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Nonexistence of stable solutions for quasilinear Schrödinger equation
Boundary Value Problems volume 2018, Article number: 168 (2018)
Abstract
In this paper, we study the nonexistence of stable solutions for the quasilinear Schrödinger equation
where \(N\ge3\), \(q\ge5/2\) and the function \(h(x)\) is continuous and positive in \(R^{N}\). Under suitable assumptions on \(h(x)\) and q, we prove that Eq. (0.1) has no nonnegative and stable solutions.
1 Introduction and main results
In this paper we are interested in the nonexistence of stable solutions to the quasilinear Schrödinger equation
where \(N\ge3\) and \(q\ge5/2\). Equation (1.1) can be obtained as a stationary problem of the modified Schrödinger equation
where \(z: R\times R^{N} \to\mathbb{ C}\), \(W:R^{N}\to R\) is a given potential, h and l are real functions.
It is well known that the standing wave solutions of the form \(z(t,x)=\exp(-i\omega t)u(x)\) satisfy (1.2) if and only if the real function \(u(x)\) solves the equation of elliptic type
where \(V(x)=W(x)-\omega, \omega\in R\) and \(g(x,u)\equiv h(x,u^{2})u\).
Quasilinear Schrödinger equations as in (1.3) appear naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of nonlinear term l. When \(l(s)=s\), we get the superfluid film equation in plasma physics [16]:
In the case \(l(s)=(1+s)^{1/2}\), (1.3) turns into the following equation:
which models the self-channeling of a high-power ultrashort laser in matter, see [2, 10].
The existence of positive solutions for (1.5) has been extensively studied recently. In [23], the authors proved that (1.5) has a positive solution under the assumptions: \(g(x,u)=\lambda|u|^{p-1}u, \sqrt{6}-1\le p<\frac{N+2}{N-2}, \lambda>0 \) and \(0<\inf_{x\in {\mathbb {R}}^{N}}V(x)\le V(x)\le V(\infty):=\lim_{|x|\to\infty}V(x)<\infty\).
Li [20] studied the existence of quasilinear Schrödinger equations of the form
where the parameter \(\alpha\in[1,2]\) and the functions \(V(x), g(x,u)\) are 1-periodic in \(x_{j}\) for \(j=1,2,\dots, N\).
Similar works can be found in [6, 7, 11, 24, 27] and the references therein. It is noted that, in the above works, one always assumes that the potential function \(V(x)\ge0\) and \(V(x)\not\equiv 0\) in \({\mathbb {R}}^{N}\).
On the other hand, the nonexistence of solutions and either the stable or unstable solutions for Lane–Emden problems are investigated to some extent. The results can be found in [1, 3, 8, 14, 15, 18, 19, 21, 25, 26, 28], and the references therein. To the best of our knowledge, there are no results on the nonexistence of solutions for (1.1). Motivated by [4, 5, 9, 13, 17, 22], our purpose in this paper is to study the nonexistence of nonnegative and stable solutions of (1.1) under some assumptions on the weighted function \(h(x)\) and the exponent q.
Usually, we make the change of variables \(z=f^{-1}(u)\), where f is defined by
and by \(f(t)=-f(-t)\) on \((-\infty, 0]\).
Lemma 1.1
The function \(f(t)\) satisfies the following properties:
- \((f_{1})\) :
-
f is uniquely defined, odd, increasing, invertible and \(C^{\infty}\) in \(R=(-\infty, +\infty)\),
- \((f_{2})\) :
-
\(0< f'(t)\le1\), \(\forall t\in R\),
- \((f_{3})\) :
-
\(|f(t)|\le|t|\), \(\forall t\in R\),
- \((f_{4})\) :
-
\(\frac{f(t)}{t}\to1\) as \(t\to0\),
- \((f_{5})\) :
-
\(f(t)\le2 tf'(t)\le2 f(t)\), \(\forall t\in R^{+}=[0, \infty)\),
- \((f_{6})\) :
-
\(\lim_{t\to+\infty}\frac{f(t)}{t}=\sqrt{\frac{2}{3}}\).
If we take \(u=f(z)\) or \(z=f^{-1}(u)\), then (1.1) becomes the following semilinear elliptic equation:
As usual, we study the existence and the nonexistence of weak solutions of (1.1) via (1.8).
Definition 1.1
([12])
The function \(z\in C_{\mathrm{loc}}^{1,\delta}({\mathbb {R}}^{N})\) \((0<\delta<1)\) is said to be a weak solution of (1.8) if
where (and in the sequel) \(g(z)=f'(z)|f(z)|^{q-1}f(z)\). A weak solution z of (1.8) is stable if
In other words, the stability condition translates into the fact that the second variation of the energy functional is nonnegative. Thus, all the minima of the functional are stable solutions of (1.8).
Definition 1.2
A function \(u\in\mathcal{D}^{1,2}({\mathbb {R}}^{N})\) is called a weak solution of (1.1) if \(z=f^{-1}(u)\) is a weak solution of (1.8). A weak solution u of (1.1) is stable in \({\mathbb {R}}^{N}\) if \(z=f^{-1}(u)\) is a stable solution of (1.8).
So in order to prove the nonexistence of stable solutions for (1.1), it is sufficient to prove that there is no nonnegative and stable weak solution to (1.8).
The main result in this paper is as follows.
Theorem 1.2
Suppose that the positive function \(h(x)\in C^{1}_{\mathrm{loc}}({\mathbb {R}}^{N})\) is such that there exist \(a>-2\), \(a_{0}>0\) and \(R_{0}>0\) such that
Denote
Let \(q_{c}\) be the positive root of the equation \(N=X(q)\) with
and
Moreover, let \(u\in C_{\mathrm{loc}}^{1,\delta}({\mathbb {R}}^{N})\) (\(0<\delta<1\)) be a nonnegative and stable solution of (1.1). Then, we have \(u\equiv0\) in \({\mathbb {R}}^{N}\) if one of the following conditions is satisfied:
- \((A_{1})\) :
-
\(q\ge\frac{5}{2}\) and \(3\le N\le\gamma_{0}{(a)}\);
- \((A_{2})\) :
-
\(q>q_{c}\) and \(\gamma_{0}{(a)}< N<\gamma_{\infty}{(a)}\).
Remark 1.3
When \(N\in(\gamma_{0}, \gamma_{\infty})\), it is not difficult to get
The condition \(\gamma_{0}< N<\gamma_{1}\) implies that \(d\in(0,2)\) and \(q_{c}>5/2\).
2 Proof of Theorem 1.2
In order to prove the nonexistence of solution to (1.8), we use the test function method, which has been used in [5, 9] to deal with the m-Laplace equation. The proof is by contradiction which involves obtaining an a priori estimate for a solution of (1.8) by carefully choosing a special test function and then applying the scaling argument. We first establish:
Lemma 2.1
Assume \(q>1\). Let \(f(t)\) be the function defined by (1.7) and \(k>1\). Suppose \(g(t)=f'(t)|f(t)|^{q-1}f(t)\) and
Then there exist \(d_{1},d_{1}'>0\) such that
Furthermore, if \(q\ge2\), we have for all \(t\ge0\),
Proof
Let \(f=f(t)\ (t\ge0)\). Direct computation shows that
and
From \(f(0)=0\), as well as \((f_{4})\) and \((f_{6})\) in Lemma 1.1, it follows that
Furthermore, noticing the fact that the function \(H(t)=t^{-k-1}G^{2}(t)\) is positive and continuous on any bounded interval \([c, d]\subset(0, +\infty)\), we obtain (2.2) from (2.6).
In the following we prove (2.3). First, we have from (2.5) that
Differentiating (2.7) with respect to t gives
Since
we have from (2.5), (2.8) and (2.9) that
where
and
Using the property \((f_{5})\) in Lemma 1.1, we have
and then
On the other hand, directive computation gives
Then an application of (2.12) and (2.15) yields \(\gamma(t)\le\frac{3}{2q}\beta^{2}(t)\), and so
Furthermore, we see from \((f_{5})\) in Lemma 1.1 that
and then
We now claim
In fact, we have
and
Clearly, it is not difficult to verify (2.19) by the use of (2.20) and (2.21). Then, from (2.10), (2.14), (2.16), (2.18) and (2.19), we obtain
and the proof of Lemma 2.1 is completed. □
On the other hand, for the function \(g(t)\) defined in Lemma 2.1, we have the following result.
Lemma 2.2
Assume \(q>1\). Let \(f(t)\) be the function defined by (1.7) and \(k>1\). Suppose \(g(t)=f'(t)|f(t)|^{q-1}f(t)\). Then, there exists \(M_{1}>0\) such that
Proof
Obviously, the function \(Y(t)=\frac{t^{q}}{g(t)}\) is continuous on \((0, +\infty)\). From \((f_{4})\) and \((f_{6})\) in Lemma 1.1, we derive
and
Then the conclusion (2.23) follows, and the proof of Lemma 2.2 is completed. □
Lemma 2.3
Let \(z\in C_{\mathrm{loc}}^{1,\delta}({\mathbb {R}}^{N})\) \((0<\delta<1)\) be a nonnegative and stable weak solution of (1.8) with \(q\ge5/2\). Then for every \(k\in(1, k_{0}(q))\), where \(k_{0}(t)\) is defined by (1.14), there exists a constant \(C=C(q,k)\) such that
where \(\varphi=\varphi(x)\in C_{0}^{1}({\mathbb {R}}^{N})\) is a nonnegative cut-off function, in which \(\varphi(x)=\varphi_{0}(\frac{|x|}{R})\) with \(R>0\), \(\varphi_{0}(s)\in C_{0}^{1}({\mathbb {R}}^{+})\), \(0\le\varphi_{0}(s)\le1\), and it is defined by
with \(m=\frac{q+k}{q-1}>1\).
Remark 2.4
It is not difficult to verify that \(0\le\varphi_{0}(s)\le1\) and \(|\varphi'_{0}(s)|\le\beta_{0}\varphi_{0}^{1-1/m}(s)\) with \(\beta_{0}=2^{1/m}m\).
Proof
Let \(z\in C^{1,\delta}_{\mathrm{loc}}({\mathbb {R}}^{N})\) be a nonnegative and stable weak solution of (1.8) and \(k>1\). Setting \(\zeta=z^{k}\varphi^{2}\) in (1.9), we find
Applying Young’s inequality with parameter \(\epsilon\in(0,1)\), we obtain
Here and in the sequel, let C be a positive constant depending on ϵ and \(q, k\), which may vary from line to line. Then it follows from (2.28) and (2.29) that
On the other hand, taking \(\zeta=G(z)\varphi\) in (1.10), we find
and then
By Young’s inequality with \(\epsilon>0\), one derives
Then, it follows from (2.2), (2.3), (2.30) and (2.32) that
Then one sees from (2.34) that
where
Clearly,
Moreover, the elementary inequality \(\delta_{0}>0\) implies that \(k\in (1, \frac{k_{0}(q)}{2})\), where \(k_{0}(t)\) is defined by (1.14). Now, an application of (2.35) yields
Let \(\lambda=\frac{q+k}{k+1}\), \(\lambda'=\frac{q+k}{q-1}\). Then it follows from Lemma 2.2 that, if \(z=z(x)\) is nonnegative in \({\mathbb {R}}^{N}\), the function \(\psi(x)=z^{(k+1)\lambda'}(g(z)z^{k})^{-\frac{\lambda'}{\lambda }}=(\frac{z^{q}}{g(z)})^{\frac{k+1}{q-1}}\) is nonnegative and bounded in \({\mathbb {R}}^{N}\). Moreover, we obtain from the Hölder inequality, (2.23) and (2.38) that
Obviously, inequality (2.39) implies
and (2.26) follows. □
Proof of Theorem 1.2
Setting \(x=R\xi\) in (2.26), we get
where assumption (1.11) on \(h(x)\) has been used, C is a positive constant independent of R, and
Clearly, if \(\theta<0\), the desired result follows by letting \(R\to\infty\) in (2.41). In the following, we will show that some appropriate \(k=k(q)\) can be chosen such that \(\theta<0\). Let \(X(t)\) be the function defined in (1.13). Obviously, we have
Note that \(X(t)=2+(2+a)Y(t)\ (t\ge5/2)\) with
and
So, the function \(X(t)\) is increasing and \(6+2a< X(t)<10+4a\) for \(t>5/2\).
Therefore, if \(N\le6+2a\), we have \(N< X(t)\) for any \(t>5/2\). Hence, if we fix \(k\in(1, k_{0}(t))\) suitably near \(k_{0}(t)\), we obtain
For this reason, the desired result follows by letting \(R\to\infty\) in (2.41).
Assume now \(\gamma_{0}(a)< N<\gamma_{\infty}(a)\). Since \(X(t)\) is increasing, we get in this case a critical value \(q_{c}>5/2\) such that \(N< X(q)\) for \(q_{c}< q\). From this, the desired result follows again by letting \(R\to\infty\) in (2.41). Clearly, \(q_{c}\) may be determined from the equation \(N=X(q_{c})\). Then we complete the proof of Theorem 1.2. □
3 Concluding remarks
In this paper, we have considered a model described by the quasilinear Schrödinger equation. Nonexistence of stable solutions is proved.
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The authors thank the reviewers for careful reading of the paper.
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Support from the Fundamental Research Funds for the Central Universities of China (2015B31014) and by National Natural Science Foundation of China (No. 11571092) as well as by the China Postdoctoral Science Foundations (Grant No. 2017M611664) and the Nature Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJB110030) is acknowledged.
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Chen, L., Chen, C., Yang, H. et al. Nonexistence of stable solutions for quasilinear Schrödinger equation. Bound Value Probl 2018, 168 (2018). https://doi.org/10.1186/s13661-018-1087-7
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DOI: https://doi.org/10.1186/s13661-018-1087-7