Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain such that $0\in \mathrm{\Omega }$, $L:=(-\mathrm{\Delta }\cdot -\mu \frac{\cdot }{{|x|}^2})$, $2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $\overline{\mu }:={(\frac{N-2}{2})}^2$ is the best Hardy constant and $H:=H_0^1(\mathrm{\Omega })$ denotes the completion of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with respect to the norm ${\parallel u\parallel }_0={({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx)}^{\frac{1}{2}}$ and $H_{\mu }=H_{\mu }(\mathrm{\Omega })$ is defined as the completion of the $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with respect to the norm defined by ${\parallel u\parallel }_{\mu }={({\int }_{\mathrm{\Omega }}({|\mathrm{\nabla }u|}^2-\mu \frac{u^2}{{|x|}^2})dx)}^{\frac{1}{2}}$ for $\mu <\overline{\mu }$.

Definitions of strongly and weakly coupled terms are as follows.

The terms ${|u|}^{\alpha }$ and ${|v|}^{\beta }$ ($\alpha ,\beta >0$) are weakly coupled, ${|Lu|}^{\alpha }{|Kv|}^{\beta }$ ($\alpha ,\beta >0$) is strongly coupled when L or K is a derivative operator. Thus, ${\eta }_1{|u|}^{{\alpha }_1}{|v|}^{{\beta }_1}+{\eta }_2{|u|}^{{\alpha }_2}{|v|}^{{\beta }_2}$ is strongly coupled when ${\eta }_1$ and ${\eta }_2$ are positive.

The parameters in (1.1) satisfy the following assumption.

(ℋ) $N\ge 3$, $0\le \mu <\overline{\mu }$, $1<q<2$, ${\eta }_1+{\eta }_2>0$, $0\le {\eta }_i<+\mathrm{\infty }$, ${\sigma }_i>0$, ${\alpha }_i,{\beta }_i>1$, ${\alpha }_i+{\beta }_i=2^{\ast }$, $i=1,2$.

Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of $J(u,v)$ [1].

Set ${\alpha }_1={\beta }_1$, ${\alpha }_2={\beta }_2$, ${\sigma }_1={\sigma }_2$ and $u=v$. Then (1.1) reduces to the semilinear scalar problems that have been extensively investigated by many authors. See [4–6] and the references therein.

Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in [7] an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [8–16]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in [12]. In this paper, only the case $1<q<2$ of (1.1) involving multiple strongly-coupled critical terms is considered.

Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters ${\sigma }_1$, ${\sigma }_2$, μ and q are allowable.

Theorem 1.1 Suppose that (ℋ) holds and ${\sigma }_1+{\sigma }_2<{\mathrm{\Upsilon }}_1$. Then problem (1.1) has at least one positive solution.

Theorem 1.2 Suppose that (ℋ) holds, $N>4$, $\mu <\overline{\mu }-{(\frac{N}{q}-\sqrt{\overline{\mu }})}^2$ and $\frac{N}{N-2}<q<2$. Then there exists ${\mathrm{\Upsilon }}_2>0$ such that problem (1.1) has at least two positive solutions for all ${\sigma }_1$ and ${\sigma }_2$ satisfying ${\sigma }_1+{\sigma }_2<{\mathrm{\Upsilon }}_2$.

This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.

${\mathrm{\Lambda }}_1$ denotes the first eigenvalue of the operator L, $\parallel z\parallel ={\parallel z\parallel }_{H\times H}={\parallel (u,v)\parallel }_{H\times H}={({\parallel u\parallel }_H^2+{\parallel v\parallel }_H^2)}^{1/2}$ means the norm of the space $E:=H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })$, $E^{-1}$ is the dual space of E. $tz=t(u,v)=(tu,tv)$ for all $z\in E$ and $t\in \mathbb{R}$. $z=(u,v)$ is said to be nonnegative in Ω if $u\ge 0$ and $v\ge 0$ in Ω. $z=(u,v)$ is said to be positive in Ω if $u>0$ and $v>0$ in Ω. $B_r(0)=\{x\in {\mathbb{R}}^N\mid |x|<r\}$ is a ball in ${\mathbb{R}}^N$. $O({\epsilon }^t)$ denotes a quantity satisfying $|O({\epsilon }^t)|/{\epsilon }^t\le C$, $o({\epsilon }^t)$ means $|o({\epsilon }^t)|/{\epsilon }^t\to 0$ as $\epsilon \to 0$ and $o(1)$ is a generic infinitesimal value. In particular, the quantity $O_1({\epsilon }^t)$ means that there exist the constants $C_1,C_2>0$ such that $C_1{\epsilon }^t\le O_1({\epsilon }^t)\le C_2{\epsilon }^t$ as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.

Lemma 2.1 If $\{z_n\}\subset E$ is a (PS) _c -sequence of J with $z_n\rightharpoonup z$ in E, then $J^{\prime }(z)=0$ and $J(z)\ge F({\tau }_0^{(1)},{\tau }_0^{(2)})$, where

Thus, the proof is complete. □

Lemma 2.2 If $\{z_n\}\subset E$ is a (PS) _c -sequence of the functional J, then $\{z_n\}$ is bounded in E.

Proof See Hsu [[12], Lemma 2.2]. □

which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □

Lemma 3.1 The functional J is coercive and bounded below on ${\mathcal{M}}_{\sigma }$.

Thus, J is coercive and bounded below on ${\mathcal{M}}_{\sigma }$. □

We split ${\mathcal{M}}_{\sigma }$ into three parts:

Lemma 3.2 Suppose that $z\in E$ is a local minimizer of J on ${\mathcal{M}}_{\sigma }$ and $z\notin {\mathcal{M}}_{\sigma }^0$. Then $J^{\prime }(z)=0$ in $E^{-1}$.

Proof The proof is similar to that of [19] and the details are omitted. □

Lemma 3.3 ${\mathcal{M}}_{\sigma }^0=\mathrm{\varnothing }$ for all ${\sigma }_1+{\sigma }_2\in (0,{\mathrm{\Upsilon }}_1)$.

which is a contradiction. □

where $d_0=d_0({\sigma }_1,{\sigma }_2,q,N,{\mathrm{\Lambda }}_1,S_{\eta ,\beta }(\mu ),|\mathrm{\Omega }|)$ is a positive constant. □

$J(t^{+}z)={min}_{0\le t\le t_{\max }}J(tz)$ and $J(t^{-}z)={max}_{t\ge t_{\max }}J(tz)$.

Proof The proof is similar to that of [20] and is omitted. □

Then we have the following lemma.

Proof The proof is almost the same as that in [[20], Lemma 2.7] and is omitted here. □

Proof The proof is similar to that of [21] and is omitted. □

Lemma 4.2 Suppose that ${\sigma }_1+{\sigma }_2\in (0,{\mathrm{\Upsilon }}_1)$. Then J has a minimizer $z^{(1)}\in {\mathcal{M}}_{\sigma }^{+}$ such that $z^{(1)}$ is a positive solution of (1.1) and $J(z^{(1)})={\alpha }_{\sigma }={\alpha }_{\sigma }^{+}<0$.

Proof By Lemma 4.1(i), there exists a (PS) -sequence $\{z_n\}\subset {\mathcal{M}}_{\sigma }$ of J such that

$J(z_n)={\alpha }_{\sigma }+o(1)\text{and}{J}^{\prime }(z_n)=o(1)\text{in }{E}^{-1}.$

(4.1)

Therefore, $z^{(1)}\in {\mathcal{M}}_{\sigma }$ is a nontrivial solution of (1.1).

which is a contradiction. Since $J(z^{(1)})=J(|z^{(1)}|)$ and $|z^{(1)}|\in {\mathcal{M}}_{\sigma }^{+}$, by Lemma 3.2 we may assume that $z^{(1)}$ is a nontrivial nonnegative solution of (1.1).