Citation: Ya-Lei Li, Da-Bin Wang, Jin-Long Zhang. Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity[J]. AIMS Mathematics, 2020, 5(3): 2100-2112. doi: 10.3934/math.2020139
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In this article, we are consider the existence of the ground state sign-changing solution for the following p-Laplacian Kirchhoff-type equation
{−(a+b∫Ω|∇u|pdx)Δpu=|u|q−2ulnu2,x∈Ωu=0, x∈∂Ω, | (1.1) |
where Ω⊂RN is a smooth bounded domain, a,b>0, 4≤2p<q<p∗ and N>p, Δp denote the p-Laplacian operator defined by Δpu=div(|∇u|p−2⋅∇u).
Problem (1.1) stem from following Kirchhoff equations
−(a+b∫Ω|∇u|2dx)Δu=f(x,u), | (1.2) |
where Ω⊂RN is a bounded domain or Ω=RN, a>0,b>0 and u satisfies some boundary conditions.
Problem (1.2) is related to the following stationary analogue of the equation of Kirchhoff type
utt−(a+b∫Ω|∇u|2dx)Δu=f(x,u), | (1.3) |
which was introduced by Kirchhoff [1] as a generalization of the well-known D'Alembert wave equation
ρ∂2u∂t2−(p0h+E2L∫L0|∂u∂x|2dx)∂2u∂x2=f(x,u), | (1.4) |
for free vibration of elastic strings.
After the pioneer work of Lions [2], where a functional analysis approach was proposed to (1.3) with Dirichlet boundary condition, a lot of interesting results for (1.2) or similar problems are obtained in last decades.
Recently, many authors pay their attentions to find sign-changing solutions to problem (1.2) or similar equations, and indeed some interesting results were obtained, see for examples, [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein.
On the other hand, the problem (1.1) derive from the following Logarithmic Schrödinger equation
{−Δu+V(x)u=|u|q−2ulnu2,x∈Ωu∈H10(Ω). | (1.5) |
Recently, there are many results about Logarithmic Schrödinger equation like (1.5), see [26,29,30,31,32,33,34,35,36,37] and references therein. Moreover, some scholars considered sign-changing solutions to Logarithmic Schrödinger equation like (1.5) [26,33].
Motivated by the works mentioned above, especially by [9,10,26], in the present paper, we consider the existence of ground state sign-changing solutions for problem (1.1).
Denote W1,p0(Ω) the usual Sobolev space equipped with the norm
‖u‖p=∫Ω|∇u|pdx. |
The usual Lp(Ω) norm is denote by |u|pp=∫Ω|u|pdx. And we define the energy functional of problem (1.1) as follow:
Φ(u)=ap∫Ω|∇u|pdx+b2p(∫Ω|∇u|pdx)2+2q2∫Ω|u|qdx−1q∫Ω|u|qlnu2dx, |
for any u∈W1,p0(Ω).
Moreover, under our conditions, Φ(u) belongs to C1, and the Fréchet derivative of Φ is
⟨Φ′(u),v⟩=a∫Ω|∇u|p−2∇u⋅∇vdx+b(∫Ω|∇u|pdx)(∫Ω|∇u|p−2∇u⋅∇vdx)−∫Ω|u|q−2uvlnu2dx, |
for any u,v∈W1,p0(Ω).
The solution of problem (1.1) is the critical point of the functional Φ(u). Furthermore, if u∈W1,p0(Ω) is a solution of problem (1.1) and u±≠0, then u is a sign-changing solution of problem (1.1), where
u+=max{u(x),0},u−=min{u(x),0}. |
It is noticed that
Φ(u)=Φ(u+)+Φ(u+)+bp‖u+‖p‖u−‖p, if u±≢ |
The main results can be stated as follows.
Theorem 1.1. The problem (1.1) has a sign-changing u_{0}\in\mathcal{M} with precisely two nodal domains such that \Phi(u_{0}) = inf_{\mathcal{M}}\Phi: = m, where
\mathcal{M} = \{u\in {W}^{1, p}_{0}(\Omega), u^{\pm}\neq 0, ~and ~\langle \Phi'(u), u^{+} \rangle = \langle \Phi'(u), u^{-} \rangle = 0\}. |
Theorem 1.2. The problem (1.1) has a solutions u_{0}\in\mathcal{N} such that \Phi(u_{0}) = inf_{\mathcal{N}}\Phi: = c, where \mathcal{N} = \{u\in {W}^{1, p}_{0}(\Omega), u\neq 0, and \langle \Phi'(u), u\rangle = 0\} . Moreover, m\geq2c .
Remark 1.1. The Kirchhoff function M(t) = a+bt^{n} (n > 0, a > 0, b\geq0) can be regarded as the special case of the function M:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+} satisfying the following conditions:
(M1) M\in C(\mathbb{R}^{+}) satisfies \inf_{t\in\mathbb{R}^{+}}M(t)\geq m_{0} > 0 , where m_{0} is a constant;
(M2) There exists \theta\geq1 such that \theta \mathbb{M} = \theta\int_{0}^{t}M(\tau)d\tau\geq M(t)t for any t\geq0.
In this paper, we consider this problem under condition M(t) = a+bt. From details in proof of Lemma 2.1 (see later), we can not obtain result similar as Lemma 2.1 for the case of Kirchhoff equations of the general forms. However, Lemma 2.1 play an important role in proof of main resuls. So, for the case of Kirchhoff equations of the general forms, the methods in this paper seems not valid.
Lemma 2.1. For all u\in {W}^{1, p}_{0}(\Omega) and s, t\geq 0 there holds
\begin{align} \Phi(u)&\geq \Phi(su^{+}+tu^{-})+\frac{1-s^{q}}{q}\langle \Phi'(u), u^{+}\rangle+\frac{1-t^{q}}{q}\langle \Phi'(u), u^{-}\rangle+a(\frac{1-s^{p}}{p}-\frac{1-s^{q}}{q})\|u^{+}\|^p\\ &+a(\frac{1-t^{p}}{p}-\frac{1-t^{q}}{q})\|u^{-}\|^p+b[(\frac{1-s^{2p}}{2p}-\frac{1-s^{q}}{q})\|u^{+}\|^{2p}+(\frac{1-t^{2p}}{2p}-\frac{1-t^{q}}{q})\| u^{-}\|^{2p}]\\ &+b[\frac{s^{2p}+t^{2p}-2s^{p}t^{p}}{2p}]\|u^{+}\|^{p}\|u^{-}\|^{p}. \end{align} | (2.1) |
Proof. Since (2.1) holds when u = 0 , in the following, we always assume that u\neq0 . It is easy to see that
\begin{align} 2(1-\tau^{q})+q\tau^{q}\ln\tau^{2} \gt 0, \forall\tau\in(0, 1)\cup(1, \infty). \end{align} | (2.2) |
Let \Omega^{+} = \{x\in\Omega:u(x) > 0\} and \Omega^{-} = \{x\in\Omega:u(x) < 0\} . For any u\in {W}^{1, p}_{0}(\Omega)\setminus\{0\} , we have that
\begin{align} \int_{\Omega}&|su^{+}+tu^{-}|^{q}\ln(su^{+}+tu^{-})^{2}dx\\ & = \int_{\Omega^{+}}|su^{+}+tu^{-}|^{q}\ln(su^{+}+tu^{-})^{2}dx+\int_{\Omega^{-}}|su^{+}+tu^{-}|^{q}\ln(su^{+}+tu^{-})^{2}dx\\ & = \int_{\Omega^{+}}|su^{+}|^{q}\ln(su^{+})^{2}dx+\int_{\Omega^{-}}|su^{-}|^{q}\ln(su^{-})^{2}dx\\ & = \int_{\Omega}[|su^{+}|^{q}\ln(su^{+})^{2}]+|tu^{-}|^{q}\ln(tu^{-})^{2}dx\\ & = \int_{\Omega}[|su^{+}|^{q}(\ln s^{2}+\ln(u^{+})^{2})+|tu^{-}|^{q}(\ln t^{2}+\ln(u^{-})^{2})]dx. \end{align} | (2.3) |
Combining (2.2) with (2.3), we obtain
\begin{align} \Phi(u)- &\Phi(su^{+}+tu^{-}) = \frac{a}{p}(\|u\|^{p}-\|su^{+}+tu^{-}\|^{p})+\frac{b}{2p}(\|u\|^{2p}-\|su^{+}+tu^{-}\|^{2p})\\ &+\frac{2}{q^{2}}\int_{\Omega}|u|^{q}-|su^{+}+tu^{-}|^{q}dx-\frac{1}{q}\int_{\Omega}|u|^{q}\ln u^{2}-|su^{+}+tu^{-}|^{q}\ln(su^{+}+tu^{-})^{2}dx\\ & = \frac{a}{p}(\|u\|^{p}-s^{p}\|u^{+}\|^{p}-t^{p}\|u^{-}\|^{p})+\frac{b}{2p}(\|u\|^{2p}-s^{2p}\|u^{+}\|^{2p}-t^{2p}\|u^{-}\|^{2p})\\ &-2s^{p}t^{p}\|u^{+}\|^{p}\|u^{-}\|^{p})+\frac{2}{q^{2}}\int_{\Omega}|u^{+}|^{q}+|u^{-}|^{q}-s^{q}|u^{+}|^{q}-t^{q}|u^{-}|^{q}dx -\frac{1}{q}\int_{\Omega}(|u^{+}|^{q}\ln(u^{+})^{2}\\ &+|u^{-}|^{q}\ln(u^{-})^{2}-|su^{+}|^{q}\ln s^{2}-|su^{+}|^{q}\ln (u^{+})^{2}-|tu^{-}|^{q}\ln t^{2} -|tu^{-}|^{q}\ln (u^{-})^{2})dx\\ & = \frac{a}{p}(1-s^{p})\| u^{+}\|^{p}+\frac{a}{p}(1-t^{p})\| u^{-}\|^{p}+\frac{b}{2p}(1-s^{2p})\|u^{+}\|^{2p}\\ &+\frac{b}{2p}(1-t^{2p})\|u^{-}\|^{2p}+\frac{b}{p}(1-s^{p}t^{p})\|u^{+}\|^{p}\| u^{-}\|^{p}+\frac{2}{q^{2}}\int_{\Omega}(|u^{+}|^{q}-|su^{+}|^{q}+|u^{-}|^{q}-|tu^{-}|^{q})dx\\ &-\frac{1}{q}\int_{\Omega}(|u^{+}|^{q}\ln(u^{+})^{2}-|su^{+}|^{q}\ln(su^{+})^{2})dx-\frac{1}{q}\int_{\Omega}(|u^{-}|^{q}\ln(u^{-})^{2}-|tu^{-}|^{q}\ln(tu^{-})^{2})dx\\ & = \frac{1-s^{q}}{q}\langle\Phi'(u), u^{+}\rangle+\frac{1-t^{q}}{q}\langle \Phi'(u), u^{-}\rangle+a(\frac{1-s^{p}}{p}-\frac{1-s^{q}}{q})\|u^{+}\|^p\\ &+a(\frac{1-t^{p}}{p}-\frac{1-t^{q}}{q})\|u^{-}\|^p+b((\frac{1-s^{2p}}{2p}-\frac{1-s^{q}}{q})\|u^{+}\|^{2p}+(\frac{1-t^{2p}}{2p}-\frac{1-t^{q}}{q})\| u^{-}\|^{2p})\\ &+b(\frac{1-s^{p}t^{p}}{p}-\frac{1-s^{q}}{q}-\frac{1-t^{q}}{q})\|u^{+}\|^{p}\|u^{-}\|^{p}+\frac{2(1-s^{q})+qs^{q}\ln s^{2}}{q^{2}}|u^{+}|_{q}^{q}\\ &+\frac{2(1-t^{q})+qt^{q}\ln t^{2}}{q^{2}}|u^{-}|_{q}^{q}\\ &\geq\frac{1-s^{q}}{q}\langle \Phi'(u), u^{+}\rangle+\frac{1-t^{q}}{q}\langle \Phi'(u), u^{-}\rangle+a(\frac{1-s^{p}}{p}-\frac{1-s^{q}}{q})\|u^{+}\|^p\\ &+a(\frac{1-t^{p}}{p}-\frac{1-t^{q}}{q})\|u^{-}\|^p+b((\frac{1-s^{2p}}{2p}-\frac{1-s^{q}}{q})\|u^{+}\|^{2p}+(\frac{1-t^{2p}}{2p}-\frac{1-t^{q}}{q})\| u^{-}\|^{2p})\\ &+b[\frac{s^{2p}+t^{2p}-2s^{p}t^{p}}{2p}]\|u^{+}\|^{p}\|u^{-}\|^{p}, \end{align} | (2.4) |
which implies that (2.2) holds.
According to Lemma 2.1, we can obtain the following corollaries.
Corollary 2.1. For all u\in {W}^{1, p}_{0}(\Omega) and t\geq 0 , we have that
\begin{align} \Phi(u)&\geq\Phi(tu)+\frac{1-t^{q}}{q}\langle \Phi'(u), u\rangle+a(\frac{1-t^{p}}{p}-\frac{1-t^{q}}{q})\|u\|^p. \end{align} | (2.5) |
Corollary 2.2. For any u\in \mathcal{M} , we have that
\Phi(u) = \max\limits_{s, t\geq0}\Phi(su^{+}+tu^{-}). |
Corollary 2.3. For any u\in \mathcal{N} , we have that
\Phi(u) = \max\limits_{t\geq0}\Phi(tu). |
Lemma 2.2. For any u\in W^{1, p}_{0}(\Omega) with u^{\pm}\neq 0 , there exists an unique pair (s_{u}, t_{u}) of positive numbers such that s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M} .
Proof. Firstly, for any u\in W^{1, p}_{0}(\Omega) with u^{\pm}\neq 0 , we prove the existence of (s_{u}, t_{u}) . Let
\begin{align} G(s, t)& = \langle \Phi'(su^{+}+tu^{-}), su^{+} \rangle = as^{p}\|u^{+}\|^{p}+bs^{2p}\|u^{+}\|^{2p}\\ &+bs^{p}t^{p}\|u^{+}\|^{p}\|u^{-}\|^{p}-\int_{\Omega}|su^{+}|^{q}\ln(su^{+})^{2}dx, \end{align} | (2.6) |
and
\begin{align} H(s, t)& = \langle \Phi'(su^{+}+tu^{-}), tu^{-} \rangle = at^{p}\|u^{+}\|^{p}+bt^{2p}\|u^{-}\|^{2p}\\ &+bs^{p}t^{p}\|u^{+}\|^{p}\|u^{-}\|^{p}-\int_{\Omega}|tu^{-}|^{q}\ln(tu^{-})^{2}dx. \end{align} | (2.7) |
From assumptions, we have that
\lim_{t\rightarrow0}\frac{|t|^{q-1}\ln t^{2}}{|t|^{p-1}} = 0;\; \lim_{t\rightarrow {\infty}}\frac{|t|^{q-1}\ln t^{2}}{|t|^{r-1}} = 0, r\in(q, p^{*}). |
Then for any \varepsilon > 0 , there exists C_{\varepsilon} > 0 such that
\begin{align} |t|^{q-1}\ln t^{2}\leq\varepsilon|t|^{p-1}+C_{\varepsilon}|t|^{r-1}. \end{align} | (2.8) |
Since 4\leq 2p < q < p^{*} , it follows from (2.8) that
G(s, s) \gt 0 ~{\rm and}~ H(s, s) \gt 0 for s \gt 0 ~{\rm small ~enough}, \\ G(t, t) \gt 0~ {\rm and} ~H(t, t) \gt 0~ {\rm for}~ t \gt 0~{\rm large ~enough}. |
Thus, there exist 0 < \alpha < \beta such that
\begin{align} G(\alpha, \alpha) \gt 0, \; H(\alpha, \alpha) \gt 0;\; G(\beta, \beta) \lt 0, \; H(\beta, \beta) \lt 0. \end{align} | (2.9) |
Thanks to (2.6), (2.7) and (2.9), we have that
\begin{align} G(\alpha, t) \gt 0, \; G(\beta, t) \lt 0, \; \forall t\in[\alpha, \beta] \end{align} | (2.10) |
and
\begin{align} H(s, \alpha) \gt 0, \; H(s, \beta) \lt 0, \; \forall s\in[\alpha, \beta]. \end{align} | (2.11) |
So, combining (2.10), (2.11) with Miranda's Theorem [38], there exists some point (s_{u}, t_{u}) with \alpha < s_{u}, t_{u} < \beta such that
G(s_{u}, t_{u}) = H(s_{u}, t_{u}) = 0. |
That is, there exists a pair (s_{u}, t_{u}) of positive numbers such that s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M} .
Secondly, we prove that (s_{u}, t_{u}) is unique.
Arguing by contradiction, we assume that there exist two pair (s_{i}, t_{i}) , i = 1, 2 such that s_{1}u^{+}+t_{1}u^{-}\in\mathcal{M} and s_{2}u^{+}+t_{2}u^{-}\in\mathcal{M} .
According to corollary 2.2, we have that
\begin{align} \Phi(s_{1}u^{+}+t_{1}u^{-})\geq \Phi(s_{2}u^{+}+t_{2}u^{-})+as^{p}_{1}(\frac{1-(\frac{s_{2}}{s_{1}})^{p}}{p}-\frac{1-(\frac{s_{2}}{s_{1}})^{q}}{q})\|u^{+}\|^p, \end{align} | (2.12) |
\begin{align} \Phi(s_{2}u^{+}+t_{2}u^{-})\geq \Phi(s_{1}u^{+}+t_{1}u^{-})+as^{p}_{2}(\frac{1-(\frac{s_{1}}{s_{2}})^{p}}{p}-\frac{1-(\frac{s_{1}}{s_{2}})^{q}}{q})\|u^{+}\|^p. \end{align} | (2.13) |
It is noticed that
h(x) = \frac{1-s^{x}}{x} is monotonically decreasing on (0, \infty) for s \gt 0 and s\neq1. |
Therefore, by (2.12) and (2.13), we have that (s_{1}, t_{1}) = (s_{2}, t_{2}). That is, (s_{u}, t_{u}) is unique.
Lemma 2.3. For any u\in {W}^{1, p}_{0}(\Omega) with u\neq0 , there exists an unique t_{u} > 0 such that t_{u}u\in\mathcal{N} .
Proof. Since the proof is similar to that of Lemma 2.2, we omit detail here.
Through the standard discussions, we have following result.
Lemma 2.4. The following minimax characterization hold
c = \inf\limits_{u\in {W}^{1, p}_{0}(\Omega), u\neq 0} \max\limits_{t\geq 0}\Phi(tu), \; m = \inf\limits_{u\in {W}^{1, p}_{0}(\Omega), u\neq 0} \max\limits_{s, t\geq 0}\Phi(su^{+}+tu^{-}). |
Lemma 2.5. m > 0 is achieved.
Proof. For any u\subset\mathcal{M} , we have \langle \Phi'(u), u\rangle = 0 and then
\begin{align} a\|u\|^{p}&\leq a\|u\|^{p}+b\|u\|^{2p} = \int_{\Omega}|u|^{q}\ln u^{2}dx\\ &\leq C_{1}\|u\|^{q}+C_{2}\|u\|^{r}. \end{align} | (2.14) |
Since q, r > p , there exists a constant \rho > 0 such that \|u\|\geq \rho for any u\subset\mathcal{M} .
Let {u_{n}}\subset\mathcal{M} be such that \Phi(u_{n})\rightarrow m , then {u_{n}} is bounded in W^{1, p}_{0} . Thus, there exists u_{0} , in subsequence sense, such that
u_{n}^{\pm}\rightharpoonup u_{0}^{\pm}~ {\rm in} ~W^{1, p}_{0}(\Omega)\\ u_{n}^{\pm}\rightarrow u_{0}^{\pm}~ {\rm in} ~L^{s}(\Omega), p\leq s \lt p^{*} |
Since u_{n}\subset\mathcal{M} , one has \langle \Phi'(u_{n}), u_{n}^{\pm}\rangle = 0 , that is
\begin{align} a\rho^{p}\leq a\|u_{n}^{\pm}\|^{p}&\leq a\|u_{n}^{\pm}\|^{p}+b\|u_{n}^{\pm}\|^{2p}+b\|u_{n}^{+}\|^{p}\|u_{n}^{-}\|^{p} = \int_{\Omega}|u_{n}^{\pm}|^{q}\ln (u^{\pm}_{n})^{2}dx\\ &\leq \varepsilon\int_{\Omega}|u_{n}^{\pm}|^{q}dx+C_{\varepsilon}\int_{\Omega}|u_{n}^{\pm}|^{r}dx\leq C_{4}\int_{\Omega}|u_{n}^{\pm}|^{r}dx \end{align} | (2.15) |
By the compactness of the embedding W^{1, p}_{0}(\Omega)\hookrightarrow L^{r}(\Omega) , we get
C_{5}\rho^{p}\leq\int_{\Omega}|u_{0}^{\pm}|^{r}dx, |
which implies u_{0}^{\pm}\neq0 .
By the Lebesgue dominated convergence theorem and the weak semicontinuity of norm, we have
\begin{align*} \nonumber a\|u_{0}^{\pm}\|^{p}+b\|u_{0}^{+}\|^{2p}+b\|u_{0}^{+}\|^{p}\|u_{0}^{-}\|^{p}&\leq\liminf_{n\rightarrow\infty}(a\|u_{n}^{\pm}\|^{p}+b\|u_{n}^{+}\|^{2p}+b\|u_{n}^{+}\|^{p}\|u_{n}^{-}\|^{p})\\ \nonumber & = \liminf_{n\rightarrow\infty}\int_{\Omega}|u_{n}^{\pm}|^{q}\ln (u_{n}^{\pm})^{2}dx\\ & = \int_{\Omega}|u_{0}^{\pm}|^{q}\ln (u_{0}^{\pm})^{2}dx, \end{align*} |
that is,
\langle \Phi'(u_{0}), u_{0}^{+}\rangle\leq0 and \langle \Phi'(u_{0}), u_{0}^{-}\rangle\leq0. |
Since u_{0}^{\pm}\neq0 , it follows from Lemma 2.2 that there exist constants s, t > 0 such that su_{0}^{+}+tu_{0}^{-}\in\mathcal{M} .
From corollary 2.2, corollary 2.3 and the weak semicontinuity of norm, we have that
\begin{align*} \nonumber m& = \lim_{n\rightarrow\infty}[\Phi(u_{n})-\frac{1}{q}\langle \Phi'(u_{n}), u_{n}\rangle]\\ \nonumber & = \lim_{n\rightarrow\infty}[(\frac{a}{p}-\frac{a}{q})\|u_{n}\|^{p}+(\frac{b}{2p}-\frac{b}{q})\| u_{n}\|^{2p}+\frac{2}{q^{2}}|u_{n}|_{q}^{q}]\\ \nonumber &\geq(\frac{a}{p}-\frac{a}{q})\|u_{0}\|^{p}+(\frac{b}{2p}-\frac{b}{q})\|u_{0}\|^{2p}+\frac{2}{q^{2}}|u_{0}|_{q}^{q}\\ \nonumber & = \Phi(u_{0})-\frac{1}{q}\langle\Phi'(u_{0}, u_{0}\rangle\\ \nonumber &\geq \Phi(su_{0}^{+}+tu_{0}^{-})+\frac{1-s^{q}}{q}\langle \Phi'(u_{0}), u_{0}^{+}\rangle+\frac{1-t^{q}}{q}\langle \Phi'(u_{0}), u_{0}^{-}\rangle-\frac{1}{q}\langle \Phi'(u_{0}, u_{0}\rangle\\ &\geq m-\frac{s^{q}}{q}\langle \Phi'(u_{0}), u_{0}^{+}\rangle-\frac{t^{q}}{q}\langle \Phi'(u_{0}), u_{0}^{-}\rangle\\ &\geq m, \end{align*} |
which asserts
\langle \Phi'(u_{0}), u_{0}^{\pm}\rangle = 0, \; \Phi(u_{0}) = m. |
Furthermore, thanks to u_{0}^{\pm}\neq0 and (2.1), we have that
\begin{align*} \nonumber m& = \Phi(u_{0})\geq a(\frac{1}{p}-\frac{1}{q})\|u_{0}^{+}\|^p+a(\frac{1}{p}-\frac{1}{q})\|u_{0}^{-}\|^p \gt 0. \end{align*} |
\mathbf{Proof\; of\; Theorem\; 1.1:}
Proof. Firstly, thanks to Lemma 2.5, we prove the minimizer u_{0} of \inf_{\mathcal{M}}\Phi is critical point of \Phi .
Arguing by contradiction, we assume that \Phi'(u_{0})\neq0 . Then there exist \delta > 0 and \varsigma > 0 such that \|(\Phi'(u)\|\geq\varsigma , for all \|u-u_{0}\|\leq3\delta and u\in {W}^{1, p}_{0}(\Omega) .
Let D: = (\frac{1}{2}, \frac{3}{2})\times(\frac{1}{2}, \frac{3}{2}) , by Lemma 2.1, one has
\begin{align} \epsilon: = \max\limits_{(s, t)\in\partial D}\Phi(su_{0}^{+}+tu_{0}^{-}) \lt m. \end{align} | (3.1) |
For \varepsilon: = \min{(m-\epsilon)/3}, \delta\varsigma/8\} and S_{\delta}: = B(u_{0}, \delta) , according to Lemma 2.3 in [39], there exists a deformation \eta\in C([0, 1]\times {W}^{1, p}_{0}(\Omega), {W}^{1, p}_{0}(\Omega)) such that
(a) \eta(1, v) = v if v \notin\Phi^{-1}([m-2\varepsilon, m+2\varepsilon])\cap S_{2\delta} ;
(b) \eta(1, (\Phi^{m+\varepsilon}\cap S_{\delta})\subset \Phi^{m-\varepsilon} , where \Phi^{c} = \{u\in {W}^{1, p}_{0}(\Omega):\Phi(u)\leq c\} ;
(c) \Phi(\eta(1, v))\leq \Phi(v) for all v\in {W}^{1, p}_{0}(\Omega) .
By Lemma 2.1 and (c) , we have
\begin{align} \Phi(\eta(1, su_{0}^{+}+tu_{0}^{-})&\leq \Phi(su_{0}^{+}+tu_{0}^{-}) \lt \Phi(u_{0})\\ & = m, \; \; \; \forall s, t\geq0, |s-1|^{2}+|t-1|^{2}\geq\delta^{2}/\|u_{0}\|^{2}. \end{align} | (3.2) |
On the other hand, by Corollary 2.3, we can obtain that \Phi(su_{0}^{+}+tu_{0}^{-})\leq \Phi(u_{0}) = m for s, t > 0 . Then it follows from (b) that
\begin{align} \Phi(\eta(1, su_{0}^{+}+tu_{0}^{-})\leq m-\varepsilon, \; \; \; \; \forall s, t\geq0, |s-1|^{2}+|t-1|^{2} \lt \delta^{2}/\|u_{0}\|^{2}. \end{align} | (3.3) |
So, thanks to (3.2) and (3.3), one has
\begin{align} \max\limits_{(s, t)\in\bar{D}}\Phi(\eta(1, su_{0}^{+}+tu_{0}^{-}) \lt m. \end{align} | (3.4) |
Let k(s, t) = su_{0}^{+}+tu_{0}^{-} , we now prove that \eta(1, k(D))\cap \mathcal{M}\neq\varnothing . Let \gamma(s, t): = \eta(1, k(s, t))
\begin{align*} \Psi_{0}(s, t)&: = (\langle \Phi'(k(s, t)), u_{0}^{+}\rangle, \langle \Phi'(k(s, t)), u_{0}^{-}\rangle)\\ & = (\langle \Phi'(su_{0}^{+}+tu_{0}^{-}), u_{0}^{+}\rangle, \langle \Phi'(su_{0}^{+}+tu_{0}^{-}), u_{0}^{-}\rangle)\\ &: = (h_{1}(s, t), h_{2}(s, t)) \end{align*} |
and
\begin{align*} \Psi_{1}(s, t): = (\frac{1}{s}\langle \Phi'(\gamma(s, t)), (\gamma(s, t))^{^{+}}\rangle, \frac{1}{t}\langle \Phi'(\gamma(s, t)), (\gamma(s, t))^{^{-}}\rangle). \end{align*} |
Clearly, \Psi_{0} is a C^{1} functions and by a direct calculation, we have
\begin{align*} \frac{\partial h_{1}(s, t)}{\partial s}|_{(1, 1)} & = a(p-1)\|u_{0}^{+}\|^{p}+b(2p-1)\|u_{0}^{+}\|^{2p}+b(p-1)\|u_{0}^{+}\|^{p}\|u_{0}^{-}\|^{p}\\ &-(q-1)\int_{\Omega}|u_{0}^{+}|^{q}\ln(u_{0}^{+})^{2}dx-2\int_{\Omega}|u_{0}^{+}|^{q}dx \\ & = bp\|u_{0}^{+}\|^{2p}-(q-p)\int_{\Omega}|u_{0}^{+}|^{q}\ln(u_{0}^{+})^{2}dx-2\int_{\Omega}|u_{0}^{+}|^{p}dx, \end{align*} |
\begin{align*} \frac{\partial h_{1}(s, t)}{\partial t}|_{(1, 1)} = bp\|u_{0}^{+}\|^{p}\|u_{0}^{-}\|^{p}. \end{align*} |
Similarly, we have
\begin{align*} \frac{\partial h_{2}(s, t)}{\partial t}|_{(1, 1)} & = bp\|u_{0}^{-}\|^{2p}-(q-p)\int_{\Omega}|u_{0}^{-}|^{q}\ln(u_{0}^{-})^{2}dx-2\int_{\Omega}|u_{0}^{-}|^{p}dx, \end{align*} |
\begin{align*} \frac{\partial h_{2}(s, t)}{\partial s}|_{(1, 1)} = bp\|u_{0}^{+}\|^{p}\|u_{0}^{-}\|^{p}. \end{align*} |
Let
M = \left[ \begin{array}{cc} \frac{\partial h_{1}(s, t)}{\partial s}|_{(1, 1)} & \frac{\partial h_{2}(s, t)}{\partial s}|_{(1, 1)} \\ \frac{\partial h_{1}(s, t)}{\partial t}|_{(1, 1)} & \frac{\partial h_{2}(s, t)}{\partial t}|_{(1, 1)} \\ \end{array} \right], |
then we have that
\begin{align*} \det M& = \frac{\partial h_{1}(s, t)}{\partial s}|_{(1, 1)}\times\frac{\partial h_{2}(s, t)}{\partial t}|_{(1, 1)} -\frac{\partial h_{1}(s, t)}{\partial t}|_{(1, 1)}\times\frac{\partial h_{2}(s, t)}{\partial s}|_{(1, 1)}\neq0. \end{align*} |
Therefore, by topological degree theory [40,41,42], we conclude that \Psi_{1}(s_{0}, t_{0}) = 0 for some (s_{0}, t_{0})\in D , so that \eta(1, k(s_{0}, t_{0})) = \gamma(s_{0}, t_{0})\in \mathcal{M} , which is contradicted to (3.4).
Next, we prove u_{0} has two nodal domains.
We assume that
u_{0} = u_{1}+u_{2}+u_{3} |
where
u_{1}\geq0, \; u_{2}\leq0, \; \Omega_{1}\cap\Omega_{2} = \emptyset, \; u_{1}\mid_{ \Omega\backslash \Omega_{1}\cup\Omega_{2}} = u_{2}\mid_{ \Omega\backslash \Omega_{1}\cup\Omega_{2}} = u_{3}\mid_{ \Omega_{1}\cup\Omega_{2}} = 0, |
\Omega_{1}: = \{x\in\Omega, u_{1}(x) \gt 0\}\; \text{and}\; \Omega_{2}: = \{x\in\Omega, u_{2}(x) \lt 0\} |
are two connected open subsets of \Omega .
Setting v: = u_{1}+u_{2} , we see that v^{+} = u_{1} and v^{-} = u_{2} , i.e., v^{\pm}\neq0 . According to \langle \Phi'(u_{0}), v^{\pm}\rangle = 0 , we have that
\begin{align} \langle\Phi'(v), v^{\pm}\rangle = -b\|v^{\pm}\|^{p}\|u_{3}\|^{p}. \end{align} | (3.5) |
Thanks to (2.1) and (3.5), we have that
\begin{align*} \nonumber m& = \Phi(u_{0}) = \Phi(u_{0})-\frac{1}{q}\langle \Phi'(u_{0}), u_{0}\rangle\\ \nonumber & = \Phi(v)+\Phi(u_{3})+\frac{b}{p}\|u_{3}\|^{p}\|v\|^{p}-\frac{1}{q}(\langle \Phi'(v), v\rangle+\langle \Phi'(u_{3}), u_{3}\rangle+2b\|u_{3}\|^{p}\|v\|^{p})\\ \nonumber &\geq\sup_{s, t\geq0}(\Phi(sv^{+}+tv^{-})+\frac{1-s^{q}}{q}\langle \Phi'(v), v^{+}\rangle+\frac{1-t^{q}}{q}\langle \Phi'(v), v^{-}\rangle)\\ \nonumber &+\Phi(u_{3})-\frac{1}{q}\langle \Phi'(v), v\rangle-\frac{1}{q}\langle \Phi'(u_{3}), u_{3}\rangle\\ \nonumber &\geq\sup_{s, t\geq0}(\Phi(sv^{+}+tv^{-})+\frac{b}{q}s^{p}\|v^{+}\|^{p}\|u_{3}\|^{p}+\frac{b}{q}t^{p}\|v^{-}\|^{p}\|u_{3}\|^{p})\\ \nonumber &+a(\frac{1}{p}-\frac{1}{q})\|u_{3}\|^{p}+b(\frac{1}{2p}-\frac{1}{q})\|u_{3}\|^{2p}+\frac{2}{q^{2}}|u_{3}|_{q}^{q}\\ &\geq m+a(\frac{1}{p}-\frac{1}{q})\|u_{3}\|^{p}, \end{align*} |
which implies u_{3} = 0 . That is, u_{0} has two nodal domains.
\mathbf{Proof\; of\; Theorem\; 1.2:}
Proof. Similar as the proof of Lemma 2.5, there exists v_{0}\in \mathcal{N} such that \Phi(v_{0}) = c > 0 . By arguments similar to that of Theorem 1.1, the critical points of the functional \Phi on \mathcal{N} are critical points of \Phi in W^{1, p}_{0}(\Omega) and we obtain (\Phi)'(v_{0}) = 0 . It follows from definitions of c = inf_{\mathcal{N}}\Phi and \mathcal{N} = \{u\in {W}^{1, p}_{0}(\Omega), u\neq 0, and \langle \Phi'(u), u\rangle = 0\} that v_{0} is a ground state solution of (1.1).
According to Theorem 1.1, there exists a u_{0}\in\mathcal{M} such that \Phi(u_{0}) = m, \Phi'(u_{0}) = 0 . Therefore, by Corollary 2.2 and Lemma 2.4, we have that
\begin{align} m& = \Phi(u_{0}) = \sup\limits_{s, t\geq0} \Phi(su_{0}^{+}+tu_{0}^{-})\\ & = \sup\limits_{s, t\geq0}(\Phi(su_{0}^{+})+\Phi(tu_{0}^{-})+\frac {b}{p}s^{p}t^{p}\|u_{0}^{+}\|^{p}\|u_{0}^{-}\|^{p})\\ &\geq\sup\limits_{s\geq0}\Phi(su_{0}^{+})+\sup\limits_{t\geq0}\Phi(tu_{0}^{-})\geq2c \gt 0. \end{align} |
The proof of Theorem 2.2 is completed.
In this paper, by using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains for a class of p -Laplacian Kirchhoff-type problem (special form) with logarithmic nonlinearity. However, for the case of Kirchhoff equations of the general forms, the methods in this paper seems not valid. So, we will continuous discuss sign-changing solutions for general Kirchhoff equations with logarithmic nonlinearity in the follow-up work.
This research was supported by the Natural Science Foundation of China (No.11561043, 11961043). Authors are grateful to the reviewers and editors for their suggestions and comments to improve the manuscript.
The authors declare no conflict of interest in this paper.
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