Nodal solutions to ( p,q ) -Laplacian equations with critical growth

Abstract

In this paper, a class of $(p, q)$ -Laplacian equations with critical growth is taken into consideration: $\begin{matrix} \{\begin{array}{ll} - Δ_{p} u - Δ_{q} u + (| u |^{p - 2} + | u |^{q - 2}) u + λ ϕ | u |^{q - 2} u = μ g (u) + | u |^{q^{*} - 2} u, & x \in R^{3}, \\ - Δ ϕ = | u |^{q}, & x \in R^{3}, \end{array} \end{matrix}$ where $Δ_{ξ} u = div (| \nabla u |^{ξ - 2} \nabla u)$ is the ξ-Laplacian operator $(ξ = p, q)$ , $\frac{3}{2} < p < q < 3$ , λ and μ are positive parameters, $q^{*} = 3 q / (3 - q)$ is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results.

Keywords

Poisson equation Critical growth Variational methods Nodal solutions

1. Introduction and main result

This paper examines whether the following critical $(p, q)$ -Laplacian equation has nodal solutions: $\begin{matrix} (1.1) & \{\begin{array}{ll} - Δ_{p} u - Δ_{q} u + (| u |^{p - 2} + | u |^{q - 2}) u + λ ϕ | u |^{q - 2} u = μ g (u) + | u |^{q^{*} - 2} u, & x \in R^{3}, \\ - Δ ϕ = | u |^{q}, & x \in R^{3}, \end{array} \end{matrix}$ where $Δ_{ξ} u = div (| \nabla u |^{ξ - 2} \nabla u)$ is the ξ-Laplacian operator $(ξ = p, q)$ , $\frac{3}{2} < p < q < 3$ , λ and μ are positive parameters, $q^{*} = 3 q / (3 - q)$ is the Sobolev critical exponent. We assume function $g \in C^{1} (R, R)$ satisfying the following hypotheses:

${lim}_{u \to 0} \frac{| g (u) |}{u^{p - 1}} = 0$ ;

There exists $ζ \in (2 q, q^{*})$ such that ${lim}_{u \to \infty} \frac{| g (u) |}{u^{ζ - 1}} = 0$ ;

$\frac{g (u)}{| u |^{2 q - 1}}$ is a nondecreasing function of $u \in R ∖ {0}$ ;

Problem (1.1) is derived from a general reaction-diffusion system $\begin{matrix} (1.2) & u_{t} = div [D (u) \nabla u] + h (x, u) . \end{matrix}$ This system has a solid physical foundation given that it has been widely applied to several allied sciences, such as solid state physics [20], quantum and plasma physics [33], biophysics [14], and chemical reaction design [4]. In those situations, the function u represents a concentration; the first term on the right side of system (1.2) refers to a diffusion with a (commonly non-constant) diffusion coefficient $D (u)$ , while the second term is the reaction which has to do with source and loss processes and normally has a polynomial form in relation to the concentration u in chemical and biological applications. The case we analyse is stationary solutions to system (1.2) with a diffusion coefficient which has power-like dependence on $D (u) = (| \nabla u |^{p - 2} + | \nabla u |^{q - 2})$ and a reaction term.

When $p = q = 2$ , $Δ_{p} = Δ_{q} = Δ$ is the classical Laplacian operator, and problem (1.1) is reduced to a special form ( $V = 1$ ) of the stationary Schrödinger-Poisson system $\begin{matrix} \{\begin{array}{ll} - Δ u + V (x) u + ϕ u = f (u), & x \in R^{3}, \\ - Δ ϕ = u^{2}, & x \in R^{3}, \end{array} \end{matrix}$ which depicts how quantum particles interact with the electromagnetic field produced by the motion, and is widely studied given its strong physical background. We refer readers interested in the background of such models to [7,24,25], and recall specially the seminal study on bounded domains by Brézis and Nirenberg [8] for the Laplacian, since it leads to several quasilinear extensions.

Another special case of (1.1) is the p-Laplacian problem arising when $p = q \neq 2$ . Du et al. in [12] applied mountain pass theorem to obtain the existence of nontrivial solutions to problem (1.1) in subcritical case for the first time. See for example [1,15,16] for more results on such problems.

In this study, we search for solutions to problem (1.1) for a more general case $p \neq q$ . Interest in this topic is mostly sparked by two factors. One is its physical interpretation in applied sciences, while the other is the challenge from a mathematical perspective, more specifically, the challenge presented by lack of compactness of Sobolev’s embeddings on $R^{3}$ and by nonhomogeneity of $(p, q)$ -Laplacian operator. Via variational methods, some results of problem (1.1) are obtained. Interested readers are referred to [5] for multiple results of subcritical nonlinearity, to [22] for multiple results with high perturbations, to [11] for concentration of positive solutions in critical case, and to [23] for ground state solution in critical case without Ambrosetti-Rabinowitz condition.

However, there are relatively few results of nodal solutions to problem (1.1). Cheng and Wang in [10] attained nodal solutions to the fractional $p & q$ -Laplacian equation with quasicritical growth through constrained variational methods, the quantitative deformation lemma and Brouwer degree theory. Inspired by those works, we wonder whether the same phenomenon of nodal solutions still occurs in critical case, and what if a nonlocal term $ϕ_{u} (x) u$ is involved? The hurdles provoked by the critical $(p, q)$ -Laplacian model with nonlocal term will make this problem very intriguing.

In fact, Wang et al. in [29] employed constrained variational method and the quantitative deformation lemma to establish a least-energy sign-changing solution to Schrödinger-Poisson system with critical growth in $R^{3}$ , which provides some insight into the model we are considering. For more findings of nodal solutions, see [2,3,6,18,21,30,31,35,36] and the references therein.

Considering the above, we take one more step into demonstrating the nodal solutions to problem (1.1) with both critical growth and nonlocal term $ϕ_{u} (x) u$ involved. Our work generalize the existing results.

Prior to outlining our key findings, first define $\begin{matrix} W_{r}^{1, ξ} (R^{3}) : = {u \in W^{1, ξ} (R^{3}) : u (x) = u (| x |), ξ = p, q} \end{matrix}$ with respect to the norm $\begin{matrix} ‖ u ‖_{W_{r}^{1, ξ} (R^{3})} = {(\int_{R^{3}} (| \nabla u |^{ξ} + | u |^{ξ}) d x)}^{\frac{1}{ξ}} . \end{matrix}$ Let W be the subspace of $W_{r}^{1, ξ} (R^{3})$ , defined as $\begin{matrix} W = W_{r}^{1, p} (R^{3}) \cap W_{r}^{1, q} (R^{3}), \end{matrix}$ endowed with the norm $\begin{matrix} ‖ u ‖ = ‖ u ‖_{W_{r}^{1, p} (R^{3})} + ‖ u ‖_{W_{r}^{1, q} (R^{3})} . \end{matrix}$ For any $1 ⩽ t < \infty$ , $L^{t} (R^{3})$ denotes the Lebesgue space with the norm $| u |_{t} : = {(\int_{R^{3}} | u |^{t} d x)}^{\frac{1}{t}}$ . With $1 < η < \infty$ , we can deduce from Banach space $W_{r}^{1, η}$ being separable and reflexive that W is also a separable reflexive Banach space, and we know the embedding $W ↪ L^{ξ} (R^{3})$ is continuous. In particular, the embedding $W ↪ L^{s} (R^{3})$ is compact for $s \in [2 q, q^{*})$ , the best Sobolev constant for which is defined by $\begin{matrix} S = inf_{u \in W ∖ {0}} \frac{‖ u ‖^{q}}{{(\int_{R^{3}} | u |^{q^{*}} d x)}^{\frac{q}{q^{*}}}} . \end{matrix}$

It is easy to check via Lax-Milgram Theorem that for any given $u \in W$ , there exists a unique $ϕ_{u} \in D^{1, 2} (R^{3})$ solving weakly the equation $- Δ ϕ_{u} = | u |^{q}$ , and further, $\begin{matrix} (1.3) & ϕ_{u} (x) = \frac{1}{4 π} \int_{R^{3}} \frac{| u (y) |^{q}}{| x - y |} d y ⩾ 0 . \end{matrix}$ By substituting (1.3) into (1.1), we can give the following definition.

Definition 1.1.
We declare $u \in W$ being a weak solution to problem (1.1), if $\begin{aligned} \int_{R^{3}} (| \nabla u |^{p - 2} \nabla u \nabla φ + | u |^{p - 2} u φ) d x + \int_{R^{3}} (| \nabla u |^{q - 2} \nabla u \nabla φ + | u |^{q - 2} u φ) d x \\ + λ \int_{R^{3}} ϕ_{u} | u |^{q - 2} u φ d x \\ = μ \int_{R^{3}} g (u) φ d x + \int_{R^{3}} | u |^{q^{} - 2} u φ d x, \end{aligned}$ for any $φ \in W$ .

The terminology weak will be removed throughout this paper for convenience. Accordingly, the energy functional $I_{λ}^{μ} : W \to R$ associated with problem (1.1) is defined as $\begin{array}{rcl} I_{λ}^{μ} (u) & = & \frac{1}{p} \int_{R^{3}} (| \nabla u |^{p} + | u |^{p}) d x + \frac{1}{q} \int_{R^{3}} (| \nabla u |^{q} + | u |^{q}) d x \\ + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u} | u |^{q} d x - μ \int_{R^{3}} G (u) d x - \frac{1}{q^{}} \int_{R^{3}} | u |^{q^{}} d x, \end{array}$ where $G (t) = \int_{0}^{t} g (u) d u$ . $I_{λ}^{μ} \in C^{1} (W, R)$ is easily visible, and the critical points of $I_{λ}^{μ}$ are in fact the solutions to problem (1.1). Furthermore, every solution $u \in W$ to problem (1.1) with $u^{\pm} \neq 0$ is a nodal solution, if every u can be written as $\begin{matrix} u^{+} (x) = max {u (x), 0} and u^{-} (x) = min {u (x), 0} . \end{matrix}$

Finding the least energy nodal solutions to problem (1.1) is the primary goal in this paper. Previous research on these problems heavily relies on the following decompositions of the energy functional $I$ : $\begin{array}{c} (1.4) & I (u) = I (u^{+}) + I (u^{-}), \\ (1.5) & ⟨ I^{'} (u), u^{+} ⟩ = ⟨ I^{'} (u^{+}), u^{+} ⟩ and ⟨ I^{'} (u), u^{-} ⟩ = ⟨ I^{'} (u^{-}), u^{-} ⟩ . \end{array}$ With the occurrence of $λ ϕ_{u} (x) u$ , however, it is impossible to accomplish the same decomposition of energy functional $I_{λ}^{μ}$ as in (1.4) and (1.5). In fact, we have $\begin{array}{r} I_{λ}^{μ} (u) = I_{λ}^{μ} (u^{+}) + I_{λ}^{μ} (u^{-}) + \frac{λ}{2 q} \int_{R^{3}} (ϕ_{u^{+}} {| u^{-} |}^{q} + ϕ_{u^{-}} {| u^{+} |}^{q}) d x . \end{array}$ When $u^{+} ≢ 0$ , $\begin{array}{r} ⟨ {(I_{λ}^{μ})}^{'} (u), u^{+} ⟩ = ⟨ {(I_{λ}^{μ})}^{'} (u^{+}), u^{+} ⟩ + λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x > ⟨ {(I_{λ}^{μ})}^{'} (u^{+}), u^{+} ⟩; \end{array}$ when $u^{-} ≢ 0$ , $\begin{array}{r} ⟨ {(I_{λ}^{μ})}^{'} (u), u^{-} ⟩ = ⟨ {(I_{λ}^{μ})}^{'} (u^{-}), u^{-} ⟩ + λ \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x > ⟨ {(I_{λ}^{μ})}^{'} (u^{-}), u^{-} ⟩ . \end{array}$ As a result, the conventional approaches for getting nodal solutions to such problems appear to be inapplicable to problem (1.1). Instead, we use the technique in [6] in this paper. Define the following constrained set $\begin{matrix} N_{λ}^{μ} = {u \in W | ⟨ {(I_{λ}^{μ})}^{'} (u), u^{\pm} ⟩ = 0, u^{\pm} \neq 0}, \end{matrix}$ followed by a study on minimization of $I_{λ}^{μ}$ on $N_{λ}^{μ}$ . The fact that $N_{λ}^{μ} \neq \emptyset$ when the nonlocal term is present has already been establsihed by Shuai [26] using implicit theorem and parametric technique. Nevertheless, the nonhomogeneity of $(p, q)$ -Laplacian operator and the critical nonlinearity involved make it more of a challenge for us to solve the problem. For one thing, the complexity of the decompositions (1.4) and (1.5) corresponding to $I_{λ}^{μ}$ imposes technical barriers to demonstrating that $N_{λ}^{μ}$ is nonempty, whereas the difficulty of the nonlocal problem prevents the application of the implicit theorem and parametric technique to problem (1.1). For another, the loss of compactness induced by the critical exponent in problem (1.1) hinders us from employing the method in [3,9,27] to achieve ${inf}_{u \in N_{λ}^{μ}} I_{λ}^{μ} (u)$ . As a result of being inspired by [2], we take a different path, specifically by applying a changed Miranda’s theorem (see [19]). Moreover, the quantitative deformation lemma and degree theory are crucial in proving the minimizer of the constrained manifold is a nodal solution.

We can now provide our initial key findings.
Theorem 1.1.
With prerequisites* $(g_{1})$ – $(g_{3})$ fulfilled, there exists $μ_{1} > 0$ such that for all $μ ⩾ μ_{1}$ , problem ( 1.1 ) possesses a least energy nodal solution $u_{λ} \in N_{λ}^{μ}$ , and $u_{λ}$ has exactly two nodal domains.

The so-called energy doubling property (cf. [32]) is a further goal of this paper. That is, any nodal solution to problem (1.1) has energy strictly larger than twice the ground state energy.
Theorem 1.2.
With prerequisites $(g_{1})$ – $(g_{3})$ fulfilled, there exists $μ_{2} > 0$ such that for all $μ ⩾ μ_{2}$ , $c^{} : = {inf}_{u \in M_{λ}^{μ}} I_{λ}^{μ} (u) > 0$ is achieved and* $I_{λ}^{μ} (u_{λ}) > 2 c^{}$ . Here* $M_{λ}^{μ} = {u \in W ∖ {0} | ⟨ {(I_{λ}^{μ})}^{'} (u), u ⟩ = 0}$ , and $u_{λ}$ is the least energy nodal solution attained in Theorem 1.1 . Particularly, $c^{} > 0$ is established either by a positive or a negative function.*

The energy of the nodal solution $u_{λ}$ attained in Theorem 1.1 is evidently dependent on λ. In what follows, we demonstrate a convergence feature of $u_{λ}$ as $λ \to 0$ .
Theorem 1.3.
With prerequisites $(g_{1})$ – $(g_{3})$ fulfilled, there exists $μ_{3} > 0$ such that for all $μ ⩾ μ_{3}$ , and for any sequence ${λ_{n}}$ with $λ_{n} \to 0$ as $n \to \infty$ , there exists a subsequence, still denoted by ${λ_{n}}$ , such that ${u_{λ_{n}}} \to u_{0}$ strongly in W as $n \to \infty$ . Here $u_{0}$ is a least energy nodal solution to the following problem $\begin{matrix} (1.6) & - Δ_{p} u - Δ_{q} u + (| u |^{p - 2} + | u |^{q - 2}) u = μ g (u) + | u |^{q^{*} - 2} u in R^{3} . \end{matrix}$

The following is the outline of this paper: the least energy for the constrained problem (1.1) is established in Section 2, and proofs of the key theorems in Section 3.
2. Some technical lemmas

We start this section by introducing some properties of $ϕ_{u}$ .

Proposition 2.1 ([13]).

For any $u \in W_{r}^{1, q} (R^{3})$ , the following properties hold true:

$ϕ_{u} ⩾ 0$ , $\forall x \in R^{3}$ ;

For any $t > 0$ , $ϕ_{t u} = t^{q} ϕ_{u}$ , and $ϕ_{u_{t}} (x) = t^{k q - 2} ϕ_{u} (t x)$ where $u_{t} (x) = t^{k} u (t x)$ ;

There exists $C > 0$ such that $‖ ϕ_{u} ‖_{D^{1, 2}} ⩽ C ‖ u ‖^{q}$ , with C independent of u;

If $u_{n} ⇀ u$ in $W_{r}^{1, q} (R^{3})$ , then $ϕ_{u_{n}} ⇀ ϕ_{u}$ in $D^{1, 2} (R^{3})$ , and $\begin{matrix} \int_{R^{3}} ϕ_{u_{n}} | u_{n} |^{q - 2} u_{n} φ d x \to \int_{R^{3}} ϕ_{u} | u |^{q - 2} u φ d x, \forall φ \in W_{r}^{1, q} (R^{3}) . \end{matrix}$

Then, presume conditions $(g_{1})$ – $(g_{3})$ valid for the function g, unless stated otherwise, and fix $u \in W$ with $u^{\pm} \neq 0$ . Now, we define respectively function $φ : [0, \infty) \times [0, \infty) \to R$ and mapping $D : [0, \infty) \times [0, \infty) \to R^{2}$ as $\begin{matrix} (2.1) & φ (σ, τ) = I_{λ}^{μ} (σ u^{+} + τ u^{-}) \end{matrix}$ and $\begin{matrix} (2.2) & D (σ, τ) = (⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), σ u^{+} ⟩, ⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), τ u^{-} ⟩) . \end{matrix}$

Lemma 2.1.
The following traits apply to φ:
The pair $(σ, τ)$ is a critical point of φ with $σ, τ > 0$ if and only if $σ u^{+} + τ u^{-} \in N_{λ}^{μ}$ ;

There exists a unique pair $(σ_{u}, τ_{u})$ of φ on $R_{+} \times R_{+}$ such that $σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ ;

The unique pair $(σ_{u}, τ_{u})$ is also the maximum point of φ on $[0, \infty) \times [0, \infty)$ ;

$0 < σ_{u}$ , $τ_{u} ⩽ 1$ , if $⟨ {(I_{λ}^{μ})}^{'} (u), u^{\pm} ⟩ ⩽ 0$ .

Proof.
Conclusion (i) is clear by definition of φ.

(ii) First, we establish $σ_{u}$ and $τ_{u}$ exist.

The following inequality is plain from conditions $(g_{1})$ and $(g_{2})$ $\begin{matrix} (2.3) & | g (u) | ⩽ ε | u |^{p - 1} + C_{ε} | u |^{ζ - 1}, \forall u \in R . \end{matrix}$ Choosing $ε > 0$ such that $(1 - μ ε C) > 0$ , via Sobolev Embedding Theorem, we have $\begin{aligned} ⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), σ u^{+} ⟩ \\ = {‖ σ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{σ u^{+} + τ u^{-}} {| σ u^{+} |}^{q} d x - μ \int_{R^{3}} g (σ u^{+}) σ u^{+} d x \\ - \int_{R^{3}} {| σ u^{+} |}^{q^{}} d x \\ ⩾ σ^{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + σ^{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + σ^{q} τ^{q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ - μ ε σ^{p} \int_{R^{3}} {| u^{+} |}^{p} d x - μ C_{ε} σ^{ζ} \int_{R^{3}} {| u^{+} |}^{ζ} d x - σ^{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x \\ ⩾ (1 - μ ε C) σ^{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + σ^{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + σ^{q} τ^{q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x - μ C_{ε} σ^{ζ} \int_{R^{3}} {| u^{+} |}^{ζ} d x - σ^{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x \\ > 0, \end{aligned}$ for σ sufficiently small and all $τ ⩾ 0$ , since $q^{} > ζ > 2 q$ . Analogously, we have $⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), τ u^{-} ⟩ > 0$ , for τ sufficiently small and all $σ ⩾ 0$ .

On the other hand, it is easy to verify from $(g_{1})$ and $(g_{3})$ that $\begin{matrix} (2.4) & g (u) u > 0, u \neq 0; G (u) ⩾ 0, u \in R . \end{matrix}$ It follows that $\begin{aligned} ⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), σ u^{+} ⟩ ⩽ & σ^{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + σ^{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + σ^{q} τ^{q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x - σ^{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x < 0, \end{aligned}$ for σ large enough, and that $⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + τ u^{-}), τ u^{-} ⟩ < 0$ for τ large enough.

To conclude, there exist $r > 0$ and $R > r$ such that for all $σ, τ ⩾ 0$ , $\begin{matrix} (2.5) & ⟨ {(I_{λ}^{μ})}^{'} (r u^{+} + τ u^{-}), r u^{+} ⟩ > 0 and ⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + r u^{-}), r u^{-} ⟩ > 0, \end{matrix}$ while for all $σ, τ \in [r, R]$ , $\begin{matrix} (2.6) & ⟨ {(I_{λ}^{μ})}^{'} (R u^{+} + τ u^{-}), R u^{+} ⟩ < 0 and ⟨ {(I_{λ}^{μ})}^{'} (σ u^{+} + R u^{-}), R u^{-} ⟩ < 0 . \end{matrix}$ It can be inferred from Miranda’s Theorem [19], (2.5) and (2.6) that there exists $(σ_{u}, τ_{u}) \in [0, \infty) \times [0, \infty)$ such that $D (σ_{u}, τ_{u}) = (0, 0)$ , i.e., $σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ .

Second, we prove the pair $(σ_{u}, τ_{u})$ unique in two cases.
Case 1. $u \in N_{λ}^{μ}$ .
In this case, $\begin{matrix} ⟨ {(I_{λ}^{μ})}^{'} (u), u^{+} ⟩ = ⟨ {(I_{λ}^{μ})}^{'} (u), u^{-} ⟩ = 0, \end{matrix}$ namely $\begin{aligned} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ (2.7) & = μ \int_{R^{3}} g (u^{+}) u^{+} d x + \int_{R^{3}} {| u^{+} |}^{q^{}} d x \end{aligned}$ and $\begin{aligned} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{u^{-}} {| u^{-} |}^{q} d x + λ \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x \\ (2.8) & = μ \int_{R^{3}} g (u^{-}) u^{-} d x + \int_{R^{3}} {| u^{-} |}^{q^{}} d x . \end{aligned}$

We know from the first step that there exists at least one positive pair $(σ_{0}, τ_{0})$ such that $σ_{0} u^{+} + τ_{0} u^{-} \in N_{λ}^{μ}$ . In the following, we verify $(σ_{0}, τ_{0}) = (1, 1)$ is the unique pair.

Assume without loss of generality that $σ_{0} ⩽ τ_{0}$ . Then, we have $\begin{aligned} {‖ σ_{0} u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ_{0} u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ_{0}^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + σ_{0}^{2 q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ (2.9) & ⩽ μ \int_{R^{3}} g (σ_{0} u^{+}) σ_{0} u^{+} d x + \int_{R^{3}} {| σ_{0} u^{+} |}^{q^{}} d x \end{aligned}$ and $\begin{aligned} {‖ τ_{0} u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ τ_{0} u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + τ_{0}^{2 q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{-} |}^{q} d x + τ_{0}^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x \\ (2.10) & ⩾ μ \int_{R^{3}} g (τ_{0} u^{-}) τ_{0} u^{-} d x + \int_{R^{3}} {| τ_{0} u^{-} |}^{q^{}} d x . \end{aligned}$ It follows from (2.7)–(2.10) and $(g_{3})$ that $\begin{aligned} (σ_{0}^{p - 2 q} - 1) {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + (σ_{0}^{- q} - 1) {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ (2.11) & ⩽ μ \int_{R^{3}} (\frac{g (σ_{0} u^{+})}{{(σ_{0} u^{+})}^{2 q - 1}} - \frac{g (u^{+})}{{(u^{+})}^{2 q - 1}}) {(u^{+})}^{2 q} d x + (σ_{0}^{q^{} - 2 q} - 1) \int_{R^{3}} {| u^{+} |}^{q^{}} d x \end{aligned}$ and $\begin{aligned} (τ_{0}^{p - 2 q} - 1) {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + (τ_{0}^{- q} - 1) {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ (2.12) & ⩾ μ \int_{R^{3}} (\frac{g (τ_{0} u^{-})}{{(τ_{0} u^{-})}^{2 q - 1}} - \frac{g (u^{-})}{{(u^{-})}^{2 q - 1}}) {(u^{-})}^{2 q} d x + (τ_{0}^{q^{} - 2 q} - 1) \int_{R^{3}} {| u^{-} |}^{q^{}} d x . \end{aligned}$

If $σ_{0} < 1$ , the left side of the inequality (2.11) is positive while the right negative, which is absurd, i.e. $1 ⩽ σ_{0} ⩽ τ_{0}$ . We have, analogously, from (2.12) that $σ_{0} ⩽ τ_{0} ⩽ 1$ . Hence, $σ_{0} = τ_{0} = 1$ .
Case 2. $u \notin N_{λ}^{μ}$ .

Suppose indirectly that there exist two other positive point pairs $(σ_{1}, τ_{1})$ and $(σ_{2}, τ_{2})$ such that $\begin{matrix} γ_{1} = σ_{1} u^{+} + τ_{1} u^{-} \in N_{λ}^{μ} and γ_{2} = σ_{2} u^{+} + τ_{2} u^{-} \in N_{λ}^{μ} . \end{matrix}$ Thus, $\begin{matrix} τ_{2} = (\frac{σ_{2}}{σ_{1}}) σ_{1} u^{+} + (\frac{τ_{2}}{τ_{1}}) τ_{1} u^{-} = (\frac{σ_{2}}{σ_{1}}) τ_{1}^{+} + (\frac{τ_{2}}{τ_{1}}) τ_{1}^{-} \in N_{λ}^{μ} . \end{matrix}$ Evidently, $\begin{matrix} \frac{σ_{2}}{σ_{1}} = \frac{τ_{2}}{τ_{1}} = 1, \end{matrix}$ for $γ_{1} \in N_{λ}^{μ}$ , i.e. $σ_{1} = σ_{2}$ , $τ_{1} = τ_{2}$ .

(iii) Since $(σ_{u}, τ_{u})$ is the unique critical point of φ on $[0, \infty) \times [0, \infty)$ , combining (2.4), we have $\begin{aligned} φ (σ, τ) = & I_{λ}^{μ} (σ u^{+} + τ u^{-}) \\ = & \frac{σ^{p}}{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{σ^{q}}{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{σ^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + \frac{σ^{q} τ^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x + \frac{σ^{q} τ^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x \\ + \frac{τ^{p}}{p} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{τ^{q}}{q} {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{τ^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{-} |}^{q} d x \\ - μ \int_{R^{3}} G (σ u^{+}) d x - μ \int_{R^{3}} G (τ u^{-}) d x - \frac{σ^{q^{}}}{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x - \frac{τ^{q^{}}}{q^{}} \int_{R^{3}} {| u^{-} |}^{q^{}} d x \\ ⩽ & \frac{σ^{p}}{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{σ^{q}}{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{σ^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + \frac{σ^{q} τ^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x + \frac{σ^{q} τ^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x \\ + \frac{τ^{p}}{p} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{τ^{q}}{q} {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{τ^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{-} |}^{q} d x \\ - \frac{σ^{q^{}}}{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x - \frac{τ^{q^{}}}{q^{}} \int_{R^{3}} {| u^{-} |}^{q^{}} d x . \end{aligned}$ That indicates ${lim}_{| (σ, τ) | \to \infty} φ (σ, τ) = - \infty$ , since $q^{} > 2 q$ . Now it is sufficient to illustrate that the maximum point cannot be achieved on the boundary of $[0, \infty) \times [0, \infty)$ .

Assume indirectly $(0, \bar{τ})$ to be the global maximum point of φ with $\bar{τ} ⩾ 0$ , then $\begin{aligned} φ (σ, \bar{τ}) = & \frac{σ^{p}}{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{σ^{q}}{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{σ^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + \frac{σ^{q} {\bar{τ}}^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x + \frac{σ^{q} {\bar{τ}}^{q} λ}{2 q} \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x \\ + \frac{{\bar{τ}}^{p}}{p} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{{\bar{τ}}^{q}}{q} {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{{\bar{τ}}^{2 q} λ}{2 q} \int_{R^{3}} ϕ_{u^{-}} {| u^{-} |}^{q} d x \\ - μ \int_{R^{3}} G (σ u^{+}) d x - μ \int_{R^{3}} G (\bar{τ} u^{-}) d x - \frac{σ^{q^{}}}{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x - \frac{{\bar{τ}}^{q^{}}}{q^{}} \int_{R^{3}} {| u^{-} |}^{q^{}} d x . \end{aligned}$ A simple computation presents $\begin{aligned} φ_{σ}^{'} (σ, \bar{τ}) = & σ^{p - 1} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + σ^{q - 1} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q - 1} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x \\ + \frac{σ^{q - 1} {\bar{τ}}^{q} λ}{2} \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ + \frac{σ^{q - 1} {\bar{τ}}^{q} λ}{2} \int_{R^{3}} ϕ_{u^{+}} {| u^{-} |}^{q} d x - μ \int_{R^{3}} g (σ u^{+}) u^{+} d x - σ^{q^{} - 1} \int_{R^{3}} {| u^{+} |}^{q^{}} d x > 0, \end{aligned}$ when σ is sufficiently small, indicating that function φ in respect of σ is increasing for σ small enough, which is a contradiction. It is likewise impossible for φ to achieve its global maximum at $(\bar{σ}, 0)$ with $\bar{σ} ⩾ 0$ .

(iv) We may assume, without loss of generality, $σ_{u} ⩾ τ_{u} > 0$ . Since $σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ , we have $\begin{aligned} {‖ σ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ ⩾ {‖ σ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + σ^{2 q} λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + σ^{q} τ^{q} λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ (2.13) & = μ \int_{R^{3}} g (σ u^{+}) σ u^{+} d x + \int_{R^{3}} {| σ u^{+} |}^{q^{}} d x . \end{aligned}$ When $⟨ {(I_{λ}^{μ})}^{'} (u), u^{+} ⟩ ⩽ 0$ , $\begin{aligned} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{u^{+}} {| u^{+} |}^{q} d x + λ \int_{R^{3}} ϕ_{u^{-}} {| u^{+} |}^{q} d x \\ (2.14) & ⩽ μ \int_{R^{3}} g (u^{+}) u^{+} d x + \int_{R^{3}} {| u^{+} |}^{q^{}} d x . \end{aligned}$ We can calculate from (2.13) and (2.14) that $\begin{aligned} (σ_{0}^{p - 2 q} - 1) {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + (σ_{0}^{- q} - 1) {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ ⩾ μ \int_{R^{3}} (\frac{g (σ_{0} u^{+})}{{(σ_{0} u^{+})}^{2 q - 1}} - \frac{g (u^{+})}{{(u^{+})}^{2 q - 1}}) {(u^{+})}^{2 q} d x + (σ_{0}^{q^{} - 2 q} - 1) \int_{R^{3}} {| u^{+} |}^{q^{}} d x . \end{aligned}$ It follows from $(g_{3})$ that $σ_{u} ⩽ 1$ . Thus, $0 < σ_{u}$ , $τ_{u} ⩽ 1$ , and the proof is complete. □
Lemma 2.2.
Provided* $c_{λ}^{μ} = {inf}_{u \in N_{λ}^{μ}} I_{λ}^{μ} (u)$ , then ${lim}_{μ \to \infty} c_{λ}^{μ} = 0$ .
Proof.
$⟨ {(I_{λ}^{μ})}^{'} (u), u ⟩ = 0$ holds true for any $u \in N_{λ}^{μ}$ , namely, $\begin{aligned} {‖ u^{\pm} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{\pm} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{u^{\pm}} {| u^{\pm} |}^{q} d x + λ \int_{R^{3}} ϕ_{u^{\mp}} {| u^{\pm} |}^{q} d x \\ = μ \int_{R^{3}} g (u^{\pm}) u^{\pm} d x + \int_{R^{3}} {| u^{\pm} |}^{q^{}} d x, \end{aligned}$ which, combined with (2.3) and Sobolev Embedding Theorem, gives that $\begin{aligned} {‖ u^{\pm} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{\pm} ‖}_{W_{r}^{1, q} (R^{3})}^{q} & ⩽ μ \int_{R^{3}} g (u^{\pm}) u^{\pm} d x + \int_{R^{3}} {| u^{\pm} |}^{q^{}} d x \\ ⩽ μ ε \int_{R^{3}} {| u^{\pm} |}^{p} d x + μ C_{ε} \int_{R^{3}} {| u^{\pm} |}^{ζ} d x + \int_{R^{3}} {| u^{\pm} |}^{q^{}} d x \\ ⩽ μ ε C_{1} {‖ u^{\pm} ‖}^{p} + μ C_{2} {‖ u^{\pm} ‖}^{ζ} + C_{3} {‖ u^{\pm} ‖}^{q^{}} . \end{aligned}$

Similarly to the proof in [28], we can confirm $\begin{matrix} (2.15) & ‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q} ⩾ 2^{1 - q} ‖ u ‖^{q} or ‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q} ⩾ 2^{- p} ‖ u ‖^{p} . \end{matrix}$ Then, it is practical to choose ε small enough such that $\begin{matrix} μ ε C_{1} {‖ u^{\pm} ‖}^{p} ⩽ \frac{1}{2} ({‖ u^{\pm} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{\pm} ‖}_{W_{r}^{1, q} (R^{3})}^{q}), \end{matrix}$ and subsequently, there exists $ρ > 0$ such that $\begin{matrix} (2.16) & {‖ u^{\pm} ‖}^{p} ⩾ ρ or {‖ u^{\pm} ‖}^{q} ⩾ ρ, \end{matrix}$ for all $u \in N_{λ}^{μ}$ , since $q^{}, ζ > p, q$ .

Moreover, we can get with ease from $(g_{3})$ that for $u \neq 0$ $\begin{matrix} (2.17) & G (u) : = g (u) u - 2 q G (u) ⩾ 0, \end{matrix}$ and that function $G (u)$ increases when $u > 0$ , while it decreases when $u < 0$ .

Consequently, by (2.15) and (2.16) $\begin{aligned} I_{λ}^{μ} (u) = & I_{λ}^{μ} (u) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (u), u ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) ‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q} + \frac{μ}{2 q} \int_{R^{3}} [g (u) u - 2 q G (u)] d x \\ + (\frac{1}{2 q} - \frac{1}{q^{}}) \int_{R^{3}} | u |^{q^{}} d x \\ ⩾ & (\frac{1}{p} - \frac{1}{2 q}) ‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q} \\ > & \frac{1}{2 q} (‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q}) > 0, \end{aligned}$ which means $I_{λ}^{μ}$ is bounded below on $N_{λ}^{μ}$ , i.e., $c_{λ}^{μ} = {inf}_{u \in N_{λ}^{μ}} I_{λ}^{μ} (u)$ is well defined.

Fix $u \in W$ with $u^{\pm} \neq 0$ . In view of Lemma 2.1, for any $μ > 0$ , there exist $σ_{μ}, τ_{μ} > 0$ such that $σ_{μ} u^{+} + τ_{μ} u^{-} \in N_{λ}^{μ}$ . Thereupon, $\begin{aligned} 0 ⩽ & c_{λ}^{μ} \\ = & inf_{u \in N_{λ}^{μ}} I_{λ}^{μ} (u) \\ ⩽ & I_{λ}^{μ} (σ_{μ} u^{+} + τ_{μ} u^{-}) \\ ⩽ & \frac{1}{p} {‖ σ_{μ} u^{+} + τ_{μ} u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{q} {‖ σ_{μ} u^{+} + τ_{μ} u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ + \frac{λ}{2 q} \int_{R^{3}} ϕ_{σ_{μ} u^{+} + τ_{μ} u^{-}} {| σ_{μ} u^{+} + τ_{μ} u^{-} |}^{q} d x \\ ⩽ & \frac{σ_{μ}^{p}}{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + \frac{τ_{μ}^{p}}{p} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} \frac{σ_{μ}^{q}}{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ + \frac{τ_{μ}^{q}}{q} {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + \frac{λ C σ_{μ}^{2 q}}{2 q} {‖ u^{+} ‖}^{2 q} + \frac{λ C τ_{μ}^{2 q}}{2 q} {‖ u^{-} ‖}^{2 q} . \end{aligned}$ We only need to demonstrate in the following that $σ_{μ} \to 0$ and $τ_{μ} \to 0$ as $μ \to \infty$ .

Let $D = {(σ_{μ}, τ_{μ}) \in [0, \infty) \times [0, \infty) : D (σ_{μ}, τ_{μ}) = (0, 0), μ > 0}$ , in which D is defined as in (2.2). By (2.3), we have $\begin{aligned} σ_{μ}^{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x + τ_{μ}^{q^{}} \int_{R^{3}} {| u^{-} |}^{q^{}} d x \\ ⩽ σ_{μ}^{q^{}} \int_{R^{3}} {| u^{+} |}^{q^{}} d x + τ_{μ}^{q^{}} \int_{R^{3}} {| u^{-} |}^{q^{}} d x + μ \int_{R^{3}} g (σ_{μ} u^{+}) σ_{μ} u^{+} d x + μ \int_{R^{3}} g (τ_{μ} u^{-}) τ_{μ} u^{-} d x \\ = {‖ σ_{μ} u^{+} + τ_{μ} u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ_{μ} u^{+} + τ_{μ} u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{σ_{μ} u^{+} + τ_{μ} u^{-}} {| σ_{μ} u^{+} + τ_{μ} u^{-} |}^{q} d x \\ ⩽ σ_{μ}^{p} {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + τ_{μ}^{p} {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + σ_{μ}^{q} {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + τ_{μ}^{q} {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} \\ + λ C σ_{μ}^{2 q} {‖ u^{+} ‖}^{2 q} + λ C τ_{μ}^{2 q} {‖ u^{-} ‖}^{2 q}, \end{aligned}$ which suggests $D$ is bounded, since $q^{} > 2 q$ . With ${μ_{n}} \subset (0, \infty)$ set to satisfy $μ_{n} \to \infty$ as $n \to \infty$ , there exist $σ_{0}$ and $τ_{0}$ such that $\begin{matrix} (2.18) & (σ_{μ_{n}}, τ_{μ_{n}}) \to (σ_{0}, τ_{0}) as n \to \infty . \end{matrix}$

Next, we prove $σ_{0} = τ_{0} = 0$ . Arguing indirectly, suppose $σ_{0} > 0$ or $τ_{0} > 0$ . That $σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-} \in N_{λ}^{μ_{n}}$ is equivalent to $\begin{aligned} {‖ σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ \int_{R^{3}} ϕ_{σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}} {| σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-} |}^{q} d x \\ (2.19) & = μ_{n} \int_{R^{3}} g (σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}) (σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}) d x + \int_{R^{3}} {| σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-} |}^{q^{}} d x, \end{aligned}$ for each $n \in N$ . Then from Lebesgue dominated convergence therorem, (2.3), (2.4), and (2.18), we can get $\begin{matrix} \int_{R^{3}} g (σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}) (σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}) d x \to \int_{R^{3}} g (σ_{0} u^{+} + τ_{0} u^{-}) (σ_{0} u^{+} + τ_{0} u^{-}) d x > 0, \end{matrix}$ as $n \to \infty$ . However, it contradicts (2.19), because $μ_{n} \to \infty$ as $n \to \infty$ and ${σ_{μ_{n}} u^{+} + τ_{μ_{n}} u^{-}}$ is bounded in W. Thereupon, $σ_{0} = τ_{0} = 0$ , which is followed by ${lim}_{μ \to \infty} c_{λ}^{μ} = 0$ . □
Lemma 2.3.
There exists* $μ_{1} > 0$ such that the infimum $c_{λ}^{μ}$ is achieved for all $μ ⩾ μ_{1}$ .
Proof.
The definition of $c_{λ}^{μ}$ enables us to set ${u_{n}} \in N_{λ}^{μ}$ such that ${lim}_{n \to \infty} I_{λ}^{μ} (u_{n}) = c_{λ}^{μ}$ , and obviously ${u_{n}}$ is bounded in W. Then, $\begin{aligned} u_{n}^{\pm} ⇀ u^{\pm} in W, \\ u_{n}^{\pm} \to u^{\pm} in L^{s} (R^{3}) for s \in [2 q, q^{}), \\ u_{n}^{\pm} \to u^{\pm} a.e. in R^{3} . \end{aligned}$ Because of Lemma 2.1, we have $\begin{matrix} I_{λ}^{μ} (σ u_{n}^{+} + τ u_{n}^{-}) ⩽ I_{λ}^{μ} (u_{n}), \forall σ, τ ⩾ 0 . \end{matrix}$ Therefore, we can deduce from Brézis–Lieb Lemma, Fatou’s Lemma, and the weak lower semicontinuity of norm that $\begin{aligned} \underset{n \to \infty}{lim inf} I_{λ}^{μ} (σ u_{n}^{+} + τ u_{n}^{-}) \\ ⩾ \frac{σ^{p}}{p} lim_{n \to \infty} ({‖ u_{n}^{+} - u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p}) \\ + \frac{τ^{p}}{p} lim_{n \to \infty} ({‖ u_{n}^{-} - u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p}) \\ + \frac{σ^{q}}{q} lim_{n \to \infty} ({‖ u_{n}^{+} - u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + {‖ u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q}) \\ + \frac{τ^{q}}{q} lim_{n \to \infty} ({‖ u_{n}^{-} - u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + {‖ u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q}) \\ + \frac{σ^{2 q} λ}{2 q} \underset{n \to \infty}{lim inf} \int_{R^{3}} ϕ_{u_{n}^{+}} {| u_{n}^{+} |}^{q} d x + \frac{σ^{q} τ^{q} λ}{2 q} \underset{n \to \infty}{lim inf} \int_{R^{3}} ϕ_{u_{n}^{+}} {| u_{n}^{-} |}^{q} d x - μ \int_{R^{3}} G (σ u^{+}) d x \\ + \frac{τ^{2 q} λ}{2 q} \underset{n \to \infty}{lim inf} \int_{R^{3}} ϕ_{u_{n}^{-}} {| u_{n}^{-} |}^{q} d x + \frac{σ^{q} τ^{q} λ}{2 q} \underset{n \to \infty}{lim inf} \int_{R^{3}} ϕ_{u_{n}^{-}} {| u_{n}^{+} |}^{q} d x - μ \int_{R^{3}} G (τ u^{-}) d x \\ - \frac{σ^{q^{}}}{q^{}} lim_{n \to \infty} ({| u_{n}^{+} - u^{+} |}_{q^{}}^{q^{}} + {| u^{+} |}_{q^{}}^{q^{}}) - \frac{τ^{q^{}}}{q^{}} lim_{n \to \infty} ({| u_{n}^{-} - u^{-} |}_{q^{}}^{q^{}} + {| u^{-} |}_{q^{}}^{q^{}}) \\ ⩾ I_{λ}^{μ} (σ u^{+} + τ u^{-}) + \frac{σ^{p}}{p} A_{p_{1}} + \frac{τ^{p}}{p} A_{p_{2}} + \frac{σ^{q}}{q} A_{q_{1}} + \frac{τ^{q}}{q} A_{q_{2}} - \frac{σ^{q^{}}}{q^{}} B_{1} - \frac{τ^{q^{}}}{q^{}} B_{2}, \end{aligned}$ where $\begin{aligned} A_{p_{1}} = lim_{n \to \infty} {‖ u_{n}^{+} - u^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p}, A_{p_{2}} = lim_{n \to \infty} {‖ u_{n}^{-} - u^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p}, \\ A_{q_{1}} = lim_{n \to \infty} {‖ u_{n}^{+} - u^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q}, A_{q_{2}} = lim_{n \to \infty} {‖ u_{n}^{-} - u^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q}, \\ B_{1} = lim_{n \to \infty} {| u_{n}^{+} - u^{+} |}_{q^{}}^{q^{}}, B_{2} = lim_{n \to \infty} {| u_{n}^{-} - u^{-} |}_{q^{}}^{q^{}} . \end{aligned}$ In other words, $\begin{matrix} (2.20) & c_{λ}^{μ} ⩾ I_{λ}^{μ} (σ u^{+} + τ u^{-}) + \frac{σ^{p}}{p} A_{p_{1}} + \frac{τ^{p}}{p} A_{p_{2}} + \frac{σ^{q}}{q} A_{q_{1}} + \frac{τ^{q}}{q} A_{q_{2}} - \frac{σ^{q^{}}}{q^{}} B_{1} - \frac{τ^{q^{}}}{q^{}} B_{2}, \end{matrix}$ for all $σ, τ ⩾ 0$ . In the following, we conduct our proof in three steps.

First, we confirm $u^{\pm} \neq 0$ .

We will only prove $u^{+} \neq 0$ , since the situation $u^{-} \neq 0$ is similar. Assume indirectly that $u^{+} = 0$ . Set $τ = 0$ in (2.20), and we can get $\begin{matrix} (2.21) & c_{λ}^{μ} ⩾ \frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1} : = ϕ (σ), \forall σ ⩾ 0 . \end{matrix}$
Case 1. $B_{1} = 0$ .

$A_{p_{1}} = A_{q_{1}} = 0$ .

Thanks to (2.15), we have $u_{n}^{+} \to u^{+}$ in W, and we know from (2.16) that $‖ u^{+} ‖ > 0$ , which contradicts our assumption.

Both $A_{p_{1}}$ and $A_{q_{1}}$ are positive.

That $c_{λ}^{μ} ⩾ \frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}}$ is absurd, due to Lemma 2.2.

Either $A_{p_{1}}$ or $A_{q_{1}}$ is positive.
Then, $\begin{matrix} c_{λ}^{μ} ⩾ \frac{σ^{p}}{p} A_{p_{1}} or c_{λ}^{μ} ⩾ \frac{σ^{q}}{q} A_{q_{1}} . \end{matrix}$ Neither is possible by Lemma 2.2.
Case 2. $B_{1} > 0$ .

Denote $S^{} = (\frac{1}{q} - \frac{1}{q^{}}) S^{\frac{q^{}}{q^{} - q}}$ . Then, the definition of S enables that there exists $μ_{1} > 0$ satisfying $c_{λ}^{μ} < S^{}$ , for all $μ > μ^{}$ . Therefore, $\begin{matrix} S^{} ⩽ max_{σ ⩾ 0} {\frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1}} ⩽ max_{σ ⩾ 0} {\frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1}} < S^{}, \end{matrix}$ which is a contradiction, and hence $u^{\pm} \neq 0$ .

Second, we confirm $B_{1} = B_{2} = 0$ .

Since the two situations are similar, we only prove $B_{1} = 0$ . Assume by contradiction that $B_{1} > 0$ .
Case 1. $B_{2} > 0$ .

From Sobolev Embedding Theorem and (2.15), we know $A_{p_{1}}, A_{p_{2}}, A_{q_{1}}, A_{q_{2}} > 0$ . Obviously, $ϕ (σ) > 0$ for σ sufficiently small, while for σ large enough $ϕ (σ) < 0$ , where $ϕ (σ)$ is given by (2.21).

Hence, given the continuity of $ϕ (σ)$ , there exist $\bar{σ} > 0$ and $\bar{τ} > 0$ such that $\begin{matrix} \frac{{\bar{σ}}^{p}}{p} A_{p_{1}} + \frac{{\bar{σ}}^{q}}{q} A_{q_{1}} - \frac{{\bar{σ}}^{q^{}}}{q^{}} B_{1} = max_{σ ⩾ 0} {\frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1}} \end{matrix}$ and $\begin{matrix} \frac{{\bar{τ}}^{p}}{p} A_{p_{2}} + \frac{{\bar{τ}}^{q}}{q} A_{q_{2}} - \frac{{\bar{τ}}^{q^{}}}{q^{}} B_{2} = max_{τ ⩾ 0} {\frac{τ^{p}}{p} A_{p_{2}} + \frac{τ^{q}}{q} A_{q_{2}} - \frac{τ^{q^{}}}{q^{}} B_{2}} . \end{matrix}$ Since $[0, \bar{σ}] \times [0, \bar{τ}]$ is compact and φ is continuous, there exists $(σ_{u}, τ_{u}) \in [0, \bar{σ}] \times [0, \bar{τ}]$ such that $\begin{matrix} φ (σ_{u}, τ_{u}) = max_{(σ, τ) \in [0, \bar{σ}] \times [0, \bar{τ}]} φ (σ, τ) . \end{matrix}$

Now we are able to prove $(σ_{u}, τ_{u}) \in (0, \bar{σ}) \times (0, \bar{τ})$ .

If τ is sufficiently small, it is evident that $\begin{matrix} φ (σ, 0) = I_{λ}^{μ} (σ u^{+}) < I_{λ}^{μ} (σ u^{+}) + I_{λ}^{μ} (τ u^{-}) ⩽ I_{λ}^{μ} (σ u^{+} + τ u^{-}) = φ (σ, τ), \end{matrix}$ for all $σ \in [0, \bar{σ}]$ . Hence, there exists $τ_{0} \in [0, \bar{τ}]$ such that $φ (σ, 0) ⩽ φ (σ, τ_{0})$ for all $σ \in [0, \bar{σ}]$ , i.e., $(σ_{u}, τ_{u}) \notin [0, \bar{σ}] \times {0}$ . Analogously, we can obtain $(σ_{u}, τ_{u}) \notin {0} \times [0, \bar{τ}]$ .

On the other hand, $\begin{array}{r} \frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1} > 0, σ \in (0, \bar{σ}] \end{array}$ and $\begin{array}{r} \frac{τ^{p}}{p} A_{p_{2}} + \frac{τ^{q}}{q} A_{q_{2}} - \frac{τ^{q^{}}}{q^{}} B_{2} > 0, τ \in (0, \bar{τ}] . \end{array}$ Then, for all $σ \in [0, \bar{σ}]$ and all $τ \in [0, \bar{τ}]$ $\begin{aligned} (2.22) & S^{} ⩽ \frac{{\bar{σ}}^{p}}{p} A_{p_{1}} + \frac{{\bar{σ}}^{q}}{q} A_{q_{1}} - \frac{{\bar{σ}}^{q^{}}}{q^{}} B_{1} + \frac{τ^{p}}{p} A_{p_{2}} + \frac{τ^{q}}{q} A_{q_{2}} - \frac{τ^{q^{}}}{q^{}} B_{2} \end{aligned}$ and $\begin{aligned} (2.23) & S^{} ⩽ \frac{{\bar{τ}}^{p}}{p} A_{p_{2}} + \frac{{\bar{τ}}^{q}}{q} A_{q_{2}} - \frac{{\bar{τ}}^{q^{}}}{q^{}} B_{2} + \frac{σ^{p}}{p} A_{p_{1}} + \frac{σ^{q}}{q} A_{q_{1}} - \frac{σ^{q^{}}}{q^{}} B_{1} . \end{aligned}$

Due to (2.20), we have $φ (\bar{σ}, τ) ⩽ 0$ , $φ (σ, \bar{τ}) ⩽ 0$ for all $σ \in [0, \bar{σ}]$ and all $τ \in [0, \bar{τ}]$ . Therefore, $(σ_{u}, τ_{u}) \notin {\bar{σ}} \times [0, \bar{τ}]$ and $(σ_{u}, τ_{u}) \notin [0, \bar{σ}] \times {\bar{τ}}$ , that is $(σ_{u}, τ_{u}) \in (0, \bar{σ}) \times (0, \bar{τ})$ . It follows from Lemma 2.1 that $(σ_{u}, τ_{u})$ is a critical point of φ, so $σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ .

Then by (2.20), (2.22), and (2.23) $\begin{aligned} c_{λ}^{μ} & ⩾ I_{λ}^{μ} (σ_{u} u^{+} + τ_{u} u^{-}) + \frac{σ_{u}^{p}}{p} A_{p_{1}} + \frac{τ_{u}^{p}}{p} A_{p_{2}} + \frac{σ_{u}^{q}}{q} A_{q_{1}} + \frac{τ_{u}^{q}}{q} A_{q_{2}} - \frac{σ_{u}^{q^{}}}{q^{}} B_{1} - \frac{τ_{u}^{q^{}}}{q^{}} B_{2} \\ > I_{λ}^{μ} (σ_{u} u^{+} + τ_{u} u^{-}) ⩾ c_{λ}^{μ}, \end{aligned}$ which is a contradiction, so $B_{1} = 0$ .
Case 2. $B_{2} = 0$ .

In this case, it is possible to maximize in $[0, \bar{σ}] \times [0, \infty)$ . In fact, there exists $τ_{0} \in [0, \infty)$ satisfying $\begin{matrix} I_{λ}^{μ} (σ u^{+} + τ u^{-}) ⩽ 0, \forall (σ, τ) \in [0, \bar{σ}] \times [τ_{0}, \infty) . \end{matrix}$ Therefore, there is $(σ_{u}, τ_{u}) \in [0, \bar{σ}] \times [0, \infty)$ such that $\begin{matrix} φ (σ_{u}, τ_{u}) = max_{(σ, τ) \in [0, \bar{σ}] \times [0, \infty)} φ (σ, τ) . \end{matrix}$

Now, we are able to prove $(σ_{u}, τ_{u}) \in (0, \bar{σ}) \times (0, \infty)$ .

Note that $φ (σ, 0) < φ (σ, τ)$ for $σ \in [0, \bar{σ}]$ and τ sufficiently small, while $φ (0, τ) < φ (σ, τ)$ for $τ \in [0, \infty)$ and σ small enough. That indicates $(σ_{u}, τ_{u}) \notin [0, \bar{σ}] \times {0}$ and $(σ_{u}, τ_{u}) \notin {0} \times [0, \infty)$ .

However, $\begin{matrix} (2.24) & S^{} ⩽ \frac{{\bar{σ}}^{p}}{p} A_{p_{1}} + \frac{{\bar{σ}}^{q}}{q} A_{q_{1}} - \frac{{\bar{σ}}^{q^{}}}{q^{}} B_{1} + \frac{τ^{p}}{p} A_{p_{2}} + \frac{τ^{q}}{q} A_{q_{2}}, \end{matrix}$ for all $τ \in [0, \infty)$ .

Thereupon, we have $φ (\bar{σ}, τ) ⩽ 0$ for all $τ \in [0, \infty)$ , which implies $(σ_{u}, τ_{u}) \notin {\bar{σ}} \times [0, \infty)$ . The above suggests $(σ_{u}, τ_{u}) \in (0, \bar{σ}) \times (0, \infty)$ , i.e., $(σ_{u}, τ_{u})$ is an inner maximizer of φ in $[0, \bar{σ}) \times [0, \infty)$ . Hence, $σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ . Then by (2.20) and (2.24), we can get $\begin{aligned} c_{λ}^{μ} & ⩾ I_{λ}^{μ} (σ_{u} u^{+} + τ_{u} u^{-}) + \frac{σ_{u}^{p}}{p} A_{p_{1}} + \frac{τ_{u}^{p}}{p} A_{p_{2}} + \frac{σ_{u}^{q}}{q} A_{q_{1}} + \frac{τ_{u}^{q}}{q} A_{q_{2}} - \frac{σ_{u}^{q^{}}}{q^{}} B_{1} \\ > I_{λ}^{μ} (σ_{u} u^{+} + τ_{u} u^{-}) ⩾ c_{λ}^{μ}, \end{aligned}$ which is a contradiction. Therefore, $B_{1} = B_{2} = 0$ .

Last, we confirm $c_{λ}^{μ}$ is achieved.

From Lemma 2.1, we know there exist $σ_{u}, τ_{u} > 0$ such that $\hat{u} : = σ_{u} u^{+} + τ_{u} u^{-} \in N_{λ}^{μ}$ . It is noticeable that $⟨ {(I_{λ}^{μ})}^{'} (u), u^{\pm} ⟩ ⩽ 0$ . Using Lemma 2.1 again, we can have $0 < σ_{u}$ , $τ_{u} ⩽ 1$ .

Since $u_{n} \in N_{λ}^{μ}$ , $\begin{matrix} I_{λ}^{μ} (σ_{u} u_{n}^{+} + τ_{u} u_{n}^{-}) ⩽ I_{λ}^{μ} (u_{n}^{+} + u_{n}^{-}) = I_{λ}^{μ} (u_{n}) . \end{matrix}$

Considering $B_{1} = B_{2} = 0$ and the semicontinuity of the norm, we have $\begin{aligned} c_{λ}^{μ} ⩽ & I_{λ}^{μ} (\hat{u}) \\ = & I_{λ}^{μ} (\hat{u}) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (\hat{u}), \hat{u} ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) ‖ \hat{u} ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ \hat{u} ‖_{W_{r}^{1, q} (R^{3})}^{q} \\ + \frac{μ}{2 q} \int_{R^{3}} [g (\hat{u}) \hat{u} - 2 q G (\hat{u})] d x + (\frac{1}{2 q} - \frac{1}{q^{}}) \int_{R^{3}} | \hat{u} |^{q^{}} d x \\ ⩽ & (\frac{1}{p} - \frac{1}{2 q}) ‖ u ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ u ‖_{W_{r}^{1, q} (R^{3})}^{q} \\ + \frac{μ}{2 q} \int_{R^{3}} [g (u) u - 2 q G (u)] d x + (\frac{1}{2 q} - \frac{1}{q^{}}) \int_{R^{3}} | u |^{q^{}} d x \\ ⩽ & \underset{n \to \infty}{lim inf} [I_{λ}^{μ} (u_{n}) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (u_{n}), u_{n} ⟩] \\ = & \underset{n \to \infty}{lim inf} I_{λ}^{μ} (u_{n}) ⩽ c_{λ}^{μ} . \end{aligned}$ Thus, we are able to announce $σ = τ_{u} = 1$ , and $c_{λ}^{μ}$ is achieved by $u_{λ} = u^{+} + u^{-} \in N_{λ}^{μ}$ . □

3. Proof of Theorems

Proof of Theorem 1.1.

Thanks to Lemma 2.3, we are able to announce the minimizer $u_{λ}$ for $c_{λ}^{μ}$ is a nodal solution to problem (1.1), indeed.

That $u_{λ} \in N_{λ}^{μ}$ is equivalent to $⟨ {(I_{λ}^{μ})}^{'} (u_{λ}), u_{λ}^{+} ⟩ = ⟨ {(I_{λ}^{μ})}^{'} (u_{λ}), u_{λ}^{-} ⟩ = 0$ . In view of Lemma 2.1, we can obtain $\begin{matrix} (3.1) & I_{λ}^{μ} (σ u_{λ}^{+} + τ u_{λ}^{-}) < I_{λ}^{μ} (u_{λ}^{+} + u_{λ}^{-}) = c_{λ}^{μ}, \end{matrix}$ for $(σ, τ) \in (R_{+} \times R_{+}) ∖ (1, 1)$ .

We will prove in the following indirectly by supposing ${(I_{λ}^{μ})}^{'} (u_{λ}) \neq 0$ . Then there exist $δ > 0$ and $γ > 0$ such that $\begin{matrix} ‖ {(I_{λ}^{μ})}^{'} (v) ‖ ⩾ γ for all ‖ v - u_{λ} ‖ ⩽ 3 δ . \end{matrix}$ Choose $β \in (0, min {1 / 2, \frac{δ}{\sqrt{2} ‖ u_{λ} ‖}})$ , and $\begin{aligned} T : = (1 - β, 1 + β) \times (1 - β, 1 + β), \\ f (σ, τ) : = σ u_{λ}^{+} + τ u_{λ}^{-} for all (σ, τ) \in T . \end{aligned}$ By (3.1), we can get $\begin{matrix} (3.2) & {\tilde{c}}_{λ}^{μ} : = max_{\partial T} (I_{λ}^{μ} \circ f) < c_{λ}^{μ} . \end{matrix}$ Let $ε : = min {(c_{λ}^{μ} - {\tilde{c}}_{λ}^{μ}) / 2, γ δ / 8}$ and $S_{δ} : = B (u_{λ}, δ)$ . Lemma 2.3 in [34] guarantees that there exists a deformation $η \in C ([0, 1] \times W, W)$ such that

$η (1, v) = v$ if $v \notin {(I_{λ}^{μ})}^{- 1} ([c_{λ}^{μ} - 2 ε, c_{λ}^{μ} + 2 ε] \cap S_{2 δ})$ ,

$η (1, {(I_{λ}^{μ})}^{c_{λ}^{μ} + ε} \cap S_{δ}) \subset {(I_{λ}^{μ})}^{c_{λ}^{μ} - ε}$ ,

$I_{λ}^{μ} (η (1, v)) ⩽ I_{λ}^{μ} (v)$ for all $v \in W$ .

It is easy to check that $\begin{matrix} (3.3) & max_{(σ, τ) \in \bar{T}} I_{λ}^{μ} (η (1, f (σ, τ))) < c_{λ}^{μ} . \end{matrix}$

Now, we claim $η (1, f (T)) \cap N_{λ}^{μ} \neq \emptyset$ to contradict the definition of $c_{λ}^{μ}$ . Above all, define $\begin{matrix} h (σ, τ) : = η (1, f (σ, τ)) \end{matrix}$ and $\begin{aligned} Φ_{1} (σ, τ) : = (⟨ {(I_{λ}^{μ})}^{'} (f (σ, τ)), u_{λ}^{+} ⟩, ⟨ {(I_{λ}^{μ})}^{'} (f (σ, τ)), u_{λ}^{-} ⟩) \\ = (⟨ {(I_{λ}^{μ})}^{'} (σ u_{λ}^{+} + τ u_{λ}^{-}), u_{λ}^{+} ⟩, ⟨ {(I_{λ}^{μ})}^{'} (σ u_{λ}^{+} + τ u_{λ}^{-}), u_{λ}^{-} ⟩), \\ Φ_{2} (σ, τ) : = (\frac{1}{σ} ⟨ {(I_{λ}^{μ})}^{'} (h (σ, τ)), {(h (σ, τ))}^{+} ⟩, \frac{1}{τ} ⟨ {(I_{λ}^{μ})}^{'} (h (σ, τ)), {(h (σ, τ))}^{-} ⟩) . \end{aligned}$

Similarly to the proof in [18], we can get from the degree theory that $deg (Φ_{1}, T, 0) = 1$ , and from (3.2) that $f (σ, τ) = h (σ, τ)$ on $\partial T$ . Thus, $deg (Φ_{2}, T, 0) = deg (Φ_{1}, T, 0) = 1$ is easily visible.

Then for some $(σ_{0}, τ_{0}) \in T$ , $Φ_{2} (σ_{0}, τ_{0}) = 0$ , so that $η (1, f (σ_{0}, τ_{0})) = h (σ_{0}, τ_{0}) \in N_{λ}^{μ}$ , which is evidently contradictory to (3.3). Therefore, ${(I_{λ}^{μ})}^{'} (u_{λ}) = 0$ , meaning $u_{λ}$ is a critical point of $I_{λ}^{μ}$ . Accordingly, $u_{λ}$ is a nodal solution to problem (1.1).

Eventually, we proceed with our proof to verify $u_{λ}$ possesses precisely two nodal domains.

Suppose indirectly that $\begin{aligned} u_{λ} = u_{1} + u_{2} + u_{3}; u_{i} \neq 0, u_{1} ⩾ 0, u_{2} ⩽ 0 \\ suppt (u_{i}) \cap suppt (u_{j}) = \emptyset; i \neq j, i, j = 1, 2, 3 \\ ⟨ {(I_{λ}^{μ})}^{'} (u_{λ}), u_{i} ⟩ = 0; i = 1, 2, 3 . \end{aligned}$

Denote $v : = u_{1} + u_{2}$ . Obviously, $v^{+} = u_{1}$ and $v^{-} = u_{2}$ , i.e., $v^{\pm} \neq 0$ . Then there exists a unique pair $(σ_{v}, τ_{v}) > 0$ such that $σ_{v} u_{1} + τ_{v} u_{2} \in N_{λ}^{μ}$ , from which we have $I_{λ}^{μ} (σ_{v} u_{1} + τ_{v} u_{2}) ⩾ c_{λ}$ .

Furthermore, we can obtain $⟨ {(I_{λ}^{μ})}^{'} (v), v^{\pm} ⟩ < 0$ , since $⟨ {(I_{λ}^{μ})}^{'} (u_{λ}), u_{i} ⟩ = 0$ . Because of Lemma 2.1, $(σ_{v}, τ_{v}) \in (0, 1] \times (0, 1]$ .

Thereupon, $\begin{aligned} 0 = & \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (u_{λ}), u_{3} ⟩ \\ = & \frac{1}{2 q} ‖ u_{3} ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ u_{3} ‖_{W_{r}^{1, q} (R^{3})}^{q} + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{1}} | u_{3} |^{q} d x + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{2}} | u_{3} |^{q} d x \\ + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{3}} | u_{3} |^{q} d x - \frac{μ}{2 q} \int_{R^{3}} g (u_{3}) u_{3} d x - \frac{1}{2 q} \int_{R^{3}} | u_{3} |^{q^{*}} d x \\ < & I_{λ}^{μ} (u_{3}) + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{1}} | u_{3} |^{q} d x + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{2}} | u_{3} |^{q} d x . \end{aligned}$ Accordingly, we can deduce from (2.17) that $\begin{aligned} c_{λ}^{μ} ⩽ & I_{λ}^{μ} (σ_{v} u_{1} + τ_{v} u_{2}) \\ = & I_{λ}^{μ} (σ_{v} u_{1} + τ_{v} u_{2}) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (σ_{v} u_{1} + τ_{v} u_{2}), σ_{v} u_{1} + τ_{v} u_{2} ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) (‖ σ_{v} u_{1} ‖_{W_{r}^{1, p} (R^{3})}^{p} + ‖ τ_{v} u_{2} ‖_{W_{r}^{1, p} (R^{3})}^{p}) + \frac{1}{2 q} (‖ σ_{v} u_{1} ‖_{W_{r}^{1, q} (R^{3})}^{q} + ‖ τ_{v} u_{2} ‖_{W_{r}^{1, q} (R^{3})}^{q}) \\ + \frac{μ}{2 q} \int_{R^{3}} [g (σ_{v} u_{1}) σ_{v} u_{1} - 2 q G (σ_{v} u_{1})] d x + \frac{μ}{2 q} \int_{R^{3}} [g (τ_{v} u_{2}) τ_{v} u_{2} - 2 q G (τ_{v} u_{2})] d x \\ + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | σ_{v} u_{1} |^{q^{*}} d x + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | τ_{v} u_{2} |^{q^{*}} d x \\ ⩽ & (\frac{1}{p} - \frac{1}{2 q}) (‖ u_{1} ‖_{W_{r}^{1, p} (R^{3})}^{p} + ‖ u_{2} ‖_{W_{r}^{1, p} (R^{3})}^{p}) + \frac{1}{2 q} (‖ u_{1} ‖_{W_{r}^{1, q} (R^{3})}^{q} + ‖ u_{2} ‖_{W_{r}^{1, q} (R^{3})}^{q}) \\ + \frac{μ}{2 q} \int_{R^{3}} [g (u_{1}) u_{1} - 2 q G (u_{1})] d x + \frac{μ}{2 q} \int_{R^{3}} [g (u_{2}) u_{2} - 2 q G (u_{2})] d x \\ + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | u_{1} |^{q^{*}} d x + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | u_{2} |^{q^{*}} d x \\ = & I_{λ}^{μ} (u_{1} + u_{2}) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (u_{1} + u_{2}), u_{1} + u_{2} ⟩ \\ < & I_{λ}^{μ} (u_{1}) + I_{λ}^{μ} (u_{2}) + I_{λ}^{μ} (u_{3}) + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{2} + u_{3}} | u_{1} |^{q} d x + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{1} + u_{3}} | u_{2} |^{q} d x \\ + \frac{λ}{2 q} \int_{R^{3}} ϕ_{u_{1} + u_{2}} | u_{3} |^{q} d x \\ = & I_{λ}^{μ} (u_{λ}) = c_{λ}^{μ}, \end{aligned}$ which is absurd, so $u_{3} = 0$ . In other words, $u_{λ}$ has precisely two nodal domains. □

Proof of Theorem 1.2.

Using the same technique as in Lemma 2.3, we can show that there exists $μ^{*} > 0$ such that for all $μ ⩾ μ^{*}$ and every $λ > 0$ , there is $v_{λ} \in M_{λ}^{μ}$ satisfying $I_{λ}^{μ} (v_{λ}) = c^{*} > 0$ . Normative approaches (see Corollary 2.13 in Ref. [17]) ensure the critical points of the functional $I_{λ}^{μ}$ on $M_{λ}^{μ}$ are critical points of $I_{λ}^{μ}$ in W. Thus, ${(I_{λ}^{μ})}^{'} (v_{λ}) = 0$ , which indicates $v_{λ}$ is a ground state solution to problem (1.1).

Based on Theorem 1.1, we consider $u_{λ}$ as a least energy nodal solution to problem (1.1), which changes sign only once if $μ ⩾ μ_{1}$ .

Set $μ_{2} = max {μ_{1}, μ^{*}}$ and $u_{λ} = u_{λ}^{+} + u_{λ}^{-}$ . Applying a similar approach to Lemma 2.1, we claim there exist $σ_{u_{λ}^{+}}, τ_{u_{λ}^{-}} \in (0, 1)$ such that $σ_{u_{λ}^{+}} u_{λ}^{+} \in M_{λ}^{μ}$ and $τ_{u_{λ}^{-}} u_{λ}^{-} \in M_{λ}^{μ}$ .

According to Lemma 2.1, we get $\begin{matrix} 2 c^{*} ⩽ I_{λ}^{μ} (σ_{u_{λ}^{+}} u_{λ}^{+}) + I_{λ}^{μ} (τ_{u_{λ}^{-}} u_{λ}^{-}) ⩽ I_{λ}^{μ} (σ_{u_{λ}^{+}} u_{λ}^{+} + τ_{u_{λ}^{-}} u_{λ}^{-}) < I_{λ}^{μ} (u_{λ}^{+} + u_{λ}^{-}) = c_{λ}^{μ} . \end{matrix}$ It follows from $I_{λ}^{μ} (u_{λ}) > 2 c^{*}$ that $c^{*} > 0$ cannot be achieved by a nodal function. □

We ascertain a least energy nodal solution $u_{λ}$ and its energy through Theorem 1.1 and Theorem 1.2. Eventually, we will end this section with the proof of Theorem 1.3, which analyses the asymptotic behaviour of $u_{λ}$ as $λ \to 0$ . As a parameter in problem (1.1), we deem $λ > 0$ in the following.

Proof of Theorem 1.3.

Via three steps, we will demonstrate the convergence property of $u_{λ}$ .

Step 1. We demonstrate ${u_{λ_{n}}}$ to be bounded in W, if $λ_{n} \to 0$ as $n \to \infty$ .

Select a nonzero function $v \in C_{c}^{\infty} (R^{3})$ with $v^{\pm} \neq 0$ . Analogously to the demonstration of Lemma 2.1, there is a pair of positive numbers $(α_{1}, β_{1})$ independent of λ, such that $\begin{matrix} ⟨ {(I_{λ}^{μ})}^{'} (α_{1} v^{+} + β_{1} v^{-}), α_{1} v^{+} ⟩ < 0 and ⟨ {(I_{λ}^{μ})}^{'} (α_{1} v^{+} + β_{1} v^{-}), β_{1} v^{-} ⟩ < 0, \end{matrix}$ for any $λ \in [0, 1]$ .

Consequently, in light of Lemma 2.1, there exists a pair $(σ_{v} (λ), τ_{v} (λ)) \in (0, 1] \times (0, 1]$ for any $λ \in [0, 1]$ , such that $\overline{v} : = σ_{v} (λ) α_{1} v^{+} + τ_{v} (λ) β_{1} v^{-} \in N_{λ}^{μ}$ . Then we have, thanks to (2.3), that for any $λ \in [0, 1]$ $\begin{aligned} I_{λ}^{μ} (u_{λ}) ⩽ & I_{λ}^{μ} (\overline{v}) \\ = & I_{λ}^{μ} (\overline{v}) - \frac{1}{2 q} ⟨ {(I_{λ}^{μ})}^{'} (\overline{v}), \overline{v} ⟩ \\ = & (\frac{1}{p} - \frac{1}{2 q}) ‖ \overline{v} ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ \overline{v} ‖_{W_{r}^{1, q} (R^{3})}^{q} + \frac{μ}{2 q} \int_{R^{3}} [g (\overline{v}) \overline{v} - 2 q G (\overline{v})] d x \\ + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | \overline{v} |^{q^{*}} d x \\ ⩽ & (\frac{1}{p} - \frac{1}{2 q}) ‖ \overline{v} ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ \overline{v} ‖_{W_{r}^{1, q} (R^{3})}^{q} + \frac{μ}{2 q} \int_{R^{3}} (C_{1} | \overline{v} |^{p} + C_{2} | \overline{v} |^{ζ}) d x \\ + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} | \overline{v} |^{q^{*}} d x \\ ⩽ & \frac{1}{2 q} ‖ \overline{v} ‖_{W_{r}^{1, q} (R^{3})}^{q} + \frac{μ}{2 q} \int_{R^{3}} (C_{1} α_{1}^{p} {| v^{+} |}^{p} + C_{1} β_{1}^{p} {| v^{-} |}^{p}) d x \\ + \frac{μ}{2 q} \int_{R^{3}} (C_{2} α_{1}^{ζ} {| v^{+} |}^{ζ} + C_{2} β_{1}^{ζ} {| v^{-} |}^{ζ}) d x \\ + (\frac{1}{p} - \frac{1}{2 q}) ‖ \overline{v} ‖_{W_{r}^{1, p} (R^{3})}^{p} + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} {| α_{1} v^{+} |}^{q^{*}} d x \\ + (\frac{1}{2 q} - \frac{1}{q^{*}}) \int_{R^{3}} {| β_{1} v^{-} |}^{q^{*}} d x \\ : = & C_{*}, \end{aligned}$ with $C_{*}$ being a positive constant independent of λ, and with that we have as $n \to \infty$ , $\begin{aligned} C_{*} + 1 & ⩾ I_{λ_{n}}^{μ} (u_{λ_{n}}) \\ = I_{λ_{n}}^{μ} (u_{λ_{n}}) - \frac{1}{2 q} ⟨ {(I_{λ_{n}}^{μ})}^{'} (u_{λ_{n}}), u_{λ_{n}} ⟩ ⩾ (\frac{1}{p} - \frac{1}{2 q}) ‖ u_{λ_{n}} ‖_{W_{r}^{1, p} (R^{3})}^{p} + \frac{1}{2 q} ‖ u_{λ_{n}} ‖_{W_{r}^{1, q} (R^{3})}^{q} . \end{aligned}$ Combined with (2.15), it follows ${u_{λ_{n}}}$ is bounded in W.

Step 2. Then, we determine problem (1.6) has one nodal solution $u_{0}$ .

Given Step 1 and the fact that ${u_{λ_{n}}}$ is bounded in W, there exists $u_{0} \in W$ such that up to a subsequence, $\begin{aligned} u_{λ_{n}} ⇀ u_{0} in W, \\ u_{λ_{n}} \to u_{0} in L^{s} (R^{3}) for p \in [2 q, q^{*}), \\ (3.4) & u_{λ_{n}} \to u_{0} a.e. in R^{3} . \end{aligned}$ When $λ = λ_{n}$ , ${u_{λ_{n}}}$ is recognized as a weak solution to (1.1), which implies $\begin{aligned} \int_{R^{3}} (| \nabla u_{λ_{n}} |^{p - 2} \nabla u_{λ_{n}} \nabla ψ + | u_{λ_{n}} |^{p - 2} u_{λ_{n}} ψ) d x + \int_{R^{3}} (| \nabla u_{λ_{n}} |^{q - 2} \nabla u_{λ_{n}} \nabla ψ + | u_{λ_{n}} |^{q - 2} u_{λ_{n}} ψ) d x \\ (3.5) & + λ_{n} \int_{R^{3}} ϕ_{u_{λ_{n}}} | u_{λ_{n}} |^{q - 2} u_{λ_{n}} ψ d x - μ \int_{R^{3}} g (u_{λ_{n}}) ψ d x - \int_{R^{3}} | u_{λ_{n}} |^{q^{*} - 2} u_{λ_{n}} ψ d x = 0, \end{aligned}$ for all $ψ \in C_{c}^{\infty} (R^{3})$ .

Taking into consideration (3.4), (3.5) and Step 1, we can obtain $\begin{aligned} \int_{R^{3}} (| \nabla u_{0} |^{p - 2} \nabla u_{0} \nabla ψ + | u_{0} |^{p - 2} u_{0} ψ) d x + \int_{R^{3}} (| \nabla u_{0} |^{q - 2} \nabla u_{0} \nabla ψ + | u_{0} |^{q - 2} u_{0} ψ) d x \\ - μ \int_{R^{3}} g (u_{0}) ψ d x - \int_{R^{3}} | u_{0} |^{q^{*} - 2} u_{0} ψ d x = 0, \end{aligned}$ for all $ψ \in C_{c}^{\infty} (R^{3})$ , indicating $u_{0}$ is a weak solution to problem (1.6). We arrive at the conclusion $u_{0}^{\pm} \neq 0$ using a technique similar to Lemma 2.2, and thus complete this step.

Step 3. Finally, we confirm problem (1.6) has a least energy nodal solution $ω_{0}$ , and that there exists a unique pair $(σ_{λ_{n}}, τ_{λ_{n}}) \in [0, \infty) \times [0, \infty)$ such that $σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-} \in N_{λ_{n}}^{μ}$ . Moreover, $(σ_{λ_{n}}, τ_{λ_{n}}) \to (1, 1)$ as $n \to \infty$ .

In a manner analogous to Theorem 1.1, there exists $μ^{* *} > 0$ such that for all $μ ⩾ μ^{* *}$ , we can determine problem (1.6) has a least energy nodal solution $ω_{0}$ , where $I_{0}^{μ} (ω_{0}) = c_{0}^{μ}$ and ${(I_{0}^{μ})}^{'} (ω_{0}) = 0$ . Set $μ_{3} = max {μ_{1}, μ^{*}, μ^{* *}}$ , and then thanks to Lemma 2.1, we can conclude with ease that there exists a unique pair $(σ_{λ_{n}}, τ_{λ_{n}}) \in R_{+} \times R_{+}$ such that $σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-} \in N_{λ_{n}}^{μ}$ .

For the rest, we simply need to validate that $(σ_{λ_{n}}, τ_{λ_{n}}) \to (1, 1)$ as $n \to \infty$ . In fact, it follows from $σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-} \in N_{λ_{n}}^{μ}$ that $\begin{aligned} {‖ σ_{λ_{n}} ω_{0}^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ_{λ_{n}} ω_{0}^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ_{n} \int_{R^{3}} ϕ_{σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-}} {| σ_{λ_{n}} ω_{0}^{+} |}^{q} d x \\ (3.6) & = μ \int_{R^{3}} g (σ_{λ_{n}} ω_{0}^{+}) σ_{λ_{n}} ω_{0}^{+} d x + \int_{R^{3}} {| σ_{λ_{n}} ω_{0}^{+} |}^{q^{*}} d x \end{aligned}$ and $\begin{aligned} {‖ τ_{λ_{n}} ω_{0}^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ τ_{λ_{n}} ω_{0}^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} + λ_{n} \int_{R^{3}} ϕ_{σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-}} {| τ_{λ_{n}} ω_{0}^{-} |}^{q} d x \\ (3.7) & = μ \int_{R^{3}} g (τ_{λ_{n}} ω_{0}^{-}) τ_{λ_{n}} ω_{0}^{-} d x + \int_{R^{3}} {| τ_{λ_{n}} ω_{0}^{-} |}^{q^{*}} d x . \end{aligned}$

Because of $(g_{3})$ , (2.15), and $λ_{n} \to 0$ as $n \to \infty$ , we can determine easily that ${σ_{λ_{n}}}$ and ${τ_{λ_{n}}}$ are bounded. Then we may assume, up to a subsequence, $σ_{λ_{n}} \to σ_{0}$ and $τ_{λ_{n}} \to τ_{0}$ . It yields from (3.6) and (3.7) that $\begin{aligned} (3.8) & {‖ σ_{0} ω_{0}^{+} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ σ_{0} ω_{0}^{+} ‖}_{W_{r}^{1, q} (R^{3})}^{q} = μ \int_{R^{3}} g (σ_{0} ω_{0}^{+}) σ_{0} ω_{0}^{+} d x + \int_{R^{3}} {| σ_{0} ω_{0}^{+} |}^{q^{*}} d x \end{aligned}$ and $\begin{aligned} (3.9) & {‖ τ_{0} ω_{0}^{-} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ τ_{0} ω_{0}^{-} ‖}_{W_{r}^{1, q} (R^{3})}^{q} = μ \int_{R^{3}} g (τ_{0} ω_{0}^{-}) τ_{0} ω_{0}^{-} d x + \int_{R^{3}} {| τ_{0} ω_{0}^{-} |}^{q^{*}} d x . \end{aligned}$

Since $ω_{0}$ is a nodal solution to problem (1.6), we can validate $\begin{aligned} (3.10) & {‖ ω_{0}^{\pm} ‖}_{W_{r}^{1, p} (R^{3})}^{p} + {‖ ω_{0}^{\pm} ‖}_{W_{r}^{1, q} (R^{3})}^{q} = μ \int_{R^{3}} g (ω_{0}^{\pm}) ω_{0}^{\pm} d x + \int_{R^{3}} {| ω_{0}^{\pm} |}^{q^{*}} d x . \end{aligned}$ Therefore, $(σ_{0}, τ_{0}) = (1, 1)$ due to (3.8)–(3.10). Thanks to Lemma 2.1, we have $\begin{matrix} I_{0}^{μ} (ω_{0}) ⩽ I_{0}^{μ} (u_{0}) = lim_{n \to \infty} I_{λ_{n}}^{μ} (u_{λ_{n}}) ⩽ lim_{n \to \infty} I_{λ_{n}}^{μ} (σ_{λ_{n}} ω_{0}^{+} + τ_{λ_{n}} ω_{0}^{-}) = I_{0}^{μ} (ω_{0}^{+} + ω_{0}^{-}) = I_{0}^{μ} (ω_{0}) . \end{matrix}$

Now, we can announce $ω_{0}$ attained in Step 2 as a least energy nodal solution to problem (1.6) and thus Theorem 1.3 has been successfully proved. □

Footnotes

Acknowledgements

S. Liang was supported by NSFC Grant (No. 12371455), the Research Foundation of Department of Education of Jilin Province (Grant No. JJKH20230902KJ), the Natural Science Foundation of Jilin Province (Grant No. YDZJ202201ZYTS582, 222614JC0106101856), and Innovation and Entrepreneurship Talent Funding Project of Jilin Province (No.2023QN21). S. Ji was supported by NSFC Grant (Nos. 12225103, 12071065 and 11871140) and the National Key Research and Development Program of China (Nos. 2020YFA0713602 and 2020YFC1808301).

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