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Volume 8, Issue 1

# Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

Cung The Anh
• Corresponding author
• Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
• Email
• Other articles by this author:
/ Bui Kim My
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/anona-2016-0165

## Abstract

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system

in a bounded domain $\mathrm{\Omega }\subset {ℝ}^{N}$ with smooth boundary $\partial \mathrm{\Omega }$. Here $p,q>1$, and ${\mathrm{\Delta }}_{\lambda }$ is the strongly degenerate operator of the form

${\mathrm{\Delta }}_{\lambda }u=\sum _{j=1}^{N}\frac{\partial }{\partial {x}_{j}}\left({\lambda }_{j}^{2}\left(x\right)\frac{\partial u}{\partial {x}_{j}}\right),$

where $\lambda \left(x\right)=\left({\lambda }_{1}\left(x\right),\mathrm{\dots },{\lambda }_{N}\left(x\right)\right)$ satisfies certain conditions.

MSC 2010: 35J70; 35D30

## 1 Introduction

In this paper, we study the existence, multiplicity and non-existence of solutions to the following semilinear degenerate elliptic system of Hamiltonian type:

$\left\{\begin{array}{cccc}& -{\mathrm{\Delta }}_{\lambda }u={|v|}^{p-1}v,\hfill & & \hfill x\in \mathrm{\Omega },\\ & -{\mathrm{\Delta }}_{\lambda }v={|u|}^{q-1}u,\hfill & & \hfill x\in \mathrm{\Omega },\\ & u=v=0,\hfill & & \hfill x\in \partial \mathrm{\Omega },\end{array}$(1.1)

where $p,q>1$, and Ω is a bounded domain in ${ℝ}^{N}$ with smooth boundary $\partial \mathrm{\Omega }$. Here ${\mathrm{\Delta }}_{\lambda }$ is the strongly degenerate elliptic operator of the form

${\mathrm{\Delta }}_{\lambda }=\sum _{j=1}^{N}\frac{\partial }{\partial {x}_{j}}\left({\lambda }_{j}^{2}\left(x\right)\frac{\partial u}{\partial {x}_{j}}\right),$

where $x=\left({x}_{1},\mathrm{\dots },{x}_{N}\right)\in {ℝ}^{N}$, and the ${\lambda }_{i}:{ℝ}^{N}\to ℝ$, $i=1,\mathrm{\dots },N$, satisfy some certain conditions. This operator was first introduced by Franchi and Lanconelli [8], and recently reconsidered and named ${\mathrm{\Delta }}_{\lambda }$-Laplacians by Kogoj and Lanconelli in [10] under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation ${\left({\delta }_{t}\right)}_{t\ge 0}$ in ${ℝ}^{N}$. We denote by Q the homogeneous dimension of ${ℝ}^{N}$ with respect to the group of dilations ${\left\{{\delta }_{t}\right\}}_{t>0}$, i.e.,

$Q:={ϵ}_{1}+\mathrm{\cdots }+{ϵ}_{N},$

see Section 2.1 for more details. The homogeneous dimension Q plays a crucial role, both in the geometry and the functional associated to the operator ${\mathrm{\Delta }}_{\lambda }$.

In the case of a single equation

the existence, non-existence and regularity of weak solutions to this problem were proved in [10]. In particular, the authors established a Pohozaev-type identity, and then used it to prove the non-existence result. Later, the existence of weak solutions to this problem was proved in [12] for the case of non-homogeneous Dirichlet boundary conditions, and in [1] for the case where the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition. See also the previous works [18, 17] for related results.

For system (1.1), as is seen in the case of the Laplace operator [4, 5, 7, 3, 14, 15, 16] and very recent results on Liouville-type theorems [2], we know that the critical hyperbola is

$\frac{1}{p+1}+\frac{1}{q+1}=\frac{Q-2}{Q}.$

For exponents $\left(p,q\right)$ lying on or above this curve, that is,

$\frac{1}{p+1}+\frac{1}{q+1}\le \frac{Q-2}{Q},$

we prove the non-existence of positive classical solutions to (1.1) in ${\delta }_{t}$-starshaped bounded domains by establishing a new Pohozaev-type identity. This new identity turns out to be a generalization of the Pohozaev-type identity in the scalar case in [10]. For $\left(p,q\right)$ below the critical hyperbola, we prove the existence of infinitely many weak solutions. To do this, we will use the variational method and the Fountain Theorem of Bartsch and de Figueiredo. It is natural to view a weak solution to (1.1) as a critical point of the corresponding functional

$\mathrm{\Phi }\left(u,v\right)={\int }_{\mathrm{\Omega }}{\nabla }_{\lambda }u\cdot {\nabla }_{\lambda }vdx-\frac{1}{p+1}{\int }_{\mathrm{\Omega }}{|v|}^{p+1}𝑑x-\frac{1}{q+1}{\int }_{\mathrm{\Omega }}{|u|}^{q+1}𝑑x.$

A natural energy space for problem (1.1) is the Hilbert space

$\stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)×\stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right).$

However, this choice of energy space will impose a strict restriction on $p,q$, namely $p,q\le \frac{Q+2}{Q-2}$, due to the Sobolev-type embedding

$\stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)↪{L}^{\frac{2Q}{Q-2}}\left(\mathrm{\Omega }\right)$

(see Proposition 2.1 below). To overcome this difficulty, following the approach introduced in [7, 9], we will use the fractional Sobolev-type spaces defined by using Fourier expansions on the eigenfunctions of $-{\mathrm{\Delta }}_{\lambda }$ (see Section 2 for details). One notes that now one of the nonlinearities may have a larger growth than ${|s|}^{\left(Q+2\right)/\left(Q-2\right)}$ provided the other nonlinearity has a suitably lower growth. Another possible way to overcome this difficulty is to use the Orlicz-space approach [6]. It is worth noticing that the results obtained in this paper are the generalizations of the corresponding results for the Laplace operator in [7, 3, 15, 13, 9, 19].

In this paper, to simplify the exposition, we only state the theorems and give the proofs for the “model problem” (1.1), although these results can be extended to a slightly more general system of the form

$\left\{\begin{array}{cccc}& -{\mathrm{\Delta }}_{\lambda }u=g\left(v\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ & -{\mathrm{\Delta }}_{\lambda }v=f\left(u\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ & u=v=0,\hfill & & \hfill x\in \partial \mathrm{\Omega },\end{array}$

under some suitable assumptions of f and g.

This paper is organized as follows: In Section 2, we recall some known results and prove some important embeddings which are necessary for studying our problem. In Section 3, we prove the non-existence of positive classical solutions to (1.1) by establishing a new Pohozaev-type identity. The existence of infinitely many nontrivial weak solutions to the problem is shown in Section 4 by using the variational method on fractional Sobolev-type spaces.

## 2.1 The ${\mathrm{\Delta }}_{\lambda }$-Laplace operator and related function spaces

As in [10], we consider the strongly degenerate operator of the form

${\mathrm{\Delta }}_{\lambda }u=\sum _{j=1}^{N}\frac{\partial }{\partial {x}_{j}}\left({\lambda }_{j}^{2}\left(x\right)\frac{\partial u}{\partial {x}_{j}}\right),$

where $x=\left({x}_{1},\mathrm{\dots },{x}_{N}\right)\in {ℝ}^{N}$, the ${\lambda }_{j}:{ℝ}^{N}\to ℝ$ are continuous and ${\lambda }_{j}>0$, $j=1,\mathrm{\dots },N$, in ${C}^{1}\left({ℝ}^{N}\setminus \prod \right)$, where

$\prod =\left\{\left({x}_{1},\mathrm{\dots },{x}_{N}\right)\in {ℝ}^{N}:\prod _{i=1}^{N}{x}_{i}=0\right\}.$

We assume the following conditions:

• (i)

${\lambda }_{1}\left(x\right)\equiv 1$, ${\lambda }_{i}\left(x\right)={\lambda }_{i}\left({x}_{1},\mathrm{\dots },{x}_{i-1}\right)$, $i=2,\mathrm{\dots },N$.

• (ii)

${\lambda }_{i}\left(x\right)={\lambda }_{i}\left({x}^{*}\right)$ holds for every $x\in {ℝ}^{N}$ and $i=1,\mathrm{\dots },N$, where

• (iii)

There exists a constant $\rho \ge 0$ such that

and for every .

• (iv)

For $t>0$ there exists a group of dilations

${\delta }_{t}:{ℝ}^{N}\to {ℝ}^{N},{\delta }_{t}\left(x\right)={\delta }_{t}\left({x}_{1},\mathrm{\dots },{x}_{N}\right)=\left({t}^{{ϵ}_{1}}{x}_{1},\mathrm{\dots },{t}^{{ϵ}_{N}}{x}_{N}\right),$

where $1\le {ϵ}_{1}\le {ϵ}_{2}\le \mathrm{\cdots }\le {ϵ}_{N}$, such that ${\lambda }_{i}$ is ${\delta }_{t}$-homogeneous of degree ${ϵ}_{i}-1$, i.e.,

This implies that the operator ${\mathrm{\Delta }}_{\lambda }$ is ${\delta }_{t}$-homogeneous of degree two, i.e.,

We denote by Q the homogeneous dimension of ${ℝ}^{N}$ with respect to the group of dilations ${\left\{{\delta }_{t}\right\}}_{t>0}$, i.e.,

$Q:={ϵ}_{1}+\mathrm{\cdots }+{ϵ}_{N}.$

This operator ${\mathrm{\Delta }}_{\lambda }$ is called the ${\mathrm{\Delta }}_{\lambda }$-Laplacian; for more details on properties and examples of this operator we refer the reader to the papers [8, 10, 11].

We now recall some function spaces related to the ${\mathrm{\Delta }}_{\lambda }$-Laplace operator. Denote by $\stackrel{̊}{W}{}_{\lambda }{}^{1,p}\left(\mathrm{\Omega }\right)$, $p\ge 1$, the closure of ${C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ in the norm

${\parallel u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,p}}={\left({\int }_{\mathrm{\Omega }}{|{\nabla }_{\lambda }u|}^{p}𝑑x\right)}^{\frac{1}{p}}.$

We define ${W}_{\lambda }^{2,p}\left(\mathrm{\Omega }\right)$ as the space of all functions u such that

$u\in {L}^{p}\left(\mathrm{\Omega }\right),{\lambda }_{i}\left(x\right)\frac{\partial u}{\partial {x}_{i}}\in {L}^{p}\left(\mathrm{\Omega }\right),{\lambda }_{i}\left(x\right)\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\left(x\right)\frac{\partial u}{\partial {x}_{j}}\right)\in {L}^{p}\left(\mathrm{\Omega }\right),i,j=1,2,\mathrm{\dots },N,$

with the norm

${\parallel u\parallel }_{{W}_{\lambda }^{2,p}}={\left({\int }_{\mathrm{\Omega }}\left[{|u|}^{p}+{|{\nabla }_{\lambda }u|}^{p}+\sum _{i,j=1}^{N}{\left[{\lambda }_{i}\left(x\right)\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\left(x\right)\frac{\partial u}{\partial {x}_{j}}\right)\right]}^{p}\right]𝑑x\right)}^{\frac{1}{p}}.$

It is easy to see that ${W}_{\lambda }^{2,p}\left(\mathrm{\Omega }\right)$ and $\stackrel{̊}{W}{}_{\lambda }{}^{1,p}\left(\mathrm{\Omega }\right)$ are Banach spaces. In particular, ${W}_{\lambda }^{2,2}\left(\mathrm{\Omega }\right)$ and $\stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)$ are Hilbert spaces with the following inner products:

${\left(u,v\right)}_{{W}_{\lambda }^{2,2}}={\left(u,v\right)}_{{L}^{2}}+\sum _{i=1}^{N}{\left({\lambda }_{i}\frac{\partial u}{\partial {x}_{i}},{\lambda }_{i}\frac{\partial v}{\partial {x}_{i}}\right)}_{{L}^{2}}+\sum _{i,j=1}^{N}{\left({\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right),{\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial v}{\partial {x}_{j}}\right)\right)}_{{L}^{2}}$

and

${\left(u,v\right)}_{\stackrel{̊}{W}{}_{\lambda }^{1,2}}=\sum _{i=1}^{N}{\left({\lambda }_{i}\frac{\partial u}{\partial {x}_{i}},{\lambda }_{i}\frac{\partial v}{\partial {x}_{i}}\right)}_{{L}^{2}}.$

The following result was established in [10].

#### Proposition 2.1.

The embedding

is continuous. Moreover, the embedding

$\stackrel{̊}{W}{}_{\lambda }{}^{1,p}\left(\mathrm{\Omega }\right)↪{L}^{\gamma }\left(\mathrm{\Omega }\right)$

is compact for every $\gamma \mathrm{\in }\mathrm{\left[}\mathrm{1}\mathrm{,}{p}_{\lambda }^{\mathrm{*}}\mathrm{\right)}$.

We now prove the following important result.

#### Lemma 2.2.

The embedding

${W}_{\lambda }^{2,2}\left(\mathrm{\Omega }\right)\cap \stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)↪{L}^{\gamma }\left(\mathrm{\Omega }\right)$

is continuous if $\mathrm{1}\mathrm{\le }\gamma \mathrm{\le }\frac{\mathrm{2}\mathit{}Q}{Q\mathrm{-}\mathrm{4}}$.

#### Proof.

For any $u\in {C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, we have

${\parallel u\parallel }_{{W}_{\lambda }^{2,2}}={\left({\parallel u\parallel }_{{L}^{2}}^{2}+\sum _{j=1}^{N}{\parallel {\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\parallel }_{{L}^{2}}^{2}+\sum _{i,j=1}^{N}{\parallel {\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)\parallel }_{{L}^{2}}^{2}\right)}^{\frac{1}{2}}$

and

${\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)\in {L}^{2}\left(\mathrm{\Omega }\right),i,j=1,2,\mathrm{\dots },N.$

Thus,

${\parallel {\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,2}}^{2}={\parallel {\nabla }_{\lambda }\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)\parallel }_{{L}^{2}}^{2}=\sum _{i=1}^{N}{\int }_{\mathrm{\Omega }}{|{\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)|}^{2}𝑑x<+\mathrm{\infty }.$(2.1)

Hence, ${\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\in \stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)$, $j=1,\mathrm{\dots },N$, and by Proposition 2.2 we have $\stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)↪{L}^{{2}_{\lambda }^{*}}\left(\mathrm{\Omega }\right)$, where ${2}_{\lambda }^{*}:=\frac{2Q}{Q-2}$, so

${\parallel {\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\parallel }_{{L}^{{2}_{\lambda }^{*}}}\le C{\parallel {\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,2}}.$(2.2)

Therefore, we get ${\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\in {L}^{{2}_{\lambda }^{*}}\left(\mathrm{\Omega }\right)$, $j=1,\mathrm{\dots },N$, and by the definition of the space $\stackrel{̊}{W}{}_{\lambda }{}^{1,{2}_{\lambda }^{*}}\left(\mathrm{\Omega }\right)$ we infer that $u\in \stackrel{̊}{W}{}_{\lambda }{}^{1,{2}_{\lambda }^{*}}\left(\mathrm{\Omega }\right)$. Applying Proposition 2.2 once again, we obtain

${\parallel u\parallel }_{{L}^{\left({2}_{\lambda }^{*}Q\right)/\left(Q-{2}_{\lambda }^{*}\right)}}\le C{\parallel u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,{2}_{\lambda }^{*}}}.$(2.3)

From (2.1)–(2.3) we obtain

${\parallel u\parallel }_{{L}^{\frac{2Q}{Q-4}}}\le C{\parallel u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,{2}_{\lambda }^{*}}}={\parallel {\nabla }_{\lambda }u\parallel }_{{L}^{{2}_{\lambda }^{*}}}$$\le C{\parallel {\nabla }_{\lambda }u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,2}}$$\le C{\left(\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{|{\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)|}^{2}𝑑x\right)}^{\frac{1}{2}}$$\le C{\left({\parallel u\parallel }_{{L}^{2}}^{2}+\sum _{j=1}^{N}{\parallel {\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\parallel }_{{L}^{2}}^{2}+\sum _{i,j=1}^{N}{\parallel {\lambda }_{i}\frac{\partial }{\partial {x}_{i}}\left({\lambda }_{j}\frac{\partial u}{\partial {x}_{j}}\right)\parallel }_{{L}^{2}}^{2}\right)}^{\frac{1}{2}}$$=C{\parallel u\parallel }_{{W}_{\lambda }^{2,2}}.\mathit{∎}$

## 2.2 Functional setting of the problem

We now define some functional spaces which are used to study problem (1.1).

We consider the operator

$A:{W}_{\lambda }^{2,2}\left(\mathrm{\Omega }\right)\cap \stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)\to {L}^{2}\left(\mathrm{\Omega }\right),$

where $A=-{\mathrm{\Delta }}_{\lambda }$ with the homogeneous Dirichlet boundary condition. Then A is linear, positive, self-adjoint and has a compact inverse. Consequently, there exists an orthonormal basis of ${L}^{2}\left(\mathrm{\Omega }\right)$ consisting of eigenfunctions ${\phi }_{j}\in \stackrel{̊}{W}{}_{\lambda }{}^{1,2}\left(\mathrm{\Omega }\right)$, $j=1,2,\mathrm{\dots }$, of A with eigenvalues

We denote ${E}^{s}=D\left({A}^{s}\right)$, with $s>0$, the space with the inner product

${\left(u,v\right)}_{{E}^{s}}={\int }_{\mathrm{\Omega }}{A}^{s}u{A}^{s}v𝑑x,u,v\in {E}^{s},$

where

$D\left({A}^{s}\right)=\left\{\phi =\sum _{j=1}^{\mathrm{\infty }}{a}_{j}{\phi }_{j}:{a}_{j}\in ℝ,\sum _{j=1}^{\mathrm{\infty }}{\mu }_{j}^{2s}{a}_{j}^{2}<+\mathrm{\infty }\right\},$${A}^{s}\phi =\sum _{j=1}^{\mathrm{\infty }}{a}_{j}{\mu }_{j}^{s}{\phi }_{j}.$

We notice that, as a consequence of Lemma 2.2 and interpolation theorems, we have the following important embeddings which play an important role for our investigation.

#### Lemma 2.3.

The embeddings

${E}^{s}↪{L}^{q+1}\left(\mathrm{\Omega }\right)\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{E}^{t}↪{L}^{p+1}\left(\mathrm{\Omega }\right)$

are continuous if $\frac{\mathrm{1}}{q\mathrm{+}\mathrm{1}}\mathrm{\ge }\frac{\mathrm{1}}{\mathrm{2}}\mathrm{-}\frac{\mathrm{2}\mathit{}s}{Q}$ and $\frac{\mathrm{1}}{p\mathrm{+}\mathrm{1}}\mathrm{\ge }\frac{\mathrm{1}}{\mathrm{2}}\mathrm{-}\frac{\mathrm{2}\mathit{}t}{Q}$, respectively. Moreover, these embeddings are compact if the corresponding inequalities are strict.

#### Proof.

By Lemma 2.2 and since $\frac{2Q}{Q-4}>2$, we obtain

${\parallel u\parallel }_{{L}^{2}}\le C{\parallel u\parallel }_{{W}_{\lambda }^{2,2}}.$

On the other hand, by Proposition 2.1, we have

${\parallel u\parallel }_{{L}^{2}}\le C{\parallel u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,2}}.$

Therefore,

${\parallel u\parallel }_{{L}^{2}}\le C\mathrm{max}\left\{{\parallel u\parallel }_{\stackrel{̊}{W}{}_{\lambda }^{1,2}},{\parallel u\parallel }_{{W}_{\lambda }^{2,2}}\right\}=C{\parallel u\parallel }_{D\left(A\right)}.$

Hence, by interpolation, the injection

is continuous. Moreover, this embedding is compact if

$\frac{1}{q+1}>\frac{1-2s}{2}+\frac{2s}{{2}_{\lambda }^{*}}=\frac{1}{2}-\frac{2s}{Q}.$

This completes the proof of the first statement. The second one is proved similarly. ∎

For $s,t\ge 0$ such that $s+t=1$, we consider $E={E}^{s}×{E}^{t},$ a Hilbert space with the inner product

and the bilinear form

$B\left(\left(u,v\right),\left(\phi ,\psi \right)\right)={\int }_{\mathrm{\Omega }}\left({A}^{s}u{A}^{t}\psi +{A}^{s}\phi {A}^{t}v\right)𝑑x.$

It is easy to see that B is symmetric and continuous and there exists a self-adjoint bounded linear operator $L:E\to E$ with

Associated to B and L we define the quadratic form $P:E\to ℝ$ by

Using arguments as in [7], one can prove that the operator L only has two eigenvalues $±1$, and the eigenspaces ${E}^{+}$ and ${E}^{-}$ are

${E}^{+}=\left\{\left(u,{A}^{s-t}u\right):u\in {E}^{s}\right\}\mathit{ }\text{and}\mathit{ }{E}^{-}=\left\{\left(u,-{A}^{s-t}u\right):u\in {E}^{s}\right\}.$

Let ${\left\{{e}_{j}\right\}}_{j=1}^{\mathrm{\infty }}$ be an orthonormal basis of ${E}^{s}$. Thus ${\left\{{A}^{s-t}{e}_{j}\right\}}_{j=1}^{\mathrm{\infty }}$ is also an orthonormal basis of ${E}^{t}$. We denote

${E}_{n}^{s}:=\mathrm{span}\left\{{e}_{j}:j=1,\mathrm{\dots },n\right\}\mathit{ }\text{and}\mathit{ }{E}_{n}^{t}:=\mathrm{span}\left\{{A}^{s-t}{e}_{j}:j=1,\mathrm{\dots },n\right\}.$

We can check that E is presented as a direct sum $E={E}^{+}\oplus {E}^{-}$, ${E}_{n}={E}_{n}^{+}\oplus {E}_{n}^{-}={E}_{n}^{s}×{E}_{n}^{t}$, and

$\overline{\bigcup _{n=1}^{\mathrm{\infty }}{E}_{n}^{+}}={E}^{+},\overline{\bigcup _{n=1}^{\mathrm{\infty }}{E}_{n}^{-}}={E}^{-}.$

Therefore, for every $z={z}^{+}+{z}^{-}\in E={E}^{+}\oplus {E}^{-}$, we have ${z}^{+}=\left(u,{A}^{s-t}u\right)$ and ${z}^{-}=\left(u,-{A}^{s-t}u\right)$. This implies that

and

$P\left({z}^{+}\right)-P\left({z}^{-}\right)=\frac{1}{2}{\parallel z\parallel }_{E}^{2}.$

Now we define the functional $\mathrm{\Phi }:E={E}^{s}×{E}^{t}\to ℝ$ associated to the problem (1.1) by

$\mathrm{\Phi }\left(z\right)=P\left(z\right)-{\int }_{\mathrm{\Omega }}H\left(u,v\right)𝑑x,$

where

$H\left(u,v\right)=\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}.$

One can check that Φ is well-defined on E and $\mathrm{\Phi }\in {C}^{1}\left(E,ℝ\right)$ with

${\mathrm{\Phi }}^{\prime }\left(u,v\right)\left(\varphi ,\psi \right)={\int }_{\mathrm{\Omega }}\left({A}^{s}u{A}^{t}\psi +{A}^{t}v{A}^{s}\varphi \right)𝑑x+{\int }_{\mathrm{\Omega }}\left({u}^{q}\varphi +{v}^{p}\psi \right)𝑑x.$

One also sees that the critical points of Φ are the weak solutions of problem (1.1) in the following sense.

#### Definition 2.4.

We say that $z=\left(u,v\right)\in E={E}^{s}×{E}^{t}$ is a weak solution of (1.1) if

## 2.3 Some results of abstract critical points theory

We will use the Fountain Theorem established by Bartsch and de Figueiredo in [3] to prove the existence of infinitely many weak solutions to problem (1.1).

#### Definition 2.5.

Let X be a Hilbert space and a functional $\mathrm{\Phi }\in {C}^{1}\left(X,ℝ\right)$. Given a sequence $\mathcal{ℱ}=\left({X}_{n}\right)$ of finite dimensional subspaces of X such that ${X}_{n}\subset {X}_{n+1}$, $n=1,2,\mathrm{\dots }$, and

$\overline{\bigcup _{n=1}^{\mathrm{\infty }}{X}_{n}}=X.$

We say that

• (i)

a sequence $\left({z}_{k}\right)\subset X$ with ${z}_{k}\in {X}_{{n}_{k}}$, ${n}_{k}\to \mathrm{\infty }$, is a ${\left(\mathrm{PS}\right)}_{c}^{\mathcal{ℱ}}$-sequence if

$\mathrm{\Phi }\left({z}_{k}\right)\to c\mathit{ }\text{and}\mathit{ }\left(1+\parallel {z}_{k}\parallel \right)\left({{\mathrm{\Phi }}^{\prime }|}_{{X}_{{n}_{k}}}\right)\left({z}_{k}\right)\to 0;$

• (ii)

Φ satisfies ${\left(\mathrm{PS}\right)}_{c}^{\mathcal{ℱ}}$ at level $c\in ℝ$ if every sequence ${\left(\mathrm{PS}\right)}_{c}^{\mathcal{ℱ}}$-sequence has a subsequence converging to a critical point of Φ.

#### Theorem 2.6 ([3]).

Assume $\mathrm{\Phi }\mathrm{:}E\mathrm{\to }\mathrm{R}$ is ${C}^{\mathrm{1}}\mathit{}\mathrm{\left(}E\mathrm{,}\mathrm{R}\mathrm{\right)}$ and satisfies the following conditions:

• (Φ1)

Φ satisfies ${\left(\mathrm{PS}\right)}_{c}^{\mathcal{ℱ}}$ with $\mathcal{ℱ}=\left({E}_{n}\right)$, $n=1,2,\mathrm{\dots }$ and $c>0$.

• (Φ2)

There exists a sequence ${r}_{k}>0$, $k=1,2,\mathrm{\dots }$ , such that for some $k\ge 2$,

• (Φ3)

There exists a sequence of isomorphisms ${T}_{k}:E\to E$, $k=1,2,\mathrm{\dots }$ , with ${T}_{k}\left({E}_{n}\right)={E}_{n}$ for all k and n , and there exists a sequence ${R}_{k}>0$, $k=1,2,\mathrm{\dots }$ , such that, for $z={z}^{+}+{z}^{-}\in {E}_{k}^{+}\oplus {E}^{-}$ and ${R}_{k}=\mathrm{max}\left\{\parallel {z}^{+}\parallel ,\parallel {z}^{-}\parallel \right\}$ , one has

$\parallel {T}_{k}z\parallel >{r}_{k}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\mathrm{\Phi }\left({T}_{k}z\right)<0,$

where ${r}_{k}$ is obtained in (Φ2).

• (Φ4)

${d}_{k}:=sup\left\{\mathrm{\Phi }\left({T}_{k}\left({z}^{+}+{z}^{-}\right)\right):{z}^{+}\in {E}_{k}^{+},{z}^{-}\in {E}^{-},\parallel {z}^{+}\parallel ,\parallel {z}^{-}\parallel \le {R}_{k}\right\}<+\mathrm{\infty }$.

• (Φ5)

Φ is even, i.e., $\mathrm{\Phi }\left(z\right)=\mathrm{\Phi }\left(-z\right)$.

Then Φ has an unbounded sequence of critical values.

As is noticed in [3], condition (Φ4) holds if the functional Φ maps bounded sets in E to bounded sets in $ℝ$.

## 3 Non-existence of positive classical solutions

In this section, we prove the non-existence result for our problem when the domain Ω is ${\delta }_{t}$-starshaped in the following sense.

#### Definition 3.1 ([10]).

A domain Ω is called ${\delta }_{t}$-starshaped with respect to the origin if $0\in \mathrm{\Omega }$ and $〈T,\nu 〉\ge 0$ at every point of $\partial \mathrm{\Omega }$, where ν is the outward normal vector and $〈\cdot ,\cdot 〉$ denotes the inner product in ${ℝ}^{N}$.

As mentioned above, we consider the vector field

$T:=\sum _{i=1}^{N}{ϵ}_{i}{x}_{i}\frac{\partial }{\partial {x}_{i}},$(3.1)

and this field is the generator of the group of dilation ${\left({\delta }_{t}\right)}_{t\ge 0}$ (here, a function u is ${\delta }_{t}$-homogeneous of degree m if and only if $Tu=mu$).

As in [10], we will denote by ${\mathrm{\Lambda }}^{2}\left(\overline{\mathrm{\Omega }}\right)$ the linear space of the functions $u\in C\left(\overline{\mathrm{\Omega }}\right)$ such that ${X}_{j}u$ and ${X}_{j}^{2}u$, for $j=1,\mathrm{\dots },N$, exist in the weak sense of distributions in Ω and can be continuously extended to $\overline{\mathrm{\Omega }}$. Here ${X}_{j}:={\lambda }_{j}\frac{\partial }{\partial {x}_{j}}$.

#### Lemma 3.2.

For any $u\mathrm{,}v\mathrm{\in }{\mathrm{\Lambda }}^{\mathrm{2}}\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{\right)}$, we have

${\int }_{\mathrm{\Omega }}\left[T\left(u\right){\mathrm{\Delta }}_{\lambda }v+T\left(v\right){\mathrm{\Delta }}_{\lambda }u\right]𝑑x$$={\int }_{\partial \mathrm{\Omega }}\left[T\left(u\right)〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉+T\left(v\right)〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉\right]𝑑S-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+\left(Q-2\right){\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x,$(3.2)

where T is the vector field in (3.1), ν is the outward normal to Ω and

${\nu }_{\lambda }=\left({\lambda }_{1}{\nu }_{1},\mathrm{\dots },{\lambda }_{N}{\nu }_{N}\right),{\nabla }_{\lambda }=\left({\lambda }_{1}{\partial }_{{x}_{1}},\mathrm{\dots },{\lambda }_{N}{\partial }_{{x}_{N}}\right).$

#### Remark 3.3.

If $u=v$, we recover from (3.2) the Pohozaev-type identity in the scalar case, which was obtained in [10, Theorem 2.1].

#### Proof.

Integrating by parts, we have

${\int }_{\mathrm{\Omega }}T\left(u\right){\mathrm{\Delta }}_{\lambda }v𝑑x=\sum _{i,j=1}^{N}{\int }_{\partial \mathrm{\Omega }}{ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}{\nu }_{j}𝑑S-{\int }_{\mathrm{\Omega }}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{\partial }{\partial {x}_{j}}\left({ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}\right)𝑑x$$={I}_{1}+{I}_{2},$(3.3)

where

${I}_{1}=\sum _{i,j=1}^{N}{\int }_{\partial \mathrm{\Omega }}{ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}{\nu }_{j}𝑑S$$=\sum _{i=1}^{N}{\int }_{\partial \mathrm{\Omega }}{ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉𝑑S$$={\int }_{\partial \mathrm{\Omega }}T\left(u\right)〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉𝑑S$(3.4)

and

${I}_{2}=-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\left({ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}\right)𝑑x$$=-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}\left[{\delta }_{ij}{ϵ}_{i}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{\partial u}{\partial {x}_{i}}+{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{{\partial }^{2}u}{\partial {x}_{i}{x}_{j}}\right]𝑑x$$=-\sum _{j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{j}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{\partial u}{\partial {x}_{j}}𝑑x-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{{\partial }^{2}u}{\partial {x}_{i}{x}_{j}}𝑑x$$={I}_{2,1}+{I}_{2,2}.$

Integrating by parts in ${I}_{2,2}$ gives

${I}_{2,2}=-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{{\partial }^{2}u}{\partial {x}_{i}{x}_{j}}𝑑x$$=-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+Q{\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x+\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{j}\partial {x}_{i}}𝑑x$$+\sum _{j=1}^{N}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial {x}_{j}}\frac{\partial v}{\partial {x}_{j}}T{\lambda }_{j}^{2}𝑑x.$(3.5)

Since ${\lambda }_{j}$ is ${\delta }_{t}$-homogeneous of degree ${ϵ}_{j}-1$, we have $T{\lambda }_{j}^{2}=2{\lambda }_{j}T{\lambda }_{j}=2\left({ϵ}_{j}-1\right){\lambda }_{j}^{2}$. Therefore, from (2.1) we get

${I}_{2,2}=-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+Q{\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x+\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{j}\partial {x}_{i}}𝑑x$$+\sum _{j=1}^{N}{\int }_{\mathrm{\Omega }}2\left({ϵ}_{j}-1\right)\frac{\partial u}{\partial {x}_{j}}\frac{\partial v}{\partial {x}_{j}}{\lambda }_{j}^{2}𝑑x$$=-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+\left(Q-2\right){\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x+\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{j}\partial {x}_{i}}𝑑x$$+2\sum _{j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{j}\frac{\partial u}{\partial {x}_{j}}\frac{\partial v}{\partial {x}_{j}}{\lambda }_{j}^{2}𝑑x.$(3.6)

It follows from (3.3)–(3.6) that

${\int }_{\mathrm{\Omega }}T\left(u\right){\mathrm{\Delta }}_{\lambda }v𝑑x={\int }_{\partial \mathrm{\Omega }}T\left(u\right)〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉𝑑S-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+\left(Q-2\right){\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x$$-{I}_{2,1}+\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{j}\partial {x}_{i}}𝑑x.$(3.7)

By similar computations, we also obtain

${\int }_{\mathrm{\Omega }}T\left(v\right){\mathrm{\Delta }}_{\lambda }u𝑑x={\int }_{\partial \mathrm{\Omega }}T\left(v\right)〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉𝑑S-\sum _{j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{j}{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}\frac{\partial u}{\partial {x}_{j}}𝑑x-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{i}{x}_{j}}𝑑x$$={\int }_{\partial \mathrm{\Omega }}T\left(v\right)〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉𝑑S+{I}_{2,1}-\sum _{i,j=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}{x}_{i}{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{{\partial }^{2}v}{\partial {x}_{i}{x}_{j}}𝑑x.$(3.8)

Combining (3.7) and (3.8), we obtain (3.2). ∎

The following theorem is the main result of this section.

#### Theorem 3.4.

Assume $N\mathrm{\ge }\mathrm{3}$, and let $p\mathrm{,}q\mathrm{>}\mathrm{1}$ satisfy

$\frac{1}{p+1}+\frac{1}{q+1}\le \frac{Q-2}{Q}.$(3.9)

If Ω is bounded and ${\delta }_{t}$-starshaped with respect to the origin, then problem (1.1) has no nontrivial nonnegative solution $u\mathrm{\in }{\mathrm{\Lambda }}^{\mathrm{2}}\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{\right)}$.

#### Proof.

From (1.1) we have

${\int }_{\mathrm{\Omega }}\left[T\left(v\right){\mathrm{\Delta }}_{\lambda }u+T\left(u\right){\mathrm{\Delta }}_{\lambda }v\right]𝑑x=-{\int }_{\mathrm{\Omega }}\left[T\left(v\right){v}^{p}+T\left(u\right){u}^{q}\right]𝑑x$$=-\sum _{i=1}^{N}{\int }_{\partial \mathrm{\Omega }}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right){ϵ}_{i}{x}_{i}𝑑S+\sum _{i=1}^{N}{\int }_{\mathrm{\Omega }}{ϵ}_{i}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right)𝑑x$$=Q{\int }_{\mathrm{\Omega }}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right)𝑑x.$(3.10)

From (3.2) and (3.10) we infer that

$Q{\int }_{\mathrm{\Omega }}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right)𝑑x$$={\int }_{\partial \mathrm{\Omega }}\left[〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉T\left(v\right)+〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉T\left(u\right)\right]𝑑S-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S+\left(Q-2\right){\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x.$(3.11)

On the other hand, integrating by parts, we get

${\int }_{\mathrm{\Omega }}{|v|}^{q+1}𝑑x=-{\int }_{\mathrm{\Omega }}v{\mathrm{\Delta }}_{\lambda }u𝑑x={\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x,$(3.12)

and

${\int }_{\mathrm{\Omega }}{|u|}^{q+1}𝑑x=-{\int }_{\mathrm{\Omega }}u{\mathrm{\Delta }}_{\lambda }v𝑑x={\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x.$(3.13)

Therefore, we obtain from (3.12) and (3.13) the following identity:

${\int }_{\mathrm{\Omega }}\left[\left(1-\theta \right){|u|}^{q+1}+\theta {|v|}^{q+1}\right]𝑑x={\int }_{\mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉𝑑x.$(3.14)

From (3.11) and (3.14) we get

$Q{\int }_{\mathrm{\Omega }}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right)𝑑x-\left(Q-2\right){\int }_{\mathrm{\Omega }}\left[\left(1-\theta \right){|u|}^{q+1}+\theta {|v|}^{q+1}\right]𝑑x$$={\int }_{\partial \mathrm{\Omega }}\left[〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉T\left(v\right)+〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉T\left(u\right)\right]𝑑S-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S.$

This is equivalent to

${\int }_{\mathrm{\Omega }}\left[\left(\frac{Q}{p+1}-\theta \left(Q-2\right)\right){|v|}^{p+1}+\left(\frac{Q}{q+1}-\left(1-\theta \right)\left(Q-2\right)\right){|u|}^{q+1}\right]𝑑x$$={\int }_{\partial \mathrm{\Omega }}\left[〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉T\left(v\right)+〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉T\left(u\right)\right]𝑑S-{\int }_{\partial \mathrm{\Omega }}〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉𝑑S.$(3.15)

Now, since $u,v\in {C}^{1}\left(\overline{\mathrm{\Omega }}\setminus \mathrm{\Pi }\right)$, the condition $u=v=0$ on $\partial \mathrm{\Omega }$ implies that $\frac{\partial u}{\partial {x}_{j}}=\frac{\partial u}{\partial \nu }{\nu }_{j}$, $\frac{\partial v}{\partial {x}_{j}}=\frac{\partial v}{\partial \nu }{\nu }_{j}$ at any point of $\partial \mathrm{\Omega }\setminus \mathrm{\Pi }$ for any $j=1,\mathrm{\dots },N$. Therefore, on $\partial \mathrm{\Omega }\setminus \mathrm{\Pi }$ we have

$\left[〈{\nabla }_{\lambda }u,{\nu }_{\lambda }〉T\left(v\right)+〈{\nabla }_{\lambda }v,{\nu }_{\lambda }〉T\left(u\right)\right]-〈{\nabla }_{\lambda }u,{\nabla }_{\lambda }v〉〈T,\nu 〉$$=\sum _{i,j=1}^{N}\left[{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}{\nu }_{j}{ϵ}_{i}{x}_{i}\frac{\partial v}{\partial {x}_{i}}+{\lambda }_{j}^{2}\frac{\partial v}{\partial {x}_{j}}{\nu }_{j}{ϵ}_{i}{x}_{i}\frac{\partial u}{\partial {x}_{i}}-{\lambda }_{j}^{2}\frac{\partial u}{\partial {x}_{j}}\frac{\partial v}{\partial {x}_{j}}{ϵ}_{i}{x}_{i}{\nu }_{i}\right]$$=\sum _{i,j=1}^{N}\frac{\partial u}{\partial \nu }\frac{\partial v}{\partial \nu }{\lambda }_{j}^{2}{\nu }_{j}^{2}{ϵ}_{i}{x}_{i}{\nu }_{i}$$=\frac{\partial u}{\partial \nu }\frac{\partial v}{\partial \nu }{|{\nu }_{\lambda }|}^{2}〈T,\nu 〉.$

Substituting this identity into (3.15), we obtain

${\int }_{\mathrm{\Omega }}\left[\left(\frac{Q}{p+1}-\theta \left(Q-2\right)\right){|v|}^{p+1}+\left(\frac{Q}{q+1}-\left(1-\theta \right)\left(Q-2\right)\right){|u|}^{q+1}\right]𝑑x={\int }_{\partial \mathrm{\Omega }}\frac{\partial u}{\partial \nu }\frac{\partial v}{\partial \nu }{|{\nu }_{\lambda }|}^{2}〈T,\nu 〉𝑑S.$(3.16)

Choosing $\theta =\frac{Q}{\left(Q-2\right)\left(p+1\right)}$ yields $\frac{Q}{p+1}-\theta \left(Q-2\right)=0$, and from (3.9) we get

$\frac{Q}{q+1}-\left(1-\theta \right)\left(Q-2\right)\le 0.$

We infer from (3.16) that

${\int }_{\partial \mathrm{\Omega }}\frac{\partial u}{\partial \nu }\frac{\partial v}{\partial \nu }{|{\nu }_{\lambda }|}^{2}〈T,\nu 〉𝑑S\le 0.$(3.17)

Since Ω is ${\delta }_{t}$-starshaped, we have $〈T,\nu 〉\ge 0$ on $\partial \mathrm{\Omega }$; along with $\frac{\partial u}{\partial \nu },\frac{\partial v}{\partial \nu }\le 0$, we arrive at

${\int }_{\partial \mathrm{\Omega }}\frac{\partial u}{\partial \nu }\frac{\partial v}{\partial \nu }{|{\nu }_{\lambda }|}^{2}〈T,\nu 〉𝑑S\ge 0.$(3.18)

Then from (3.17) and (3.18) we get

We have

and from (3.16) we get $u\equiv 0$ and hence $v\equiv 0$.

We have

Since $〈T,\nu 〉\ne 0$ on $\partial \mathrm{\Omega }$, we have $\frac{\partial u}{\partial \nu }=0$ or $\frac{\partial v}{\partial \nu }=0$ at some point of $\partial \mathrm{\Omega }$. Moreover, $-{\mathrm{\Delta }}_{\lambda }u,-{\mathrm{\Delta }}_{\lambda }v,u,v\ge 0$ in Ω and $u=v=0$ on $\partial \mathrm{\Omega }$. This implies that $u\equiv 0$ or $v\equiv 0$; hence $u\equiv v\equiv 0$. ∎

## 4 Existence of infinitely many weak solutions

#### Theorem 4.1.

If $p\mathrm{,}q\mathrm{>}\mathrm{1}$, Ω is a smooth and bounded domain in ${\mathrm{R}}^{N}$ and

$\frac{1}{p+1}+\frac{1}{q+1}>\frac{Q-2}{Q},$(4.1)

then problem (1.1) has infinitely many weak solutions.

#### Proof.

We notice that, by the fact that $H\left(-u,-v\right)=H\left(u,v\right)$ for all $\left(u,v\right)\in ℝ×ℝ$, with

$H\left(u,v\right)=\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1},$

the functional $I\left(u,v\right)$ is even, and therefore (Φ5) is satisfied. We will check conditions (Φ1)–(Φ4) in Theorem 2.6, and the proof is divided into four steps.

Step 1: Checking condition (Φ1). By using [3, Remark 2.1], it suffices to check that a ${\left(\mathrm{PS}\right)}_{c}^{\mathcal{ℱ}}$-sequence in E is bounded. This is the content of the following lemma.

Step 2: Checking condition (Φ2). We claim that if $\mathrm{\Phi }\in {C}^{1}\left(E,ℝ\right)$ and (4.1) is satisfied, then there exists a sequence ${r}_{k}>0$, $k=1,2,\mathrm{\dots }$, such that for some $k\ge 2$,

(4.6)

Indeed, for any $z=\left(u,v\right)\in E$ we have

${\int }_{\mathrm{\Omega }}H\left(u,v\right)𝑑x={\int }_{\mathrm{\Omega }}\left(\frac{1}{q+1}{|u|}^{q+1}+\frac{1}{p+1}{|v|}^{p+1}\right)𝑑x$$\le C\left({\parallel u\parallel }_{{L}^{q+1}}^{q+1}+{\parallel v\parallel }_{{L}^{p+1}}^{p+1}+1\right)$$\le C\left({\parallel u\parallel }_{{L}^{q+1}}^{r}+{\parallel v\parallel }_{{L}^{p+1}}^{r}+1\right) \left(r=\mathrm{max}\left\{p+1,q+1\right\}>2\right)$$=C\left({\parallel z\parallel }_{{L}^{q+1}×{L}^{p+1}}^{r}+1\right).$

We prove that

(4.7)

Indeed, since $0<{\mu }_{k+1}\le {\mu }_{k}$, i.e., $\left\{{\mu }_{k}\right\}$ is decreasing and bounded from below, there exists $\mu \ge 0$ such that

On the other hand, for every $k\ge 0$ there exists ${z}_{j}\in E$ such that ${z}_{j}\perp {E}_{{k}_{j}-1}$, ${\parallel {z}_{j}\parallel }_{E}=1$ and

${\parallel {z}_{j}\parallel }_{{L}^{q+1}×{L}^{p+1}}>\frac{{\mu }_{k}}{2}.$

Since E is a Hilbert space, there exist $z\in E$ and a subsequence relabeled $\left\{{z}_{j}\right\}$ such that

that is,

Moreover, for

$\eta =\left(\varphi ,\psi \right)\in E={E}^{s}×{E}^{t}$

we have

$\varphi =\sum _{j=1}^{\mathrm{\infty }}{c}_{j}{e}_{j}\mathit{ }\text{and}\mathit{ }\psi =\sum _{j=1}^{\mathrm{\infty }}{d}_{j}{A}^{s-t}{e}_{j}.$

Thus,

$\left(z,\eta \right)=\underset{j\to \mathrm{\infty }}{lim}\left({z}_{j},\eta \right)$$=\underset{j\to \mathrm{\infty }}{lim}\left[{\left({u}_{j},\varphi \right)}_{{E}^{s}}+{\left({v}_{j},\psi \right)}_{{E}^{t}}\right]$$=\underset{j\to \mathrm{\infty }}{lim}\left[\sum _{k=1}^{\mathrm{\infty }}{c}_{k}\left({u}_{j},{e}_{k}\right)+\sum _{k=1}^{\mathrm{\infty }}{d}_{k}\left({v}_{j},{A}^{s-t}{e}_{k}\right)\right]=0.$

This implies that

and since the embedding $E↪{L}^{q+1}×{L}^{p+1}$ is compact, we get

Therefore, we obtain $\mu =0$, i.e., ${\mu }_{k}\to 0$ as $k\to \mathrm{\infty }$. Thus, (4.7) is proved.

Next, for $z\in {E}^{+}$, $z\perp {E}_{k-1}$, we have

$\mathrm{\Phi }\left(z\right)=\frac{1}{2}{\parallel z\parallel }_{E}^{2}-{\int }_{\mathrm{\Omega }}\left(\frac{{|u|}^{q+1}}{q+1}+\frac{{|v|}^{p+1}}{p+1}\right)𝑑x$$\ge \frac{1}{2}{\parallel z\parallel }_{E}^{2}-C\left({\parallel z\parallel }_{{L}^{q+1}×{L}^{p+1}}^{r}+1\right)$$=\frac{1}{2}{\parallel z\parallel }_{E}^{2}-C{\parallel \frac{z}{{\parallel z\parallel }_{E}}\parallel }_{{L}^{q+1}×{L}^{p+1}}^{r}{\parallel z\parallel }_{E}^{r}-C$$\ge \frac{1}{2}{\parallel z\parallel }_{E}^{2}-C{\mu }_{k}^{r}{\parallel z\parallel }_{E}^{r}-C$$=\left(\frac{1}{2}-C{\mu }_{k}^{r}{\parallel z\parallel }_{E}^{r-2}\right){\parallel z\parallel }_{E}^{2}-C.$

Choosing ${\parallel z\parallel }_{E}={r}_{k}$ with ${r}_{k}={\left(Cr{\mu }_{k}^{r}\right)}^{-1/\left(r-2\right)}>0$, $k\in ℕ$, we have

$\mathrm{\Phi }\left(z\right)\ge \left(\frac{1}{2}-\frac{1}{r}\right){r}_{k}^{2}-C\to +\mathrm{\infty }.$

Therefore, we obtain (4.6) and this implies that condition (Φ2) holds.

Step 3: Checking condition (Φ3). We prove that there exists a sequence ${\alpha }_{k}>0$, $k\in ℕ$, such that (Φ3) is satisfied with ${T}_{k}:={T}_{{\alpha }_{k}}$ and ${R}_{k}:={\alpha }_{k}$.

Indeed, for each $\alpha >0$, we consider ${T}_{\alpha }:E\to E$ being the isomorphism defined by ${T}_{\alpha }=\left({\alpha }^{p}u,{\alpha }^{q}v\right)$. It is easy to see that ${T}_{\alpha }{E}_{n}={E}_{n}$ for all $n=1,2,\mathrm{\dots }$.

First, with $z={z}^{+}+{z}^{-}\in {E}_{k}^{+}\oplus {E}^{-}$, we denote ${z}^{-}={z}_{1}^{-}+{z}_{2}^{-}$, where ${z}_{1}^{-}\in {E}_{k}^{-}$, ${z}_{2}^{-}\perp {E}_{k}^{-}$, and further $\overline{z}={z}^{+}+{z}_{1}^{-}$, $z=\left(u,v\right)$ and ${z}^{±}=\left({u}^{±},{v}^{±}\right)$. Then we have ${u}_{2}^{-}\perp \overline{u}$ in ${L}^{2}$, and hence, using the Hölder inequality and since Ω is bounded and $q+1,p+1>2$, we have

${\parallel \overline{u}\parallel }_{{L}^{2}}^{2}={|\left(u,\overline{u}\right)|}_{{L}^{2}}\le {\parallel u\parallel }_{{L}^{2}}{\parallel \overline{u}\parallel }_{{L}^{2}}\le {\parallel u\parallel }_{{L}^{q+1}}{\parallel \overline{u}\parallel }_{{L}^{q+1}}.$(4.8)

Now, for each $k\in ℕ$, since ${E}_{k}^{s}$ and ${E}_{k}^{t}$ are finite-dimensional subspaces, all norms corresponding on ${E}_{k}^{s}$ and ${E}_{k}^{t}$ are equivalent, so there exist positive constants ${\sigma }_{k}$, ${\sigma }_{k}^{\prime }$, ${\tau }_{k}$, ${\tau }_{k}^{\prime }$ such that

(4.9)

and

(4.10)

Using (4.8)–(4.10), we get

${\parallel u\parallel }_{{L}^{q+1}}\ge {\parallel \overline{u}\parallel }_{{L}^{2}}^{2}{\parallel \overline{u}\parallel }_{{L}^{q+1}}^{-1}\ge {\sigma }_{k}^{2}{\parallel \overline{u}\parallel }_{{E}^{s}}^{2}{\parallel \overline{u}\parallel }_{{L}^{q+1}}^{-1}\ge {\sigma }_{k}^{2}{\sigma }_{k}^{\prime }{\parallel \overline{u}\parallel }_{{E}^{s}}^{2}{\parallel \overline{u}\parallel }_{{E}^{s}}^{-1}={\overline{\sigma }}_{k}{\parallel \overline{u}\parallel }_{{E}^{s}}$(4.11)

and

${\parallel v\parallel }_{{L}^{p+1}}\ge {\overline{\tau }}_{k}{\parallel \overline{v}\parallel }_{{E}^{t}},$(4.12)

where ${\overline{\sigma }}_{k}={\sigma }_{k}^{2}{\sigma }_{k}^{\prime }$ and ${\overline{\tau }}_{k}={\tau }_{k}^{2}{\tau }_{k}^{\prime }$.

For $\overline{u}={u}^{+}+{u}_{1}^{-}$ and $\overline{v}={A}^{s-t}\left({u}^{+}-{u}_{1}^{-}\right),$ one has ${\parallel \overline{u}\parallel }_{{E}^{s}}={\parallel {u}^{+}+{u}_{1}^{-}\parallel }_{{E}^{s}}$ and

${\parallel \overline{v}\parallel }_{{E}^{t}}={\parallel {A}^{t}{A}^{s-t}\left({u}^{+}-{u}_{1}^{-}\right)\parallel }_{{L}^{2}}={\parallel {A}^{s}\left({u}^{+}-{u}_{1}^{-}\right)\parallel }_{{L}^{2}}={\parallel {u}^{+}-{u}_{1}^{-}\parallel }_{{E}^{s}}.$

By the definition of the Hamiltonian function $H\left(u,v\right)$ and by (4.11)–(4.12), for any $\alpha >0$ we have

${\int }_{\mathrm{\Omega }}H\left({T}_{\alpha }z\right)𝑑x={\int }_{\mathrm{\Omega }}H\left({\alpha }^{p}u,{\alpha }^{q}v\right)𝑑x$$\ge {\int }_{\mathrm{\Omega }}\left(\frac{1}{q+1}{\alpha }^{p\left(q+1\right)}{|u|}^{q+1}+\frac{1}{p+1}{\alpha }^{q\left(p+1\right)}{|v|}^{p+1}\right)𝑑x$$\ge c\left({\alpha }^{p\left(q+1\right)}{\parallel u\parallel }_{{L}^{q+1}}^{q+1}+{\alpha }^{q\left(p+1\right)}{\parallel v\parallel }_{{L}^{p+1}}^{p+1}\right)$$\ge c\left({\alpha }^{p\left(q+1\right)}{\overline{\sigma }}_{k}^{q+1}{\parallel \overline{u}\parallel }_{{E}^{s}}^{q+1}+{\alpha }^{q\left(p+1\right)}{\overline{\tau }}_{k}^{p+1}{\parallel \overline{v}\parallel }_{{E}^{t}}^{p+1}\right).$(4.13)

Since

${\parallel {u}^{+}\parallel }_{{E}^{s}}=\frac{1}{2}{\parallel {z}^{+}\parallel }_{{E}^{+}}=\frac{\alpha }{2},$

we see that one of the following two inequalities occurs:

${\parallel {u}^{+}+{u}_{1}^{-}\parallel }_{{E}^{s}}\ge \frac{\alpha }{2} \text{or} {\parallel {u}^{+}-{u}_{1}^{-}\parallel }_{{E}^{s}}\ge \frac{\alpha }{2}.$(4.14)

By (4.13) and (4.14),

${\int }_{\mathrm{\Omega }}H\left({\alpha }^{p}u,{\alpha }^{q}v\right)𝑑x\ge c{\delta }_{k}{\alpha }^{\left(p+1\right)\left(q+1\right)},$

where

${\delta }_{k}=\mathrm{min}\left\{{\left(\frac{{\sigma }_{k}^{2}{\sigma }_{k}^{\prime }}{2}\right)}^{q+1},{\left(\frac{{\tau }_{k}^{2}{\tau }_{k}^{\prime }}{2}\right)}^{p+1}\right\}.$

On the other hand,

$P\left({T}_{\alpha }z\right)={\int }_{\mathrm{\Omega }}{A}^{s}\left({\alpha }^{p}u\right){A}^{t}\left({\alpha }^{q}v\right)𝑑x={\alpha }^{q+p}P\left(z\right)={\alpha }^{q+p}\frac{1}{2}{\parallel z\parallel }_{E}^{2}=\frac{1}{2}{\alpha }^{q+p}\left({\parallel {z}^{+}\parallel }_{{E}^{+}}^{2}-{\parallel {z}^{-}\parallel }_{{E}^{-}}^{2}\right).$(4.15)

Therefore, for ${\parallel {z}^{+}\parallel }_{{E}^{+}}=\alpha$ we obtain

$P\left({T}_{\alpha }z\right)\le \frac{1}{2}{\alpha }^{q+p+2}.$(4.16)

Since

$\mathrm{\Phi }\left({T}_{\alpha }z\right)=P\left({T}_{\alpha }z\right)-{\int }_{\mathrm{\Omega }}H\left({T}_{\alpha }z\right)𝑑x,$

we infer from (4.15) and (4.16) that

$\mathrm{\Phi }\left({T}_{\alpha }z\right)\le \frac{1}{2}{\alpha }^{q+p+2}-{\delta }_{k}{\alpha }^{\left(q+1\right)\left(p+1\right)}.$

Since $p,q>1$ and $\left(q+1\right)\left(p+1\right)>q+p+2$, there exists an ${\alpha }_{0}\left(k\right)>0$ such that for all ${\alpha }_{k}>{\alpha }_{0}\left(k\right)$ one has $\mathrm{\Phi }\left({T}_{{\alpha }_{k}}\right)<0$. Moreover, we also obtain

${\parallel {T}_{\alpha }z\parallel }_{E}={\int }_{\mathrm{\Omega }}{A}^{s}\left({\alpha }^{p}u\right){A}^{t}\left({\alpha }^{q}v\right)𝑑x={\alpha }^{p+q}{\parallel z\parallel }_{E}^{2}\ge {\alpha }^{\mathrm{min}\left\{p,q\right\}}{\parallel z\parallel }_{E}^{2}.$

This implies that, for ${\alpha }_{k}=\mathrm{max}\left\{{\parallel {z}^{+}\parallel }_{{E}^{+}}^{2},{\parallel {z}^{-}\parallel }_{{E}^{-}}^{2}\right\}$,

Thus we can choose ${\alpha }_{k}$ to obtain

$\mathrm{\Phi }\left({T}_{{\alpha }_{k}}z\right)<0\mathit{ }\text{and}\mathit{ }{\parallel {T}_{{\alpha }_{k}}z\parallel }_{E}\ge {r}_{k}$

for ${r}_{k}>0$ is given. Therefore, condition (Φ3) in Theorem 2.6 is satisfied.

Step 4: Checking condition (Φ4). Let U be any bounded subset of E, i.e.,

Then

We have

$|\mathrm{\Phi }\left(u,v\right)|\le {\int }_{\mathrm{\Omega }}|{A}^{s}u{A}^{t}v|𝑑x+{\int }_{\mathrm{\Omega }}\left(\frac{{|v|}^{p+1}}{p+1}+\frac{{|u|}^{q+1}}{q+1}\right)𝑑x$$\le {\parallel {A}^{s}u\parallel }_{{L}^{2}}{\parallel {A}^{t}v\parallel }_{{L}^{2}}+\frac{1}{p+1}{\parallel v\parallel }_{{L}^{p+1}}^{p+1}+\frac{1}{q+1}{\parallel u\parallel }_{{L}^{q+1}}^{q+1}.$

Since ${\parallel {A}^{s}u\parallel }_{{L}^{2}}={\parallel u\parallel }_{{E}^{s}}$, ${\parallel {A}^{t}v\parallel }_{{L}^{2}}={\parallel v\parallel }_{{E}^{t}}$ and the embeddings ${E}^{s}↪{L}^{q+1}$ and ${E}^{t}↪{L}^{p+1}$ are compact, we get

$|\mathrm{\Phi }\left(u,v\right)|\le {\parallel u\parallel }_{{E}^{s}}{\parallel v\parallel }_{{E}^{t}}+c\left({\parallel u\parallel }_{{E}^{s}}^{q+1}+{\parallel v\parallel }_{{E}^{t}}^{p+1}\right).$

This proves condition (Φ4).

The proof of Theorem 4.1 is therefore complete. ∎

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Accepted: 2017-06-09

Published Online: 2017-07-28

Funding Source: National Foundation for Science and Technology Development

Award identifier / Grant number: 101.02-2015.10

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.10.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 661–678, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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