Liouville theorem for elliptic equations with mixed boundary value conditions and finite Morse indices

In this paper, we establish Liouville type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: \(-\Delta u=|u|^{p-1}u\) in \(\mathbb{R}^{N}_{+}\), \(\frac{\partial u}{\partial\nu}=|u|^{q-1}u\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=0\) on \(\Gamma_{0}\), and \(-\Delta u=|u|^{p-1}u\) in \(\mathbb{R}^{N}_{+}\), \(\frac{\partial u}{\partial\nu}=|u|^{q-1}u\) on \(\Gamma_{1}\), \(u=0\) on \(\Gamma_{0}\), where \(\mathbb{R}^{N}_{+} =\{x\in\mathbb{R}^{N}:x_{N}>0\}\), \(\Gamma_{1}=\{x\in \mathbb{R}^{N}:x_{N}=0,x_{1}<0\}\) and \(\Gamma_{0}=\{x\in\mathbb{R}^{N}:x_{N}=0,x_{1}>0\}\). The exponents p, q satisfy the conditions in Theorem 1.1.

In this paper, we establish a Liouville type theorem in the upper half space \(\mathbb{R}^{N}_{+}\) with mixed boundary conditions.

In a seminal paper [1], Gidas and Spruck considered positive solutions of the nonlinear elliptic problem

for the subcritical case: \(1< p<\frac{N+2}{N-2}\), \(N\geq3\). Although when \(p = \frac{N+2}{N-2}\), the critical case, problem (1.1) possesses a family of solutions, it was proved in [1] that problem (1.1) has no positive solutions. This is the so-called Liouville type theorem. Soon afterward, similar results were established in [2] for positive solutions of the subcritical problem in the upper half space \(\mathbb{R}^{N}_{+}\):

As soon as such results appeared, they received wide attention as regards the theory itself and its applications. Particularly, one may use it to establish a prior bound of solutions for nonlinear elliptic equations by the blow-up method, then various methods, such as topological degree, fix point theorems etc., can be used to obtain the existence of solutions for such problems; see for instance [1] and [2].

On the other hand, it is puzzling if problems (1.1) and (1.2) admit sign-changing solutions. A partial answer came from [3] by assuming additionally that solutions have finite Morse indices. It was proved in [3] that problems (1.1) and (1.2) do not possess nontrivial bounded solution with finite Morse index provided \(1< p<\frac{N+2}{N-2}\). In applications, fortunately, one may find critical points with finite Morse indices by the mountain pass theorem and saddle point theorem and so on, it allows one to establish for instance as in [4], the existence result for indefinite nonlinearities.

Recently, it is investigated by many authors various type of Liouville theorems for solutions with finite Morse indices, such as problems with Neumann boundary condition, Dirichlet-Neumann mixed boundary and nonlinear boundary conditions etc.; see [5–10] and references therein.

Inspired by previous work, we study in this paper Liouville type results for the following nonlinear Neumann mixed boundary value problem:

as well as the nonlinear Dirichlet mixed boundary value problem

where \(\mathbb{R}^{N}_{+}=\{x\in\mathbb{R}^{N}:x_{N}>0\}\), \(\Gamma_{1}=\{x\in \mathbb{R}^{N}:x_{N}=0,x_{1}<0\}\) and \(\Gamma_{0}=\{x\in\mathbb{R}^{N}:x_{N}=0,x_{1}>0\}\).

It was proved in [11] that problems (1.3) and (1.4) have no positive solutions. In this paper, we deal with solutions of problems (1.3) and (1.4) with finite Morse indices, which are possibly sign-changing. For this purpose, we first define the Morse indices of solutions to problems (1.3) and (1.4), respectively. For problem (1.3), the Morse index of a solution u is defined by

where

For a solution u of problem (1.4), we define its Morse index by

with \(\langle I^{\prime\prime}(u)\varphi,\varphi\rangle\) defined in (1.5).

Our main results are as follows.

Theorem 1.1

If \(1< p\leq\frac{N+2}{N-2}\), \(1< q\leq\frac{N}{N-2}\), and \((p,q)\neq (\frac{N+2}{N-2},\frac{N}{N-2})\), then problems (1.3) and (1.4) do not possess nontrivial bounded solution with finite Morse index.

Theorem 1.1 will be proved in the next section. We first prove that a finite Morse index implies certain integrable conditions on u. Then by the Pohozaev identity, we show the nonexistence result.

In this section, we establish the Liouville type theorem for bounded solutions of problems (1.3) and (1.4) with finite Morse indices, that is, we show that such solutions must be trivial. We assume in this section that p, q in (1.3) and (1.4) satisfying \(1< p\leq\frac{N+2}{N-2}\), \(1< q\leq\frac{N}{N-2}\), and \((p,q)\neq (\frac{N+2}{N-2},\frac{N}{N-2})\).

For \(s>2r>0\), we define a cut-off function \(\phi_{r,s}\) as

Moreover, we require \(|\nabla\phi_{r,s}|\leq\frac{2}{r}\) for \(r<|x|<2r\) and \(|\nabla\phi_{r,s}|\leq\frac{2}{s}\) for \(s<|x|<2s\). Then we have the following result; see also [4].

Lemma 2.1

Let u be a solution of (1.3) or (1.4) with finite Morse index. Then there exists \(R_{0}>0\) such that

for any \(R>2R_{0}\).

Proof

We suppose that the Morse index of u is k. Now, we prove the conclusion by contradiction. Suppose on the contrary that inequality (2.1) does not hold, there would exist \(r_{1}>0\) and \(R_{1}>2r_{1}\), such that

Similarly, we can find \(r_{2}>2R_{1}\) and \(R_{2}>2r_{2}\), such that

Iterating this procedure, we see that there exist \(r_{k+1}>2R_{k}\) and \(R_{k+1}>2r_{k+1}\), such that

By the above inequalities, we deduce that \(u\phi_{r_{i},R_{i}}\not \equiv0\) for every \(1\leq i\leq k+1\). Moreover, since \(\{u\phi _{r_{i},R_{i}}\}_{i=1}^{k+1}\) have disjoint supports, we deduce that \(\{ u\phi_{r_{i},R_{i}}\}_{i=1}^{k+1}\) are linearly independent. So the dimension of the linear space

is \(k+1\). We conclude from

for \(1\leq i\leq k+1\) that

for any \(\varphi\in M\). This implies that the Morse index of u is at least \(k+1\), which contradicts the fact that the Morse index of u is equal to k. The assertion follows. □

Next, we show that a finite Morse index implies u satisfying a certain integrable condition. More precisely, we have the following lemma.

Lemma 2.2

Assume that \(1< p\leq\frac{N+2}{N-2}\), \(1< q\leq\frac{N}{N-2}\), and \((p,q)\neq (\frac{N+2}{N-2},\frac{N}{N-2})\). If u is a bounded solution of problem (1.3) or (1.4) with finite Morse index, then we have

Proof

We only prove the results for problem (1.3). For problem (1.4), the proof can proceed similarly.

First, we prove that \(\int_{\mathbb{R}_{+}^{N}} |u|^{p+1}\,dx<\infty\). By Lemma 2.1, there exists \(R_{0}>0\), such that

for any \(R>2R_{0}\). That is,

Apparently,

Multiplying (1.3) by \(u\phi_{R_{0},R}^{2}\) and integrating by parts, we obtain

By (2.2) and (2.3), we have

In particular, we have

If \(N=2\), since u is a bounded solution, the right hand side of (2.5) is bounded by a positive constant independent of R. So we have \(\int_{\mathbb{R}_{+}^{N}} |u|^{p+1}\,dx<\infty\) by letting \(R\to \infty\) in (2.5). Now for the case \(N\geq3\), we deduce from the Hölder inequality that

Suppose that \(\int_{\mathbb{R}_{+}^{N}} |u|^{p+1}\,dx\) is infinite, then we deduce that

Let \(\alpha=N\frac{p-1}{p+1}-2\), \(\beta=\frac{2}{p+1} \), and \(J(R)=\int_{B_{R}^{+}}|u|^{p+1}\,dx\). Iterating (2.7), we obtain

where \(\gamma=1+\beta+\beta^{2}+\cdots+\beta^{k}\). The boundedness of u implies that the right hand side of (2.8) is of order \(R^{M}\) with

as \(k\to\infty\). Hence, we can choose k large enough, such that \(M<0\). Then it follows from (2.8) that

as \(R\to\infty\), which is impossible. So we get \(\int_{\mathbb{R}_{+}^{N}} |u|^{p+1}\,dx<\infty\).

Next, we prove that \(\int_{\partial\mathbb{R}_{+}^{N}} |u|^{q+1}\,dx'<\infty \). In fact, we deduce from (2.4) that

If \(N=2\), the right hand side of (2.9) is bounded by a positive constant independent of R. So we conclude that

by letting \(R\to\infty\) in (2.9). Now for \(N\geq3\), we infer from (2.9) and the Hölder inequality that

since \(\int_{\mathbb{R}_{+}^{N}} |u|^{p+1}\,dx<\infty\) and \(p\leq\frac{N+2}{N-2}\).

Finally, if we multiply (1.3) by u and integrate on \(\mathbb{R}^{N}_{+}\), then we get

This completes the proof of this lemma. □

The next lemma is the well-known local Pohozaev identity for elliptic problems with nonlinear boundary value condition.

Lemma 2.3

Let u be a solution of (1.3) or (1.4). Then the following identity holds:

where \(B_{R}^{+}=\{x:|x|< R\textit{ and }x_{N}>0\}\), \(\partial B_{R}^{+}=\{x:|x|=R\textit{ and }x_{N}>0\}\), \(\Gamma_{0}^{R}=\{x\in\Gamma_{0}: |x|< R\}\), \(\Gamma_{1}^{R}=\{x\in\Gamma_{1}: |x|< R\}\), \(\partial\Gamma_{0}^{R}=\{x\in\Gamma_{0}: |x|=R\}\), \(\partial\Gamma _{1}^{R}=\{ x\in\Gamma_{1}: |x|=R\}\).

Proof

The proof of this lemma is standard, we give it here for completeness. We deal only with problem (1.3). The proof for problem (1.4) is almost the same except that different boundary value conditions were used. We omit the details.

Multiplying (1.3) by \(\langle x,\nabla u\rangle\) and integrating in \(B_{R}^{+}\), we obtain

A direct calculation shows that

Next, we deduce

Combining the above two equations, we obtain the identity for problem (1.3). □

Proof of Theorem 1.1

We only prove the conclusion for problem (1.3), the proof for problem (1.4) is the same. By Lemma 2.1,

Hence, there exists a sequence \(R_{n}\to\infty\) such that

and

as \(n\to\infty\). So we deduce from the Pohozaev identity that

On the other hand, if we multiply (1.3) by u and integrate by parts, then we obtain

By the assumptions on the exponent p, q, we deduce from (2.13) and (2.14) that

or

hence we have \(u\equiv0\). □

The work is supported by National Natural Science Foundation of China (No:11271170) and GAN PO 555 program of Government of Jiangxi Province.

Authors and Affiliations

1. Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, P.R. China

   Xueqiao Wang & Xiongjun Zheng

Corresponding author

Correspondence to Xiongjun Zheng.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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