mu}. $$
(3.12)
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
$$ \inf \biggl\{ \frac{a}{p}\Vert \omega \Vert ^{p}+ \frac{\lambda}{p(m+1)}\Vert \omega \Vert ^{p(m+1)}-\frac{1}{q}\Vert \omega \Vert ^{q}_{q}:\omega\in E,\Vert \omega \Vert ^{q}_{q}=q \mbox{ and } \omega\geq u_{\mu}\biggr\} . $$
(3.13)
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 \(v\in X\),
$$ \bigl(a+\lambda_{n}\Vert u_{n}\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n} \nabla v+\vert u_{n}\vert ^{p-2}u_{n}v \bigr) \,dx= \int_{\mathbb{R}^{N}}\vert u_{n}\vert ^{q-2}u_{n}v \,dx. $$
(3.14)
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
$$ 0= \bigl(I_{\lambda_{n}}'(u_{n})-I_{\lambda^{*}}' \bigl(u^{*} \bigr) \bigr) (u_{n}-u)=X_{n}+Y_{n}-Z_{n}, $$
(3.15)
where
$$\begin{aligned}& X_{n}= \bigl(a+\lambda_{n}\Vert u_{n}\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u_{n}}\vert ^{p-2}\nabla{u_{n}}- \bigl\vert \nabla{u^{*}} \bigr\vert ^{p-2}\nabla{u^{*}} \bigr)\nabla \bigl(u_{n}-u^{*} \bigr)\,dx \\& \hphantom{X_{n}=} {}+ \bigl(\vert u_{n}\vert ^{p-2}u_{n}- \bigl\vert u^{*} \bigr\vert ^{p-2}u^{*} \bigr) \bigl(u_{n}-u^{*} \bigr)\,dx; \\& Y_{n}= \bigl(\lambda_{n}\Vert u_{n}\Vert ^{pm} - \lambda^{*} \bigl\Vert u^{*} \bigr\Vert ^{pm} \bigr) \int_{\mathbb{R}^{N}} \bigl\vert \nabla{u^{*}} \bigr\vert ^{p-2} \nabla{u^{*}}\nabla \bigl(u_{n}-u^{*} \bigr) \\& \hphantom{Y_{n}=} {}+ \bigl\vert u^{*} \bigr\vert ^{p-2}u^{*} \bigl(u_{n}-u^{*} \bigr)\,dx; \\& Z_{n}= \int_{\mathbb{R}^{N}} \bigl(\vert u_{n}\vert ^{q-2}u_{n}- \bigl\vert u^{*} \bigr\vert ^{q-2}u^{*} \bigr) \bigl(u_{n}-u^{*} \bigr)\,dx. \end{aligned}$$
By the Hölder inequality and compact embedding \(u_{n}\to u\) in \(L^{q}(\mathbb{R}^{N},H)\) we have
$$ \begin{aligned}[b] \vert X_{n}\vert &= \biggl\vert \int_{\mathbb{R}^{N}} \bigl(\vert u_{n}\vert ^{q-2}u_{n}- \bigl\vert u^{*} \bigr\vert ^{q-2}u^{*} \bigr) \bigl(u_{n}-u^{*} \bigr)\,dx \biggr\vert \\ &\leq \int_{\mathbb{R}^{N}} \bigl(\vert u_{n}\vert ^{q-1}+ \bigl\vert u^{*} \bigr\vert ^{q-1} \bigr) \bigl\vert u_{n}-u^{*} \bigr\vert \,dx \\ &\leq C \bigl(\Vert u_{n}\Vert ^{q-1}+ \bigl\Vert u^{*} \bigr\Vert ^{q-1} \bigr) \bigl\Vert u_{n}-u^{*} \bigr\Vert _{q}\to{0} \quad \mbox{as } n\to\infty. \end{aligned} $$
(3.16)
Next, consider the functional \(j:X\to\mathbb{R}\) defined by
$$ j(\omega)= \int_{\mathbb{R}^{N}} \bigl\vert \nabla{u^{*}} \bigr\vert ^{p-2} \nabla{u^{*}}\nabla{\omega}+ \bigl\vert u^{*} \bigr\vert ^{p-2}u^{*}\omega \,dx. $$
(3.17)
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
$$ \vert Y_{n}\vert \le \bigl(\Vert u_{n} \Vert ^{pm}+ \bigl\Vert u^{*} \bigr\Vert ^{pm} \bigr) \bigl\vert g \bigl(u_{n}-u^{*} \bigr) \bigr\vert \to{0} \quad \mbox{as } n\to \infty. $$
(3.18)
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. □

Declarations

Acknowledgements

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.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Yancheng Normal University, Yancheng, P.R. China

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