Normalized solutions of Kirchhoff equations with critical and subcritical nonlinearities: the defocusing case

Abstract

In this paper, we study the following problem

$$\begin{aligned} -\left( a + b\int _{{\mathbb {R}}^3}\arrowvert \nabla u\arrowvert ^2 \right) \Delta u =\lambda u +\arrowvert u\arrowvert ^{p-2}u+\mu \arrowvert u\arrowvert ^{q-2}u \,\,\text{ in } {\mathbb {R}}^3, \end{aligned}$$

under the constraint $\int _{{\mathbb {R}}^3}u^2=c^2$, where a, b,  and c are positive constants, $\lambda$ is a real number, $\mu <0$, $2<q<p\le 6$. We study the existence and nonexistence of solution in the subcritical and critical case in the exponent p. The result extend the preview ones for the case $\mu <0$, so called into the literature of defocusing case. To prove the existence of solution we use an appropriate minimax theorem combined with dilations which preserve the $L^2$ norm and with fiber maps. In the critical case, for nonexistence of solution the main tool is the Hadamard three spheres theorem.

1 Introduction

In this paper, we study the following problem

$$\begin{aligned} -\left( a + b\int _{{\mathbb {R}}^3}\arrowvert \nabla u\arrowvert ^2 \right) \Delta u =\lambda u +\arrowvert u\arrowvert ^{p-2}u+\mu \arrowvert u\arrowvert ^{q-2}u \,\,\text{ in } {\mathbb {R}}^3, \end{aligned}$$

(1)

under the constraint

$$\begin{aligned} \int _{{\mathbb {R}}^3}u^2=c^2, \end{aligned}$$

(2)

where a, b,  and c are positive constants; $\lambda$ is a real number; $\mu <0$; $2<q<p\le 6$.

When $b>0$ the problem (1) is called of Kirchhoff model and it has been studied by many authors. It is associated to the following evolution equation

$$\begin{aligned} u_{tt}-\left( a+b\int _{{\mathbb {R}}^3}\arrowvert \nabla u\arrowvert ^2\,dx\right) \Delta u=f(x,u) \quad \text{ in } \Omega \times (0,\infty ), \end{aligned}$$

which was initially introduced by Kirchhoff [15]. Ma and Rivera [18] studied a transmission problem concerning a system of two nonlinear elliptic equations of Kirchhoff type. See also [1, 6, 7, 9,10,11, 17, 19, 26] and references therein.

If the constraint (2) is in place, some usual techniques do not work and additional arguments are necessary to overcome the technical difficulties. As described by Jeanjean [13], (2) has physical motivation and it involves the seek of normalized solutions.

When $b=0$, the problem (1)–(2) was studied by Cazenave and Lions [5], Jeanjean et al. [14], and Soave [22, 23]. In [22], the author considered $N\ge 3$, $\mu \in {\mathbb {R}}$ and $p=\frac{2N}{N-2}$ the critical Sobolev exponent. He studied the existence and properties of ground states with constraint for the following nonlinear Schrödinger equation with combined power nonlinearities

$$\begin{aligned} i\psi _t+\Delta \psi +\mu \arrowvert \psi \arrowvert ^{q-2}\psi +\arrowvert \psi \arrowvert ^{p-2}\psi =0\,\,\text{ in } {\mathbb {R}}^N. \end{aligned}$$

The results of [22] are the first one concerning existence of normalized ground states for the Sobolev critical NLSE in the whole space ${\mathbb {R}}^N$. On the other hand, in [23], Soave studied the subcritical case also with $\mu \in {\mathbb {R}}$. Soave also give new criteria for global existence and finite time blow-up in the associated dispersive equation. In [5] also can be found a motivation to introduce the conditions (2). Jeanjean et al. [14] studied the existence of ground state solution in the case $N\ge 3$, $\mu >0$, $2<q<2+\frac{4}{N}$, and p the critical Sobolev exponent. Finally, we would like to cite the work of Wei and Wu [27] where the authors proved the existence of solutions of mountain-pass type for $N=3$, $2<q<2+\frac{4}{N}$, and p the critical exponent. Wei and Wu also studied the existence and nonexistence of ground states for $2+\frac{4}{N}\le q<2^*$ with $\mu >0$ large. See also Ilyasov [12] where the author proved orbital stability result for physical ground states of a nonlinear Schrödinger equation.

Returning to the Kirchhoff model, i.e., the case $a,b>0$, the existence and asymptotic properties of solutions to (1) with the constraint (2) was studied by Li, Luo, and Yang [16] where the authors considered the case $\mu >0$, the focusing case. The case $\mu =0$ was studied by Ye [29] and Zeng and Zhang [30].

Summarizing, to the best of our knowledge the defocusing case of the problem (1)–(2) was studied only by Soave and with $b=0$. The case $b>0$ was studied by Li, Luo, and Yang [16], but with $\mu >0$. In this paper, we consider the case $b>0$ and $\mu <0$.

We denote by $\Vert \cdot \Vert _p$ the norm in the $L^p({\mathbb {R}}^3)$ space and, in the special case $p=2$, we denote only by $\Vert \cdot \Vert$. We also define the functional $E_{\mu }:S_c\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} E_{\mu }(u)=\frac{a}{2}\Vert \nabla u\Vert ^2+\frac{b}{4}\Vert \nabla u\Vert ^4-\frac{1}{p}\Vert u\Vert _p^p-\frac{\mu }{q}\Vert u\Vert _q^q, \end{aligned}$$

where $S_c$ is the constraint space

$$\begin{aligned} S_c=\{u\in H^1({\mathbb {R}}^3):\, \Vert u\Vert =c\}. \end{aligned}$$

Therefore, the weak solutions of (1) with the constraint (2) can be obtained as critical points of the functional $E_{\mu }$.

It is possible to prove that, if $u\in H^1({\mathbb {R}}^3)$ is a weak solution of (1), then the following Pohozaev identity

$$\begin{aligned} P_{\mu }(u):= a\Vert \nabla u\Vert ^2+b\Vert \nabla u\Vert ^4-\mu \delta _q\Vert u\Vert _q^q-\delta _p\Vert u\Vert _p^p=0. \end{aligned}$$

Thus, the critical points of $E_{\mu }$ is contained in the following Pohozaev set

$$\begin{aligned} {\mathcal {P}}_{c,\mu } = \{u\in S_c:\,P_{\mu }(u)=0\}. \end{aligned}$$

To overcome some difficulties concerning with the convergence of the Palais-Smale sequence of $E_{\mu }$, it is necessary to build a sequence $(u_n)_{n\in {\mathbb {N}}}$ such that

$$\begin{aligned} P_{\mu }(u_n)\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$. Due to the constraint (2), it is necessary to define the dilations

$$\begin{aligned} (s*u)(x)=e^{\frac{3s}{2}}u(e^s x), \end{aligned}$$

which preserve the $L^2$ norm, i.e.,

$$\begin{aligned} \int _{{\mathbb {R}}^3}(s*u)^2=\int _{{\mathbb {R}}^3}u^2, \end{aligned}$$

and it is a continuous map from ${\mathbb {R}}\times H^1({\mathbb {R}}^3)$ into $H^1({\mathbb {R}}^3)$ (see Bartsch and Soave [2]). Therefore, we work with the following fiber maps

$$\begin{aligned} \Psi _u^{\mu }(s)=E_{\mu }(s*u)=\frac{ae^{2s}}{2}\Vert \nabla u\Vert ^2+\frac{be^{4s}}{4}\Vert \nabla u\Vert ^4-\frac{e^{p\delta _p s}}{p}\Vert u\Vert _p^p-\frac{\mu e^{q\delta _q s}}{q}\Vert u\Vert _q^q, \end{aligned}$$

where $\delta _q=\frac{3(q-2)}{2q}$ and $\delta _p=\frac{3(p-2)}{2p}$. The functional $\Psi _u^{\mu }$ allows us to project a function on the Pohozaev set. This ideas also was used by Soave [23] and Li, Luo, and Yang [16].

The main results of the present paper are the following

• If $\mu <0$, $2<q<6$ and $p=6$, we prove that the problem (1) with the constraint (2) does not have solution.

• If $2<q\le \frac{14}{3}<p<6$ are given constants, and $\mu <0$ satisfies an additional assumption, we prove that there exists $\lambda <0$ such that the problem (1) with the constraint (2) has a solution. Moreover, the solution is radially symmetric and a ground state on $S_c$.

The proof of our results combine the techniques used by Soave [23], Li, Luo, and Yang [16] and additional arguments to solve technical problems. As Soave and Li, Luo, and Yang, we also used an appropriate minimax theorem to prove the existence of solution, see Lemma 2.2 below.

Our paper is organized as follows. In Sect. 2 we present the notation and assumptions and we enunciate the main result. In Sect. 3 we prove the result concerning the critical case. Finally, in Sect. 4 we prove the existence of the solution in the subcritical case.

2 Preliminaries and main results

We denote by S the Sobolev constant embedding (see Talenti [25]), i.e., it is the positive constant such that

$$\begin{aligned} S=\inf _{u\in D^{1,2}({\mathbb {R}}^3)\setminus \{0\}}\frac{\Vert \nabla u\Vert ^2}{\Vert u\Vert _6^2}. \end{aligned}$$

It is well known that if $p\in (2,6)$, then it holds

$$\begin{aligned} \Vert u\Vert _p\le C_p\Vert \nabla u\Vert ^{\delta _p}\Vert u\Vert ^{1-\delta _p}, \end{aligned}$$

(3)

for all $u\in H^1({\mathbb {R}}^3)$, where $\delta _p=\frac{3(p-2)}{2p}$, (3) is called of Gagliardo-Nirenberg inequality. See Weinstein [28].

We need the result below which characterize the place of the solutions.

Lemma 2.1

Let $p,q\in (2,6]$ and $\lambda \in {\mathbb {R}}$. If $u\in H^1({\mathbb {R}}^3)$ is a weak solution of (1), then the following Pohozaev identity

$$\begin{aligned} P_{\mu }(u):= a\Vert \nabla u\Vert ^2+b\Vert \nabla u\Vert ^4-\mu \delta _q\Vert u\Vert _q^q-\delta _p\Vert u\Vert _p^p=0. \end{aligned}$$

(4)

Proof

See Jeanjean [13] and Pucci and Serrin [21].

Thus, observing Lemma 2.1, the critical points of $E_{\mu }$ is contained in the following Pohozaev set

$$\begin{aligned} {\mathcal {P}}_{c,\mu } = \{u\in S_c:\,P_{\mu }(u)=0\}. \end{aligned}$$

To look for the solutions of the problem, we split ${\mathcal {P}}_{c,\mu }$ into three disjoint sets

$$\begin{aligned} {\mathcal {P}}_{c,\mu } = {\mathcal {P}}_+^{c,\mu } \cup {\mathcal {P}}_{-}^{c,\mu } \cup {\mathcal {P}}_0^{c,\mu }, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {P}}_+^{c,\mu } = \{u\in {\mathcal {P}}_{c,\mu }:\,(\Psi _u^{\mu })''(0)>0\}, \quad {\mathcal {P}}_{-}^{c,\mu } = \{u\in {\mathcal {P}}_{c,\mu }:\,(\Psi _u^{\mu })''(0)<0\}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {P}}_0^{c,\mu } = \{u\in {\mathcal {P}}_{c,\mu }:\,(\Psi _u^{\mu })''(0)=0\}, \end{aligned}$$

here

$$\begin{aligned} (\Psi _u^{\mu })''(0) = 2a\Vert \nabla u\Vert ^2+4b\Vert \nabla u\Vert ^4-\mu q\delta _q^2\Vert u\Vert _q^q-p\delta _p^2\Vert u\Vert _p^p. \end{aligned}$$

Finally, we would like to enunciate a Lemma which is a minimax principle and it is appropriate to work with problems with constraints. We start with some definitions. Let X be a topological space and B be a closed subset of X. We say that a class F of compact subsets of X is a homotopy-stable family with extended boundary B if for any set A in F and any $\eta \in C([0,1]\times X;X)$ satisfying $\eta (t,x)=x$ for all $(t,x)\in (\{0\}\times X)\cup ([0,1]\times B)$ we have that $\eta (\{1\}\times A)\in F$. $\square$

Lemma 2.2

(Theorem 5.2 of [8]) Let $\Phi$ be a $C^1$ functional on a complete connected $C^1$ Finsler manifold X and consider a homotopy-stable family F with an extended closed boundary B. Set $m=m(\Phi ,F)$ and let F be a closed subset of X satisfying

• F1’) $(A\cap F)\setminus B\ne \emptyset$ for every $A\in F$.

• F2’) $\sup \Phi (B)\le m\le \inf \Phi (F)$. Then, for any sequence of sets $(A_n)_n$ in F such that $\lim _{n} \sup _{A_n} \Phi =m$, there exists a sequence $(x_n)_n$ in X such that

  $$\begin{aligned}&\lim _{n\rightarrow \infty }\Phi (x_n)=m, \quad \lim _{n\rightarrow \infty }\Vert d\Phi (x_n)\Vert =0, \\&\lim _{n\rightarrow \infty }dist(x_n,F)=0, \quad \lim _{n\rightarrow \infty }dist(x_n,A_n)=0. \end{aligned}$$

Now, we can enunciate the main results of the present paper.

Theorem 2.1

(Critical case) Let $\mu <0$, $2<q<6$ and $p=6$ be given constants.

• (i) If u is a critical point for ${E_{\mu }}_{\arrowvert _{S_c}}$ (not necessarily positive), then the associated Lagrange multiplier $\lambda$ is positive, and

  $$\begin{aligned} E_{\mu }(u)\ge a\frac{S\Lambda }{3}+\frac{bS^2\Lambda ^2}{12}, \end{aligned}$$

  (5)

  where

  $$\begin{aligned} \Lambda =\frac{bS^2}{2}+\sqrt{aS+\frac{b^2S^4}{4}}. \end{aligned}$$

  (6)

• (ii) The problem

  $$\begin{aligned} -\left( a + b\int _{{\mathbb {R}}^3}\arrowvert \nabla u\arrowvert ^2 \right) \Delta u = \lambda u +\arrowvert u\arrowvert ^4u +\mu \arrowvert u\arrowvert ^{q-2}u \,\,\text{ in } {\mathbb {R}}^3, \end{aligned}$$

  (7)

  with $u>0$, has no solution $u\in H^1({\mathbb {R}}^3)$, for any $\lambda >0$ and $\mu <0$.

Now, we define the constant $C_0$ by

$$\begin{aligned} C_0= \left( \frac{a}{\delta _pC_p^pc^{p(1-\delta _p)}} \right) ^{\frac{1}{p\delta _p-2}}. \end{aligned}$$

Theorem 2.2

(Subcritical case) Let $2<q\le \frac{14}{3}<p<6$ be given constants. If $\mu <0$ satisfies

$$\begin{aligned} \left( 1-\frac{1}{\delta _p} \right) (a+bC_0^2) C_0^{\frac{2-q\delta _q}{2}} +\mu \left( \frac{\delta _q}{\delta _p}-1 \right) C_q^q c^{q(1-\delta _q)} :=\varepsilon _0<0 \end{aligned}$$

(8)

then $E_{{\mu }_{\arrowvert _{S_c}}}$ has a critical point ${\tilde{u}}$ at positive level $m(c,\mu )=\inf _{u\in {\mathcal {P}}_{c,\mu }}E_{\mu }(u)>0$ satisfying: ${\tilde{u}}$ is radially symmetric, it solves (1) for some ${\tilde{\lambda }}<0$ and it is a ground state of (1) on $S_c$.

3 The critical case

In this section we prove Theorem 2.1. The main tool is the Hadamard three spheres theorem (see [20]).

Proof of Theorem 2.1

Let u be a constrained critical point of $E_{\mu }$ on $S_c$. Thus, u satisfies (1) for some $\lambda \in {\mathbb {R}}$. Multiplying (1) by u and integrating over ${\mathbb {R}}^3$, we have

$$\begin{aligned} \left( a+b\Vert \nabla u\Vert ^2 \right) \Vert \nabla u\Vert ^2 -\mu \Vert u\Vert _q^q -\Vert u\Vert _6^6 -\lambda \Vert u\Vert ^2 =0. \end{aligned}$$

(9)

On the other hand, Lemma 2.1 (with $p=6$) gives us that

$$\begin{aligned} \left( a+b\Vert \nabla u\Vert ^2 \right) \Vert \nabla u\Vert ^2 -\Vert u\Vert _6^6 =\mu \delta _q\Vert u\Vert _q^q. \end{aligned}$$

(10)

Substituting (10) in (9), we have

$$\begin{aligned} \lambda \Vert u\Vert ^2 =\mu (\delta _q-1)\Vert u\Vert _q^q. \end{aligned}$$

(11)

Since $u\in S_c$ (thus $u\ne 0$), $\mu <0$, and $\delta _q<1$, we infer that $\lambda >0$.

Now, we are going to prove the second part of item i). Using Lemma 2.1, we have

$$\begin{aligned} \left( a+b\Vert \nabla u\Vert ^2 \right) \Vert \nabla u\Vert ^2 =\mu \delta _q\Vert u\Vert _q^q +\Vert u\Vert _6^6 \le \Vert u\Vert _6^6, \end{aligned}$$

(12)

where we used that $\mu \delta _q<0$. From (12) and using the Sobolev inequality, we obtain

$$\begin{aligned} \left( a+b\Vert \nabla u\Vert ^2 \right) \Vert \nabla u\Vert ^2 \le S^{-3}\Vert \nabla u\Vert ^6. \end{aligned}$$

(13)

Now, we use the same arguments and notations of [3, 4]. Defining $L_1=a\Vert \nabla u\Vert ^2$ and $L_2=b\Vert \nabla u\Vert ^4$, from (13) we have

$$\begin{aligned} L_1+L_2 \le S^{-3}\left( \frac{L_1}{a}\right) ^3 \,\,\text{ and }\,\, L_1+L_2 \le S^{-3}\left( \frac{L_2}{b}\right) ^{\frac{3}{2}}. \end{aligned}$$

(14)

Thus,

$$\begin{aligned} L_1 \ge aS(L_1+L_2)^{\frac{1}{3}} \,\,\text{ and }\,\, L_2 \ge bS^2(L_1+L_2)^{\frac{2}{3}}. \end{aligned}$$

(15)

Consequently,

$$\begin{aligned} L_1 +L_2 \ge aS(L_1+L_2)^{\frac{1}{3}} + bS^2(L_1+L_2)^{\frac{2}{3}}. \end{aligned}$$

(16)

Denoting by $x=(L_1+L_2)^{\frac{1}{3}}$, we infer

$$\begin{aligned} x^2-bS^2x-aS\ge 0. \end{aligned}$$

(17)

Therefore,

$$\begin{aligned} (L_1 +L_2)^{\frac{1}{3}} =x \ge \Lambda , \end{aligned}$$

(18)

where $\Lambda$ is defined in (6).

On the other hand, observing the definition of $E_{\mu }(u)$ and Lemma 2.1, we have

$$\begin{aligned} E_{\mu }(u)= & {} \frac{a}{2}\Vert \nabla u\Vert ^2 +\frac{b}{4}\Vert \nabla u\Vert ^4 -\frac{\mu }{q} \Vert u\Vert _q^q -\frac{1}{6}\left( a\Vert \nabla u\Vert ^2 +b\Vert \nabla u\Vert ^4 -\mu \delta _q\Vert u\Vert ^q \right) \\= & {} \frac{a}{3}\Vert \nabla u\Vert ^2 +\frac{b}{12}\Vert \nabla u\Vert ^4 +\mu \left( \frac{\delta _q}{6}-\frac{1}{q} \right) \Vert u\Vert _q^q \end{aligned}$$

$$\begin{aligned} > \frac{a}{3}\Vert \nabla u\Vert ^2 +\frac{b}{12}\Vert \nabla u\Vert ^4. \end{aligned}$$

(19)

From this and (15), we infer

$$\begin{aligned} E_{\mu }(u) > \frac{aS}{3}(L_1+L_2)^{\frac{1}{3}} + \frac{bS^2}{12}(L_1+L_2)^{\frac{2}{3}}. \end{aligned}$$

(20)

Therefore, (18) and (20) allow us to conclude that (5) holds.

Now, we are going to prove the item ii). Suppose that u is a solution of (1). Using Brezis-Kato regularity arguments (see Struwe [24]), we have that $u,\arrowvert \nabla u\arrowvert ,\arrowvert \Delta u\arrowvert \in L^{\infty }({\mathbb {R}}^3)$. This and as $u\in L^2({\mathbb {R}}^3)$ allow us to conclude that

$$\begin{aligned} u(x)\rightarrow 0,\, \text{ as } \arrowvert x\arrowvert \rightarrow \infty . \end{aligned}$$

(21)

Thus, there exists $R_0>0$ such that

$$\begin{aligned} \Vert \nabla u\Vert ^2\le 1,\,\,\text{ for } \text{ all } \arrowvert x\arrowvert \ge R_0. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{a+b} \le \frac{1}{a+b\Vert \nabla u\Vert ^2}, \end{aligned}$$

for all $\arrowvert x\arrowvert \ge R_0$. This implies that

$$\begin{aligned} -\Delta u = \frac{1}{a+b\Vert \nabla u\Vert ^2} \left( \lambda u+\arrowvert u\arrowvert ^4u+\mu \arrowvert u\arrowvert ^{q-2}u \right) \ge \frac{\lambda }{a+b} u, \end{aligned}$$

for all $\arrowvert x \arrowvert \ge R_0$. Hence, u is superharmonic at infinity.

Now, we define

$$\begin{aligned} m(r)=\min _{\arrowvert x\arrowvert =r}\{u(x)\}. \end{aligned}$$

Since $u>0$, we have that $m(r)>0$. The Hadamard three spheres theorem (see [20]) gives us that

$$\begin{aligned} m(r) \ge \frac{ m(r_1)\left( \frac{1}{r}-\frac{1}{r_2}\right) +m(r_2)\left( \frac{1}{r_1}-\frac{1}{r}\right) }{ \frac{1}{r_1}-\frac{1}{r_2} }, \end{aligned}$$

for all $R_0<r_1<r<r_2$.

As (21) holds, we have that $m(r_2)\rightarrow 0$, as $r_2\rightarrow \infty$. Moreover, the function $r\mapsto rm(r)$ is monotone non-decreasing for $r>R_0$. Therefore,

$$\begin{aligned} m(r) \ge \frac{m(R_0)R_0}{r}, \end{aligned}$$

for all $\arrowvert x\arrowvert \ge R_0$.

Therefore, we have

$$\begin{aligned} \Vert u\Vert _{H^1({\mathbb {R}}^3)}^3 \ge C\Vert u\Vert _3^3 \ge C\int _{R_0}^{\infty } m(r)^3r^2\,dr \ge C\int _{R_0}^{\infty } \frac{1}{r}\,dr =\infty , \end{aligned}$$

where we used that $H^1({\mathbb {R}}^3)$ is continuously embedding in $L^3({\mathbb {R}}^3)$. This is enough to conclude that the problem does not have solution. $\square$

4 The subcritical case

In this section we prove Theorem 2.2. We start with some lemmas, some of the proofs can be found into the references. Others, due to the presence of Kirchhoff nonlinearity, need a new proof.

Lemma 4.1

For $u\in S_c$ and $s\in {\mathbb {R}}$, the map $\varphi \mapsto s*\varphi$ from $T_u S_c$ to $T_{s*u} S_c$ is a linear isomorphism with inverse $\psi \mapsto (-s)*\psi$, where $T_u S_c=\{\varphi \in S_c:\,\int _{{\mathbb {R}}^3}u\varphi =0\}$.

Proof

See Jeanjean [13]. $\square$

Lemma 4.2

(Compactness of PS sequences) Let $2<q<\frac{14}{3}\le p<6$ be given constants. We suppose that $(u_n)_{n\in {\mathbb {N}}}\subset S_c$ is a PS sequence for $E_{{\mu }_{\arrowvert _{S_c}}}$ at level $c\ne 0$ and it holds

• (i) $P_{\mu }(u_n)\rightarrow 0$, as $n\rightarrow \infty$.

• (ii) $\mu <0$ and (8) is in place.

Then, up to a subsequence, $u_n\rightarrow u$ strongly in $H^1({\mathbb {R}}^3)$, and $u\in S_c$ is a radial solution to (1) for some $\lambda <0$.

Proof

Since $P_{\mu }(u_n)\rightarrow 0$, as $n\rightarrow \infty$, we have

$$\begin{aligned} a\Vert \nabla u_n\Vert ^2+b\Vert \nabla u_n\Vert ^4-\mu \delta _q\Vert u_n\Vert _q^q-\delta _p\Vert u_n\Vert _p^p ={\mathcal {O}}(1), \end{aligned}$$

(22)

as $n\rightarrow \infty$. Thus,

$$\begin{aligned}&c+1 \ge E_{\mu }(u_n) \nonumber \\&\quad =a \left( \frac{1}{2} -\frac{1}{p\delta _p} \right) \Vert \nabla u_n\Vert ^2 +b \left( \frac{1}{4} -\frac{1}{p\delta _p} \right) \Vert \nabla u_n\Vert ^4 -\frac{\mu }{q} \left( 1 -\frac{q\delta _q}{p\delta _p} \right) \Vert u_n\Vert _q^q +{\mathcal {O}}(1) \nonumber \\&\quad \ge a \left( \frac{1}{2} -\frac{1}{p\delta _p} \right) \Vert \nabla u_n\Vert ^2 +b \left( \frac{1}{4} -\frac{1}{p\delta _p} \right) \Vert \nabla u_n\Vert ^4 +{\mathcal {O}}(1). \end{aligned}$$

(23)

From this and since $\Vert u_n\Vert ^2=c$, we infer that $(u_n)$ is bounded in $H^1({\mathbb {R}}^3)$. As $H^1_r({\mathbb {R}}^3)\hookrightarrow L^s({\mathbb {R}}^3)$ compactly, for $s\in (2,6)$, there exists $u\in H^1_r({\mathbb {R}}^3)$ such that

$$\begin{aligned} u_n\rightharpoonup u\,\,\text{ in } H^1_r({\mathbb {R}}^3), \quad u_n\rightarrow u\,\,\text{ in } L^s({\mathbb {R}}^3), \quad \text{ and } \quad u_n\rightarrow u\,\,\text{ a. } \text{ e. } \text{ in } {\mathbb {R}}^3, \end{aligned}$$

(24)

as $n\rightarrow \infty$.

Since $(u_n)$ is a bounded Palais-Smale sequence of $E_{{\mu }_{\arrowvert _{S_c}}}$ we can use the Lagrange multipliers rules to infer that there exist $\lambda _n\in {\mathbb {R}}$ such that

$$\begin{aligned}&a\int _{{\mathbb {R}}^3}\nabla u_n\cdot \nabla \varphi +b\Vert \nabla u_n\Vert ^2\int _{{\mathbb {R}}^3}\nabla u_n\cdot \nabla \varphi -\mu \int _{{\mathbb {R}}^3}\arrowvert u_n\arrowvert ^{q-2}u_n\varphi \nonumber \\&-\int _{{\mathbb {R}}^3}\arrowvert u_n\arrowvert ^{p-2}u_n\varphi -\lambda _n\int _{{\mathbb {R}}^3}u_n\varphi ={\mathcal {O}}(1)\Vert \varphi \Vert _{H^1}, \end{aligned}$$

(25)

for all $\varphi \in H^1({\mathbb {R}}^3)$. Taking $\varphi =u_n$, we obtain

$$\begin{aligned} (a+b\Vert \nabla u_n\Vert ^2)\Vert \nabla u_n\Vert ^2 -\mu \Vert u_n\Vert _q^q -\Vert u_n\Vert _p^p -\lambda _n\Vert u_n\Vert ^2 ={\mathcal {O}}(1)\Vert u_n \Vert _{H^1}. \end{aligned}$$

(26)

Thus,

$$\begin{aligned} \lambda _n =\frac{1}{c^2} \left( (a+b\Vert \nabla u_n\Vert ^2)\Vert \nabla u_n\Vert ^2 -\mu \Vert u_n\Vert _q^q -\Vert u_n\Vert _p^p \right) +{\mathcal {O}}(1)\Vert u_n \Vert _{H^1}. \end{aligned}$$

(27)

As $(u_n)$ is bounded in $H^1({\mathbb {R}}^3)\cap L^p({\mathbb {R}}^3)\cap L^q({\mathbb {R}}^3)$, (27) gives us that $(\lambda _n)$ is a bounded sequence. Therefore, up to subsequence, there exists $\lambda \in {\mathbb {R}}$ such that

$$\begin{aligned} \lambda _n\rightarrow \lambda , \end{aligned}$$

(28)

as $n\rightarrow \infty$.

Now, we are going to prove that $\lambda <0$. Indeed, since $P_{\mu }(u_n)\rightarrow 0$, as $n\rightarrow \infty$, we have

$$\begin{aligned} a\Vert \nabla u_n\Vert ^2+b\Vert \nabla u_n\Vert ^4=\mu \delta _q\Vert u_n\Vert _q^q+\delta _p\Vert u_n\Vert _p^p +{\mathcal {O}}(1) \le \delta _p\Vert u_n\Vert _p^p +{\mathcal {O}}(1). \end{aligned}$$

(29)

Using the Gagliardo-Nirenberg inequality, we have

$$\begin{aligned} a\Vert \nabla u_n\Vert ^2 \le \delta _p\Vert u_n\Vert _p^p +{\mathcal {O}}(1) \le \delta _pC_p^p\Vert u_n\Vert _2^{p(1-\delta _p)}\Vert \nabla u_n\Vert _2^{p\delta _p} +{\mathcal {O}}(1). \end{aligned}$$

(30)

Since $u_n\in S_c$ (thus $\Vert u_n\Vert ^2=c^2$), we have and using the weak lower semi-continuity, we obtain

$$\begin{aligned} B \ge \left( \frac{a}{\delta _pC_p^pc^{p(1-\delta _p)}} \right) ^{\frac{1}{p\delta _p-2}} =C_0, \end{aligned}$$

(31)

where $B=\lim _{n\rightarrow \infty }\Vert \nabla u_n\Vert ^2$. Thus, for n sufficiently large, we have

$$\begin{aligned} \Vert \nabla u_n\Vert ^2 \ge C_0. \end{aligned}$$

(32)

Combining (22) with (27), we have

$$\begin{aligned} \lambda _n = \frac{1}{c^2} \left[ \left( 1-\frac{1}{\delta _p} \right) (a+b\Vert \nabla u_n\Vert ^2)\Vert \nabla u_n\Vert ^2 + \mu \left( \frac{\delta _q}{\delta _p}-1 \right) \Vert u_n\Vert ^q_q \right] +{\mathcal {O}}(1). \end{aligned}$$

(33)

Using the Gagliardo-Nirenberg inequality, we have

$$\begin{aligned} \Vert u_n\Vert ^q_q \le C_q^q\Vert u_n\Vert ^{q(1-\delta _q)}\Vert \nabla u_n\Vert ^{q\delta _q}. \end{aligned}$$

(34)

Since $u_n\in S_c$ (thus $\Vert u_n\Vert _2^2=c^2$), we obtain

$$\begin{aligned} \Vert u_n\Vert ^q_q \le C_q^qc^{q(1-\delta _q)}\Vert \nabla u_n\Vert ^{q\delta _q}. \end{aligned}$$

(35)

Thus, since $1-\frac{1}{\delta _p}<0$, (31), (33), and (35) allow us to infer

$$\begin{aligned} \lambda _n\le & {} \frac{1}{c^2} \Vert \nabla u_n\Vert ^{q\delta _q} \left[ \left( 1-\frac{1}{\delta _p} \right) (a+bC_0) C_0^{\frac{2-q\delta _q}{2}}\right. \nonumber \\&\left. + \mu \left( \frac{\delta _q}{\delta _p}-1 \right) C_q^qc^{q(1-\delta _q)} \right] +{\mathcal {O}}(1). \end{aligned}$$

(36)

Using (32) and observing (8), we obtain

$$\begin{aligned} \lambda _n\le & {} \frac{1}{c^2} C_0^{q\delta _q} \left[ \left( 1-\frac{1}{\delta _p} \right) (a+bC_0) C_0^{\frac{2-q\delta _q}{2}} \right. \nonumber \\&\left. + \mu \left( \frac{\delta _q}{\delta _p}-1 \right) C_q^qc^{q(1-\delta _q)} \right] +{\mathcal {O}}(1). \end{aligned}$$

(37)

Thus, observing assumption (8), we have,

$$\begin{aligned} \lambda _n < \frac{1}{c^2} \varepsilon _0 +{\mathcal {O}}(1), \end{aligned}$$

(38)

for all n sufficiently large. Taking to the limit as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \lambda \le \frac{1}{c^2} \varepsilon _0 <0, \end{aligned}$$

(39)

This gives us that $\lambda <0$, and the claim is proved.

The last step is to prove that $u_n\rightarrow u$ strongly in $H^1({\mathbb {R}}^3)$. Indeed, taking to the limit in (25), we have

$$\begin{aligned}&a\int _{{\mathbb {R}}^3}\nabla u\cdot \nabla \varphi +bB\int _{{\mathbb {R}}^3}\nabla u\cdot \nabla \varphi -\mu \int _{{\mathbb {R}}^3}\arrowvert u\arrowvert ^{q-2}u\varphi \nonumber \\&\quad -\int _{{\mathbb {R}}^3}\arrowvert u\arrowvert ^{p-2}u\varphi -\lambda \int _{{\mathbb {R}}^3}u\varphi =0, \end{aligned}$$

(40)

for all $\varphi \in H^1({\mathbb {R}}^3)$. Combining (25) with (40) and, after this, taking $\varphi =u_n-u$, we obtain

$$\begin{aligned} (a+bB)\Vert \nabla (u_n-u)\Vert ^2-\mu \Vert u_n-u\Vert ^2\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$. Since $\mu <0$, we conclude that $(u_n)$ converges strongly in $H^1({\mathbb {R}}^3)$. $\square$

Lemma 4.3

Let $\mu <0$, and $2<q\le \frac{14}{3}<p<6$ be given constants. Then, ${\mathcal {P}}_0^{c,\mu }=\emptyset$ and ${\mathcal {P}}_{c,\mu }$ is a smooth manifold of codimension 2 in $H^1({\mathbb {R}}^3)$.

Proof

If ${\mathcal {P}}_0^{c,\mu }\ne \emptyset$, then there exists $u\in S_c$ such that

$$\begin{aligned} P_{\mu }(u)=0 \quad \text{ and } \quad (\Psi _{u}^{\mu })''(0)=0. \end{aligned}$$

Thus,

$$\begin{aligned} a\Vert \nabla u\Vert ^2+b\Vert \nabla u\Vert ^4=\mu \delta _q\Vert u\Vert _q^q+\delta _p\Vert u\Vert _p^p \end{aligned}$$

(41)

and

$$\begin{aligned} 2a\Vert \nabla u\Vert ^2+4b\Vert \nabla u\Vert ^4=\mu q\delta _q^2\Vert u\Vert _q^q-p\delta _p^2\Vert u\Vert _p^p. \end{aligned}$$

(42)

Combining (41) with (42), we have

$$\begin{aligned} (p\delta _p-2)a\Vert \nabla u\Vert ^2+(p\delta _b-4)b\Vert \nabla u\Vert ^4=\mu \delta _q(p\delta _p-q\delta _q)\Vert u\Vert _q^q <0. \end{aligned}$$

(43)

Thus,

$$\begin{aligned} \Vert \nabla u\Vert =0, \end{aligned}$$

(44)

from this and returning into (41), we obtain

$$\begin{aligned} \Vert u\Vert _q =0. \end{aligned}$$

(45)

Therefore, $u\equiv 0$, which is a contradiction, because $\Vert u\Vert ^2=c$. To prove that ${\mathcal {P}}_{c,\mu }$ is a smooth manifold of codimension 2 in $H^1({\mathbb {R}}^3)$ is analogously to the one of Lemma 5.2 of Soave [23].

Since ${\mathcal {P}}_0^{c,\mu }=\emptyset$, we observe that ${\mathcal {P}}_{c,\mu }$ is a natural constraint in the following sense $\square$

Lemma 4.4

Let $\mu <0$, and $2<q\le \frac{14}{3}<p<6$ be given constants. If $u\in {\mathcal {P}}_{c,\mu }$ is a critical point for $E_{{\mu }_{\arrowvert _{{\mathcal {P}}_{c,\mu }}}}$, then u is a critical point for $E_{{\mu }_{\arrowvert _{S_c}}}$.

Proof

See Li, Luo, and Yang [16]. $\square$

Fof each $u\in S_c$, we define

$$\begin{aligned} h(t)= \frac{a t^2}{2}\Vert \nabla u\Vert ^2+\frac{bt^4}{4}\Vert \nabla u\Vert ^4-\frac{t^{p\delta _p}}{p}\Vert u\Vert _p^p-\frac{\mu t^{q\delta _q}}{q} \Vert u\Vert _q^q \end{aligned}$$

Lemma 4.5

For every $u\in S_c$, there exists a unique $t_u\in {\mathbb {R}}$ such that $t_u*u\in P_{c,\mu }$. Moreover, $t_u$ is the unique critical point of $\Psi _u$ and it is a strict maximum at positive level. It also hold

• 1) ${\mathcal {P}}_{c,\mu }={\mathcal {P}}_{-}^{c,\mu }$.

• 2) $\Psi _u$ is strictly decreasing and concave on $(t_u,\infty )$ and $t_u<0$ implies that $P_{\mu }(u)<0$.

• 3) The function $u\in S_c\mapsto t_u\in {\mathbb {R}}$ is of class $C^1$.

• 4) If $P_{\mu }(u)<0$, then $t_u<0$.

Proof

For each $u\in S_c$, it holds

$$\begin{aligned} \lim _{s\rightarrow -\infty }\Psi _{u}^{\mu }(s)=0^+ \quad \text{ and } \quad \lim _{s\rightarrow \infty }\Psi _{u}^{\mu }(s)=-\infty . \end{aligned}$$

Thus, $\Psi _{u}^{\mu }$ has a global maximum point $t_u$ at positive level. Moreover, this is the unique critical point of $\Psi _u$. Indeed, since

$$\begin{aligned} \Psi _u(s)=h(e^s) \end{aligned}$$

we have

$$\begin{aligned} \Psi _u'(s)=h'(e^s)e^s. \end{aligned}$$

Therefore, it is enough to study the function h. The derivative of h can be write as

$$\begin{aligned} h'(t)=t^3\eta (t), \end{aligned}$$

where $\eta$ satisfies

$$\begin{aligned} \lim _{t\rightarrow 0^+}\eta (t)=\infty ,\quad \lim _{t\rightarrow \infty }\eta (t)=-\infty ,\quad \text{ and }\,\,\eta '(t)<0,\quad \text{ for } \text{ all } t>0. \end{aligned}$$

This implies that h has a unique critical point ${\tilde{t}}$ in $(0,\infty )$ and $t_u=\ln {\tilde{t}}$. Moreover, since

$$\begin{aligned} \lim _{t\rightarrow 0^+}h(t)=0^+, \quad \text{ and }\quad \lim _{t\rightarrow \infty }h(t)=-\infty \end{aligned}$$

we infer that $h({\tilde{t}})>0$.

Let $u\in {\mathcal {P}}_{c,\mu }$. Thus, $t_u=0$ and as $t_u$ is a maximum point of $\Psi _u^{\mu }$, we have that $(\Psi _u^{\mu })''(0)\le 0$. Since ${\mathcal {P}}_{c,\mu }^0=\emptyset$, we conclude that $(\Psi _u^{\mu })''(0)< 0$. therefore, ${\mathcal {P}}_{c,\mu }={\mathcal {P}}_{-}^{c,\mu }$. From the calculus above, we also infer that $\Psi _u$ is strictly decreasing and concave on $(t_u,\infty )$.

Since $(\Psi _u^{\mu })'(t)<0$ if and only if $t>t_u$, we infer that $P_{\mu }(u)=(\Psi _u^{\mu })'(0)<0$ if and only if $t_u<0$. The item 3) follows of implicit function theorem on the function $\phi (s,u)=(\Psi _u^{\mu })'(s)$. See Lemma 5.3 of Soave [23]. $\square$

Lemma 4.6

It holds

$$\begin{aligned} m(c,\mu )=\inf _{u\in {\mathcal {P}}_{c,\mu }}E_{\mu }(u)>0. \end{aligned}$$

Proof

Taking $u\in {\mathcal {P}}_{c,\mu }$, we can use Lemma 2.1 to infer that

$$\begin{aligned} \delta _p\Vert u\Vert _p^p = a\Vert \nabla u\Vert ^2+b\Vert \nabla u\Vert ^4-\mu \delta _q\Vert u\Vert _q^q. \end{aligned}$$

(46)

Since $\mu <0$, we have

$$\begin{aligned} a\Vert \nabla u\Vert ^2 \le \delta _p\Vert u\Vert _p^p. \end{aligned}$$

(47)

The Gagliardo-Niremberg inequality and (47) give us

$$\begin{aligned} a\Vert \nabla u\Vert ^2 \le \delta _pC_p^p\Vert u\Vert ^{p(1-\delta _p)}\Vert \nabla u\Vert ^{p\delta _p}. \end{aligned}$$

(48)

But $u\in S_c$, thus $\Vert u\Vert _2=c$. Consequently,

$$\begin{aligned} \Vert \nabla u\Vert ^2 \ge \left( \frac{a}{\delta _pC_p^pc^{p(1-\delta _p)}} \right) ^{\frac{1}{p\delta _p-2}} :=C_1. \end{aligned}$$

(49)

Moreover, (46) also gives us

$$\begin{aligned} E_{\mu }(u)= & {} a \left( \frac{1}{2} -\frac{1}{p\delta _p} \right) \Vert \nabla u\Vert ^2 \nonumber \\&+b \left( \frac{1}{4} -\frac{1}{p\delta _p} \right) \Vert \nabla u\Vert ^4 -\frac{\mu }{q} \left( 1 -\frac{q\delta _q}{p\delta _p} \right) \Vert u\Vert _q^q. \end{aligned}$$

(50)

Therefore, from (49) and (50), we conclude that

$$\begin{aligned} m(c,\mu ) \ge a \left( \frac{1}{2} -\frac{1}{p\delta _p} \right) C_1^2 >0. \end{aligned}$$

$\square$

Lemma 4.7

There exists $k>0$ small enough such that

$$\begin{aligned} 0<\sup _{{\overline{A}}_k}E_{\mu }(u)<m(c,\mu ) \end{aligned}$$

and if $u\in {\overline{A}}_k$, then

$$\begin{aligned} E_{\mu }(u)>0\,\,\text{ and }\,\,P_{\mu }(u)>0, \end{aligned}$$

where ${\overline{A}}_k=\{u\in S_c;\,\Vert \nabla u\Vert ^2\le k\}$.

Proof

Using the Gagliardo-Niremberg inequality and observing that $p\delta _p\ge 4$, we obtain

$$\begin{aligned} E_{\mu }(u) = \frac{a}{2} \Vert \nabla u\Vert ^2 + \frac{b}{4} \Vert \nabla u\Vert ^4 -\frac{\mu }{q} \Vert u\Vert _q^q -\frac{1}{p} \Vert u\Vert _p^p \\ \ge \frac{a}{2} \Vert \nabla u\Vert ^2 -\frac{C_p^p}{p} c^{p(1-\delta _p)} \Vert \nabla u\Vert ^{p\delta _p} >0 \end{aligned}$$

and

$$\begin{aligned} P_{\mu }(u) = a \Vert \nabla u\Vert ^2 + b \Vert \nabla u\Vert ^4 -\mu \delta _q \Vert u\Vert _q^q -\delta _p \Vert u\Vert _p^p \\ \ge a \Vert \nabla u\Vert ^2 -C_p^p c^{p(1-\delta _p)} \delta _p\Vert \nabla u\Vert ^{p\delta _p} >0, \end{aligned}$$

if $u\in {\overline{A}}_k$, with k sufficiently small. Moreover, taking k small enough, we also infer that

$$\begin{aligned} E_{\mu }(u) \le \frac{a}{2} \Vert \nabla u\Vert ^2 + \frac{b}{4} \Vert \nabla u\Vert ^4 +\frac{\arrowvert \mu \arrowvert }{q} C_q^q c^{q(1-\delta _q)} \Vert \nabla u\Vert ^{q\delta _q} < m(c,\mu ). \end{aligned}$$

$\square$

We define

$$\begin{aligned} E_{\mu }^c = \{ u\in S_c:\, E_{\mu }(u)\le c \} \end{aligned}$$

and

$$\begin{aligned} \Gamma = \{ \gamma =(\alpha ,\beta )\in C([0,1],{\mathbb {R}}\times S_c^r): \gamma (0)\in (0,{\overline{A}}_k)\,\,\text{ and }\,\, \gamma (1)\in (0,E_{\mu }^0) \}, \end{aligned}$$

where $S_c^r= S_c\cap H^1_{r}$. We also define the minimax level by

$$\begin{aligned} \sigma (c,\mu ) =\inf _{\gamma \in \Gamma } \max _{(s,u)\in \gamma ([0,1])}{\tilde{E}}(s,u), \end{aligned}$$

where

$$\begin{aligned} {\tilde{E}}(s,u)= & {} E_{\mu }(s*u)=\Psi _{u}^{\mu }(s) \\= & {} \frac{ae^{2s}}{2}\Vert \nabla u\Vert ^2+\frac{be^{4s}}{4}\Vert \nabla u\Vert ^4-\frac{e^{sp\delta _p}}{p}\Vert u\Vert _p^p-\frac{\mu e^{sq\delta _q}}{q} \Vert u\Vert _q^q . \end{aligned}$$

Proof of Theorem 2.2

We observe that, it is enough to prove that there exists a PS sequence satisfying the conditions i) and ii) of Lemma 4.2.

Since

$$\begin{aligned} \lim _{s\rightarrow -\infty }\Psi _{u}^{\mu }(s)=0 \quad \text{ and } \quad \lim _{s\rightarrow \infty }\Psi _{u}^{\mu }(s)=-\infty , \end{aligned}$$

there exist $s_1$ and $s_2$ such that

$$\begin{aligned} \Psi _{u}^{\mu }(s)\le k, \quad \text{ for } \text{ all } s<s_1 \end{aligned}$$

and

$$\begin{aligned} \Psi _{u}^{\mu }(s)\le 0, \quad \text{ for } \text{ all } s>s_2. \end{aligned}$$

We define

$$\begin{aligned} \begin{array}{rcl} \gamma _u:[0,1] &{} \rightarrow &{} {\mathbb {R}}\times S_r\\ \tau &{} \mapsto &{} (0,((1-\tau )s_1+\tau s_2)*u, \end{array} \end{aligned}$$

(51)

which is a path in $\Gamma$, thus $\sigma (c,\mu )$ is a real number.

Now, we are going to prove that for all $\gamma \in \Gamma$, there exists $\tau _{\gamma }\in (0,1)$ such that

$$\begin{aligned} \alpha (\tau _{\gamma })*\beta (\tau _{\gamma })\in {\mathcal {P}}_{-}^{c,\mu }. \end{aligned}$$

(52)

Since $\gamma (0)=(0,\beta (0))\in (0,{\overline{A}}_k)$, we have

$$\begin{aligned} t_{0*\beta (0)}=t_{\beta (0)}>0. \end{aligned}$$

Furthermore, since $E(\beta (1))={\tilde{E}}(\gamma (1))\le 0$, we infer

$$\begin{aligned} t_{\alpha (1)*\beta (1)}=t_{\beta (1)}<0. \end{aligned}$$

Moreover, using Lemma 4.5, we have that the function $u\in S_c\mapsto t_u\in {\mathbb {R}}$ is continuous. Thus, there exists $\tau _{\gamma }\in (0,1)$ such that

$$\begin{aligned} t_{\alpha (\tau _{\gamma })*\beta (\tau _{\gamma })}=0. \end{aligned}$$

Consequently, $\alpha (\tau _{\gamma })*\beta (\tau _{\gamma })=t_{\alpha (\tau _{\gamma })*\beta (\tau _{\gamma })}* \Big (\alpha (\tau _{\gamma })*\beta (\tau _{\gamma })\Big )\in {\mathcal {P}}_{c,\mu }={\mathcal {P}}^{c,\mu }_{-}$.

Using (52), we obtain that

$$\begin{aligned} \max _{\gamma ([0,1])}{\tilde{E}} \ge {\tilde{E}}(\gamma (\tau _{\gamma })) = E(\alpha (\tau _{\gamma })*\beta (\tau _{\gamma })) \ge \inf _{{\mathcal {P}}_{-}^{c,\mu }\cap S_r}E. \end{aligned}$$

Thus,

$$\begin{aligned} \sigma (c,u) \ge \inf _{{\mathcal {P}}_{-}^{c,\mu }\cap S_r}E. \end{aligned}$$

(53)

On the other hand, taking $u\in {\mathcal {P}}_{-}^{c,\mu }\cap S_r$ and $\gamma _u$ the corresponding path defined in (51), we have

$$\begin{aligned} E(u)={\tilde{E}}(0,u)=\max _{\gamma _u([0,1])}{\tilde{E}} \ge \sigma (c,u), \end{aligned}$$

from here we infer,

$$\begin{aligned} \inf _{{\mathcal {P}}_{-}^{c,\mu }\cap S_r}E \ge \sigma (c,u). \end{aligned}$$

(54)

The inequalities (53) and (54) give us that

$$\begin{aligned} \inf _{{\mathcal {P}}_{-}^{c,\mu }\cap S_r}E = \sigma (c,u). \end{aligned}$$

(55)

From this and using Lemma 4.6, we have

$$\begin{aligned} m(c,u) = \sigma (c,u). \end{aligned}$$

(56)

We also obtain that

$$\begin{aligned} \sigma (c,u) =m(c,u) >\sup _{({\overline{A}}_k\cup E^0)\cap S_r}E = \sup _{((0,{\overline{A}}_k)\cup (0,E^0))\cap ({\mathbb {R}}\times S_r)}{\tilde{E}}. \end{aligned}$$

(57)

Next task is to use Theorem 5.2 of Ghoussoub [8]. The proof is the same of Soave [23] and Li, Luo, and Yang [16], but to reader convenience we rewrite it here. We consider ${\mathcal {F}}=\{\gamma ([0,1]):\,\gamma \in \Gamma \}$ is a homotopy stable family of compact subsets of ${\mathbb {R}}\times S_r$ with extended boundary $B=(0,{\overline{A}}_k)\cup (0,E^0)$ and F’1 and F’2 of Theorem 5.2 of Ghoussoub holds with the superlevel $\{{\tilde{E}}\ge \sigma (c,\mu )\}$. Thus, taking any minimizing sequence $\{\gamma _n=(\alpha _n,\beta _n)\}\subset \Gamma$ for $\sigma (c,\mu )$ with the property that $\alpha _n\equiv 0$ and $\beta _n(\tau )\ge 0$ a.e. in ${\mathbb {R}}^3$ for every $\tau \in [0,1]$, there exists a Palais-Smale sequence $\{(s_n,w_n)\}\subset {\mathbb {R}}\times S_r$ for ${\tilde{E}}_{{\mu }_{\arrowvert _{{\mathbb {R}}\times S_{c,r}}}}$ at the level $\sigma (c,\mu )>0$ such that

$$\begin{aligned} \partial _s{\tilde{E}}_{\mu }(s_n,w_n)\rightarrow 0 \quad \text{ and } \quad \Vert \partial _u{\tilde{E}}_{\mu }(s_n,w_n)\Vert _{(T_{w_n}S_{c,r})^*}\rightarrow 0, \end{aligned}$$

(58)

as $n\rightarrow \infty$. Moreover,

$$\begin{aligned} \arrowvert s_n\arrowvert +\text{ dist}_{H^1}(w_n,B_n([0,1]))\rightarrow 0, \end{aligned}$$

(59)

as $n\rightarrow \infty$. From (58) we have

$$\begin{aligned} P_{\mu }(s_n*w_n)\rightarrow 0, \end{aligned}$$

(60)

as $n\rightarrow \infty$, and

$$\begin{aligned}&ae^{2s_n}\int _{{\mathbb {R}}^3}\nabla w_n\cdot \nabla \varphi +be^{4s_n}\Vert \nabla w_n\Vert ^2\int _{{\mathbb {R}}^3}\nabla w_n\cdot \nabla \varphi -\mu e^{q\delta _q s_n}\int _{{\mathbb {R}}^3}\arrowvert w_n\arrowvert ^{q-2}w_n\varphi \nonumber \\&\quad -e^{p\delta _p s_n}\int _{{\mathbb {R}}^3}\arrowvert w_n\arrowvert ^{p-2}w_n\varphi ={\mathcal {O}}(1)\Vert \varphi \Vert _{H^1}, \end{aligned}$$

(61)

for all $\varphi \in T_{w_n}S_{c,r}$. But, from (59), $(s_n)$ is bounded. Thus, (61) allows us to infer

$$\begin{aligned} \langle E_{\mu }'(s_n*w_n) , s_n*\varphi \rangle ={\mathcal {O}}(1)\Vert \varphi \Vert _{H^1} ={\mathcal {O}}(1)\Vert s_n*\varphi \Vert _{H^1}, \end{aligned}$$

(62)

for all $\varphi \in T_{w_n}S_{c,r}$. From (62) and using Lemma 4.1 we obtain that

$$\begin{aligned} \{u_n=s_n*w_n\}\subset S_{c,r} \end{aligned}$$

is a Palais-Smale sequence for $E_{{\mu }_{\arrowvert _{S_{c,r}}}}$ at level $\sigma (c,\mu )>0$ with

$$\begin{aligned} P_{\mu }(u_n) \rightarrow 0, \end{aligned}$$

(63)

as $n\rightarrow \infty$. Therefore, the sequence $(u_n)$ satisfies the assumptions of Lemma 4.2 and Theorem 2.2 is proved. $\square$