A Critical $p \& q$-Laplacian Problem with Shifting Subcritical Perturbation

Abstract

In this paper, we study the following critical $p \& q$-Laplacian problem with shifting subcritical perturbation nonlinearity:

where $\Omega \subset {\mathbb {R}}^{N}$ is open bounded, $k>0$, $1<q<p<N$, $p^{*}=\frac{Np}{N-p}$ is the critical Sobolev exponent and $h(x)\in C(\Omega )$ is nonnegative. Under suitable assumptions on exponents q and r, we show the existence of mountain pass solutions $u_{k}$ to ($P_{k}$) for all $k\in (0,+\infty )$. Moreover, we also investigate the behavior of the solutions $u_{k}$ for varying k and show the concentration phenomena as $k\rightarrow +\infty$.

1 Introduction

We consider the following Dirichlet $p \& q$-Laplacian nonlinear elliptic equation:

where $\Omega \subset {\mathbb {R}}^{N}$ is open bounded, $k>0$, $N>2$, $1<q<p<N$, $p^{*}=\frac{Np}{N-p}$ is the critical Sobolev exponent, $p\le r<p^{*}$ and $w^{+}=\max \{w,0\}$. The non-homogeneous differential operator $\Delta _{p} u+\Delta _{q}u$ is the so-called $p \& q$-Laplacian operator, where $\Delta _{s}u:=\text{ div }(|\nabla u|^{s-2}\nabla u)$ with $s\in \{p,q\}$.

Our problem ($P_{k}$) is motivated by the original work [8], where Gazzola considers the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u=h(x) [(u-k)^{+}]^{q-1}+u^{p^{*}-1}&{}\quad \text{ in }\,\,\Omega ,\\ u>0&{}\quad \text{ in }\,\,\Omega ,\\ u=0&{}\quad \text{ on }\,\,\partial \Omega , \end{array} \right. \end{aligned}$$

(1.1)

which can be seen as the variant of Brézis–Nirenberg problem involving the p-Laplacian and also is closely related to the following limit problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u=u^{p^{*}-1}&{}\quad \text{ in }\,\,\Omega ,\\ u\ge 0&{}\quad \text{ in }\,\,\Omega ,\\ u=0&{}\quad \text{ on }\,\,\partial \Omega . \end{array} \right. \end{aligned}$$

(1.2)

As we known, the existence of nonzero solutions of problem (1.2) depends on the geometry of the domain $\Omega$ (see, for example, [9, 16]). Therefore, there are two ways to find nonzero solutions of (1.2): modify the geometry of $\Omega$ (see [5, 15]) or perturb the critical term $u^{p^{*}-1}$ (see [2, 7]). Gazzola [8] added the low perturbation $h(x) [(u-k)^{+}]^{q-1}$ in problem (1.1) and studied not only the existence of nonzero solutions of (1.1), but also the concentration phenomena as parameter k tends to infinity. Recently, Li and Xiang [11] extended Gazzola’s results to the fractional Laplacian case. See also [4] for fractional p-Laplacian. Here, in this paper, we would like to obtain similar results to $p \& q$-Laplacian operator.

The interest in ($P_{k}$) comes from a general reaction–diffusion system

$$\begin{aligned} u_{t}=\text{ div }[D(u)\nabla u]+c(x,u),\quad D(u)=|\nabla u|^{p-2}+|\nabla u|^{q-2}. \end{aligned}$$

(1.3)

This system has widely applications in physics and related sciences such as biophysics, plasma physics and chemical reaction design. In this context, the function u describes a concentration, $\text{ div }[D(u)\nabla u]$ corresponds to diffusion with a diffusion coefficient D(u), and the term c(x, u) is the reaction and relates to source and loss processes. In general, the reaction term c(x, u) is a polynomial form with respect to the concentration u in chemical and biological applications.

On the other hand, the functional associated with the $p \& q$-Laplacian operator falls in the realm of the following double-phase energy functional

$$\begin{aligned} F_{p,q}(u;\Omega ):=\int _{\Omega }(|\nabla u|^{p}+a(x)|\nabla u|^{q}){\textrm{d}}x, \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^{N}$ is open set and $0\le a(\cdot )\in L^{\infty }(\Omega )$. Zhikov [28, 29] studied the functional $F_{p,q}(u;\Omega )$ in the setting of homogenization of strongly anisotropic materials and to obtain new examples of the Lavrentiev phenomenon. The function a(x) dictates the geometry the composite made by two different materials with hardening exponents p and q.

As previously stated, we can observe the importance of studying differential equations and variational problems with nonstandard $p \& q$-growth conditions in physics and related sciences. Furthermore, there are many mathematical challenges due to the $p \& q$-Laplacian operator is not homogeneous when $p\not =q$; therefore, usual mathematical techniques are adequate to deal with these problems, which requires the development of new techniques. This could be the central development of mathematical ideas in active areas of pure mathematics that have had a significant impact on PDEs. As a result, the related literature appears to be growing on every day, with numerous interesting publications on the subject now available. For some interesting existence and multiplicity results involving $p \& q$-Laplacian problems, we refer readers to [1, 10, 12, 14, 17,18,19,20,21,22, 24,25,26,27] for a survey of recent existence and multiplicity results for subcritical and critical $p \& q$-Laplacian problems in bounded domains.

Problem ($P_{k}$) can be seen as an interesting variant of the Brézis–Nirenberg problem for the $p \& q$-Laplacian:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u-\Delta _{q}u=\lambda |u|^{r-2}u+u^{p^{*}-1}&{}\quad \text{ in }\,\,\Omega ,\\ u\ge 0&{}\quad \text{ in }\,\,\Omega ,\\ u=0&{}\quad \text{ on }\,\,\partial \Omega . \end{array} \right. \end{aligned}$$

(1.4)

In [10], the authors proved that problem (1.4) has a non-trivial weak solution for all $\lambda >0$ under q and r satisfies (1.9). But in our article, we are interesting in the lower order perturbation $\lambda |u|^{r-2}u$ replaced by the type $h(x) [(u-k)^{+}]^{r-1}$. Then, we study the behavior of the solutions when it “shifts,” that is, the behavior of the solutions of ($P_{k}$) for varying k. Moreover, we also consider the limit problem of ($P_{k}$), that is,

We will use variational method to study problem ($P_{k}$), and then, the solutions are the critical points of related functional ${\mathcal {I}}\,:\, W_{0}^{1,p}(\Omega )\rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} {\mathcal {I}}_{k}(u)&=\frac{1}{p}\int _{\Omega }|\nabla u|^{p}{\textrm{d}}x+\frac{1}{q}\int _{\Omega }|\nabla u|^{q}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(u-k)^{+}]^{r}{\textrm{d}}x\nonumber \\&\quad -\frac{1}{p^{*}}\int _{\Omega }|u|^{p^{*}}{\textrm{d}}x, \end{aligned}$$

(1.5)

for $0<k<+\infty$ and $W_{0}^{1,p}(\Omega )$ is the usual Sobolev space with the norm

$$\begin{aligned} \Vert u\Vert _{p}=\left( \int _{\Omega }|\nabla u|^{p}{\textrm{d}}x\right) ^{1/p}. \end{aligned}$$

For $u\in W_{0}^{1,p}(\Omega )$, we also define the energy functional of the limit problem ($P_{\infty }$) by

$$\begin{aligned} {\mathcal {I}}_{\infty }(u)=\frac{1}{p}\int _{\Omega }|\nabla u|^{p}{\textrm{d}}x+\frac{1}{q}\int _{\Omega }|\nabla u|^{q}{\textrm{d}}x-\frac{1}{p^{*}}\int _{\Omega }|u|^{p^{*}}{\textrm{d}}x. \end{aligned}$$

(1.6)

Recalling that a sequence $\{u_{j}\}\subset W_{0}^{1,p}(\Omega )$ such that ${\mathcal {I}}_{k}(u_{j})\rightarrow c$ and ${\mathcal {I}}_{k}^{\prime }(u_{j})\rightarrow 0$ is called a $(PS)_{c}$ sequence. Define

$$\begin{aligned} c^{\star }=\frac{1}{N}S^{\frac{N}{p}}, \end{aligned}$$

(1.7)

where

$$\begin{aligned} S=\inf _{u\in W_{0}^{1,p}(\Omega ){\setminus }\{0\}}\frac{\Vert u\Vert _{p}^{p}}{|u|_{p^{*}}^{p}} \end{aligned}$$

(1.8)

is the best Sobolev constant and $|\cdot |_{t}$ denotes the $L^{t}(\Omega )$-norm.

In order to get the estimates, we assume that exponents q and r satisfy the following two cases:

$$\begin{aligned} \left\{ \begin{array}{ll} 1<q<\frac{N(p-1)}{N-1}\quad \text{ and } \quad \frac{N^{2}(p-1)}{(N-1)(N-p)}<r<p^{*},\\ \frac{N(p-1)}{N-1}\le q<p\quad \text{ and } \quad \frac{Nq}{N-p}<r<p^{*}. \end{array} \right. \end{aligned}$$

(1.9)

For the function h, we assume that

$$\begin{aligned} \left\{ \begin{array}{ll} h\in C(\Omega ),\quad h(x)\ge 0\,\,\text{ in }\,\,\Omega ,\quad h\not \equiv 0,\quad h\in L^{\frac{Np}{Np+(p-N)r}}(\Omega );\\ \text{ if }\,\,r=p\,\,\text{ in }\,\,(1.9), \,\,\text{ then }\,\,\frac{Np}{Np+(p-N)r}=\frac{N}{p}\,\,\text{ and } \text{ we } \text{ assume } \text{ that }\,\,|h|_{\frac{N}{p}}<S. \end{array} \right. \end{aligned}$$

(H)

In the paper, we consider nonnegative solutions and define

$$\begin{aligned} \mathcal {M}:=\{u\in W_{0}^{1,p}(\Omega ): u\ge 0\,\,\mathrm{a.e.}\,\,\text{ in }\,\,\Omega \}. \end{aligned}$$

We also can check that the set

$$\begin{aligned} \mathcal {N}:=\{u\in \mathcal {M}(\Omega ): {\mathcal {I}}_{\infty }(u)<0\} \end{aligned}$$

is not empty (see, for example, Proposition 3). Choosing $v\in \mathcal {N}$, consider the class

$$\begin{aligned} \Gamma =\{\gamma \in C([0,1],\mathcal {M}): \gamma (0)=0,\,\,\gamma (1)=v\}, \end{aligned}$$

and define the mountain pass value

$$\begin{aligned} c_{k}:=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}{\mathcal {I}}_{k}(\gamma (t)). \end{aligned}$$

(1.10)

For each $k\in (0,+\infty ]$, we say that a solution $u_{k}\in W_{0}^{1,p}(\Omega )$ of ($P_{k}$) is a mountain pass solution at the energy level $c_{k}$ if ${\mathcal {I}}_{k}(u_{k})=c_{k}$.

We first obtain a nonexistence result of problem ($P_{\infty }$).

Theorem 1

There is no mountain pass solution of problem ($P_{\infty }$).

Next, we show the existence result of problem ($P_{k}$) for each $k\in (0,+\infty )$.

Theorem 2

Under the assumptions (1.9) and (H), then there exists a mountain pass solution $u_{k}$ of problem ($P_{k}$) for all $k\in (0,+\infty )$. In addition,

• (a) there exists $x_{0}^{k}\in \Omega$ such that $u_{k}\left( x_{0}^{k}\right) >k$;

• (b) if sequence $\{k_{j}\}\subset (0,+\infty )$ converges to $k\in (0,+\infty )$ as $j\rightarrow +\infty$ and $\{u_{k_{j}}\}$ are the mountain pass solutions of problem $(P_{k_{j}})$, then there has a mountain pass solution $u_{k}$ of problem ($P_{k}$) satisfies $u_{k_{j}}\rightarrow u_{k}$ in $W_{0}^{1,p}(\Omega )$, up to a subsequence, as $j\rightarrow +\infty$.

In order to prove Theorem 2, we need some careful estimates on the dependence of the mountain pass level $c_{k}$ with respect to k, see Lemmas 5, 6 and 7. In the paper, we also consider the behavior of mountain pass solutions $u_{k}$ as $k\rightarrow +\infty$.

Theorem 3

Under the assumptions (1.9) and (H), suppose $\{u_{k}\}$ are mountain pass solutions of ($P_{k}$) for each $k\in (0,+\infty )$, and if $k\rightarrow +\infty$, then

• (1) $u_{k}\rightharpoonup 0$ in $W_{0}^{1,p}(\Omega )$.

Moreover, there exist $x_{0}\in \bar{\Omega }$ and a subsequence $\{u_{k_{j}}\}\subset \{u_{k}\}$ such that (as $k_{j}\rightarrow +\infty$):

• (2) $|\nabla u_{k_{j}}|^{p}\rightharpoonup S^{\frac{N}{p}}\delta _{x_{0}}$ in the measure topology;

• (3) $u_{k_{j}}^{p^{*}}\rightharpoonup S^{\frac{N}{p}}\delta _{x_{0}}$ in the measure topology,

where $\delta _{x_{0}}$ is the Dirac measure at $x_{0}$. Finally, if $x_{0}^{k_{j}}$ denote any of points found in Theorem 1, then we have $x_{0}^{k_{j}}\rightarrow x_{0}$ as $k_{j}\rightarrow +\infty$.

By Theorem 2, we know the following set

$$\begin{aligned} \Omega (u_{k}):=\{x\in \Omega : u_{k}(x)>k\} \end{aligned}$$

(1.11)

is not an empty set. In addition, by Theorem 3, we can see that $\Omega (u_{k})$ collapse to the single point $x_{0}$. Next, we show the precise location of $x_{0}$.

Theorem 4

Under the assumptions (1.9) and (H), suppose $\{u_{k}\}$ are mountain pass solutions of ($P_{k}$) for each $k\in (0,+\infty )$, then there exist $x_{0}^{k}\in \text{ supp }\, h$ such that $u_{k}(x_{0}^{k})>k$. In addition, if $x_{0}$ and $u_{k_{j}}$ are as in Theorem 3, then for any such $x_{0}^{k_{j}}$, we have $x_{0}^{k_{j}}\rightarrow x_{0}$ as $k_{j}\rightarrow +\infty$. In particular, $x_{0}\in \text{ supp }\, h$.

We point out that the above results are not a trivial extension of [8, 11], there are some nature difficulties, for example: (1) the proof of main results rely on variational arguments inspired by the above references. Nevertheless, owing to the absence of homogeneity caused by the presence of the $p \& q$-Laplacian operator, several arguments and estimates are more complicated (see, for example, Lemmas 2 and 3) than the single p-Laplacian case, that is $p=q$ in ($P_{k}$), see [8]. (2) the space $W_{0}^{1,p}(\Omega )$ is not a Hilbert space in general, so even if the (PS) sequence $\{u_{n}\}$ for ${\mathcal {I}}_{k}(u)$ and $u_{n}\rightharpoonup u$ weakly in $W_{0}^{1,p}(\Omega )$, it is not clear that we should have

$$\begin{aligned}&|\nabla u_{n}|^{p-2}\nabla u_{n}\rightharpoonup |\nabla u|^{p-2}\nabla u\quad \text{ weakly } \text{ in } \,\,L^{\frac{p}{p-1}}(\Omega ),\\&|\nabla u_{n}|^{q-2}\nabla u_{n}\rightharpoonup |\nabla u|^{q-2}\nabla u\quad \text{ weakly } \text{ in } \,\,L^{\frac{q}{q-1}}(\Omega ) \end{aligned}$$

or not for some renamed subsequence $\{u_{n}\}$. To overcome this difficulties, we use concentration–compactness principle (see [13]). (3) The typical difficulty in dealing problem ($P_{k}$) is the corresponding functional ${\mathcal {I}}_{k}(u)$ does not satisfy a $(PS)_{c_{k}}$ condition due to the lack of compactness of the embedding: $W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )$. We could not use the standard variational methods, and then, we need show ${\mathcal {I}}_{k}(u)$ satisfy $(PS)_{c_{k}}$ condition provides $c_{k}<c^{\star }$ (see Proposition 2). In order to obtain a mountain pass solution of ($P_{k}$), we should prove $c_{k}<c^{\star }$ for all $k\in (0,+\infty )$. However, we need more careful analysis for our problem since both p and q-Laplacian operators appear in the problem (see Lemmas 2 and 3).

We will prove some preliminary results in Sect. 2 and then prove our main results in Sect. 3.

2 Some Preliminary Facts

We begin this section by recalling the following technical lemma proved in [8, Lemma 1]. In this section, we always let $f(x,t)=h(x)(t^{+})^{r-1}$ and $F(x,t)=\frac{1}{r}f(x,t)t$.

Lemma 1

Assume that $\{k_{j}\}\subset (0,+\infty )$ is a sequence satisfies $k_{j}\rightarrow k\in (0,+\infty ]$, $\{u_{j}\}\subset W_{0}^{1,p}(\Omega )$ is a sequence satisfies $u_{j}\rightharpoonup u$ for some $u\in W_{0}^{1,p}(\Omega )$. Then

• (a) if $k<+\infty$, then

  $$\begin{aligned} f(x,u_{j}-k_{j})&\rightarrow f(x,u-k) \quad \textrm{in}\,\, L^{\frac{Np}{Np-N+p}}(\Omega ), \\ F(x,u_{j}-k_{j})&\rightarrow F(x,u-k) \quad \textrm{in}\,\, L^{1}(\Omega ); \end{aligned}$$

• (b) if $k=+\infty$, then

  $$\begin{aligned} f(x,u_{j}-k_{j})\rightarrow 0\quad \textrm{in}\,\, L^{\frac{Np}{Np-N+p}}(\Omega ), \quad F(x,u_{j}-k_{j})\rightarrow 0\quad \textrm{in}\,\, L^{1}(\Omega ); \end{aligned}$$

Proof

Since $\Vert (u_{j}-k_{j})^{+}\Vert _{p}\le \Vert u_{j}\Vert _{p}$, so $(u_{j}-k_{j})^{+}\rightharpoonup \bar{u}$ for some $\bar{u}\in W_{0}^{1,p}$, up to a subsequence. By pointwise convergence, we also know $\bar{u}=(u-k)^{+}$ if $k<+\infty$ and $\bar{u}=0$ if $k=+\infty$. Therefore,

$$\begin{aligned} (u_{j}-k_{j})^{+}\rightharpoonup (u-k)^{+} \quad \text{ if }\quad k<+\infty \end{aligned}$$

and

$$\begin{aligned} (u_{j}-k_{j})^{+}\rightharpoonup 0 \quad \text{ if }\quad k=+\infty . \end{aligned}$$

By repeating the above arguments for any subsequence of $\{u_{j}\}$, we infer that the previous convergences hold on the whole sequence.

Then, the conclusions follows by taking into account (1.9) (namely that f has subcritical growth) and by arguing as in [3, Theorem 2.2.7 ]. $\square$

The next result is the so-called the second concentration–compactness lemma of P. L. Lions (see [13]).

Proposition 1

(Concentration–Compactness Principle) Suppose $p<N$ and $\{u_{j}\}$ is a bounded sequence in $W^{1,p}({\mathbb {R}}^{N})$ converging weakly to some u such that $|\nabla u_{j}|^{p}$ converges weakly to $\mu$ and $|u_{j}|^{p^{*}}$ converges weakly to $\nu$ where $\mu ,\,\nu$ are bounded nonnegative measures on ${\mathbb {R}}^{N}$. Also assume that $|u_{j}|^{p^{*}}$ is a tight sequence, that is, there is a sequence $y_{k}\subset {\mathbb {R}}^{N}$ such that for any $\varepsilon >0$, there is an $R = R(\varepsilon )>0$ such that for any k we have $\int _{{\mathbb {R}}^{N}{\setminus } B(y_{k},R)} |u_{j}|^{p^{*}}{\textrm{d}}x< \varepsilon .$ Then we have

• (a) there exists some at most countable set I, distinct points $\{x_{i}: i\in I\}\subset {\mathbb {R}}^{N}$, $\{\mu _{i}\}\subset (0, \infty )$, $\{\nu _{i}\}\subset (0, \infty )$ such that

  $$\begin{aligned} \nu =|u|^{p^{*}}+\sum _{i\in I}\nu _{i}\delta _{x_{i}},\quad \mu \ge |\nabla u|^{p}+\sum _{i\in I}\mu _{i}\delta _{x_{i}}, \end{aligned}$$

  where $\delta _{x_{i}}$ is the Dirac measure at $x_{i}$;

• (b) moreover, $S\nu _{i}^{\frac{p}{p^{*}}}\le \mu _{i}$, where S is defined in (1.8);

• (c) if $u\equiv 0$ and

  $$\begin{aligned} \mu ({\mathbb {R}}^{N})^{\frac{1}{p}}\le S^{\frac{1}{p}}\nu ({\mathbb {R}}^{N})^{\frac{1}{p^{*}}}, \end{aligned}$$

  then I is a singleton, that is, $\nu$ is concentration at a single point.

Next, we show the functional ${\mathcal {I}}_{k}$ satisfies the $(PS)_{c}$ condition provided $0<c<c^{\star }$.

Proposition 2

Suppose $k\in (0,+\infty )$ and $c_{k}\in (0,c^{\star })$, then ${\mathcal {I}}_{k}$ satisfies the $(PS)_{c_{k}}$ condition at level $c_{k}$.

Proof

Suppose that $\{u_{j}\}\subset W_{0}^{1,p}(\Omega )$ is a (PS) sequence of ${\mathcal {I}}_{k}$, then we have

$$\begin{aligned} c_{k}+1+o(1)\Vert u_{j}\Vert&\ge {\mathcal {I}}_{k}(u_{j})-\frac{1}{r}\langle {\mathcal {I}}_{k}(u_{j}),u_{j}\rangle \nonumber \\&=\left( \frac{1}{p}-\frac{1}{r}\right) \Vert u_{j}\Vert _{p}^{p}\nonumber \\&\quad +\left( \frac{1}{q}-\frac{1}{r}\right) \Vert u_{j}\Vert _{q}^{q}+\frac{k}{r}\int _{\Omega }h(x)[(u_{j}-k)^{+}]^{r-1}{\textrm{d}}x\nonumber \\&\quad +\left( \frac{1}{r}-\frac{1}{p^{*}}\right) |u_{j}|_{p^{*}}^{p^{*}}\nonumber \\&\ge \left( \frac{1}{p}-\frac{1}{r}\right) \Vert u_{j}\Vert _{p}^{p}, \end{aligned}$$

(2.1)

if $p<r<p^{*}$ since (H). On the other hand, if $r=p$, choosing $\beta \in (p,p^{*})$ and by (H), we also have

$$\begin{aligned} c_{k}+1+o(1)\Vert u_{j}\Vert&\ge {\mathcal {I}}_{k}(u_{j})-\frac{1}{\beta }\langle {\mathcal {I}}_{k}(u_{j}),u_{j}\rangle \nonumber \\&\ge \left( \frac{1}{p}-\frac{1}{\beta }\right) \Vert u_{j}\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{\beta }\right) \Vert u_{j}\Vert _{q}^{q}\nonumber \\&\quad -\left( \frac{1}{p}-\frac{1}{\beta }\right) \int _{\Omega }h(x)[(u_{j}-k)^{+}]^{p}{\textrm{d}}x\nonumber \\&\quad +\left( \frac{1}{\beta }-\frac{1}{p^{*}}\right) |u_{j}|_{p^{*}}^{p^{*}}\nonumber \\&\ge \left( \frac{1}{p}-\frac{1}{\beta }\right) \left( 1-\frac{|h|_{\frac{N}{p}}}{S}\right) \Vert u_{j}\Vert _{p}^{p}. \end{aligned}$$

(2.2)

Hence, by (2.1) and (2.2), we know $\{u_{j}\}$ is bounded in $W_{0}^{1,p}(\Omega )$. Since $W_{0}^{1,p}(\Omega )$ is a separable and reflexive Banach space, we see that there is a $u\in W_{0}^{1,p}$ and a subsequence of $\{u_{j}\}$, still denoted by $\{u_{j}\}$ such that

$$\begin{aligned} u_{j}\rightharpoonup u\quad \text{ in } \,\,W_{0}^{1,p}(\Omega ). \end{aligned}$$

By a similar argument as [12, Lemma 2.3] (or [1, Lemma 3.5]), based on Proposition 1, one can see that

$$\begin{aligned} \left\{ \begin{array}{ll} \nabla u_{j}\rightarrow \nabla u\quad \text{ a.e. } \text{ on } \,\,\Omega ,\\ u_{j}\rightarrow u\quad \text{ strongly } \text{ in } \,\,L^{t}(\Omega )\,\,\text{ for }\,\,t\in (1,p^{*}),\\ |u_{j}|^{p^{*}-2}u_{j}\rightharpoonup |u|^{p^{*}-2}u\quad \text{ weakly } \text{ in } \,\,L^{\frac{p^{*}}{p^{*}-1}}(\Omega ),\\ |\nabla u_{j}|^{p-2}\nabla u_{j}\rightharpoonup |\nabla u|^{p-2}\nabla u\quad \text{ weakly } \text{ in } \,\,L^{\frac{p}{p-1}}(\Omega ),\\ |\nabla u_{j}|^{q-2}\nabla u_{j}\rightharpoonup |\nabla u|^{q-2}\nabla u\quad \text{ weakly } \text{ in } \,\,L^{\frac{q}{q-1}}(\Omega ). \end{array} \right. \end{aligned}$$

(2.3)

Therefore, by Brézis–Lieb lemma, we know

$$\begin{aligned} \Vert u_{j}\Vert _{s}^{p}-\Vert u\Vert _{s}^{p}=\Vert u_{j}-u\Vert _{s}^{p}+o(1)\quad \text{ for }\quad s\in \{p,q\}. \end{aligned}$$

(2.4)

In addition, since ${\mathcal {I}}_{k}^{\prime }(u_{j})\rightarrow 0$ in $W^{-1,p^{\prime }}(\Omega )$, we see that

$$\begin{aligned} \langle {\mathcal {I}}_{k}^{\prime }(u),v\rangle =\lim _{j\rightarrow +\infty }\langle {\mathcal {I}}_{k}^{\prime }(u_{j}),v\rangle =0, \end{aligned}$$

(2.5)

for any $v\in W_{0}^{1,p}(\Omega )$. So we have

$$\begin{aligned} o(1)&=\langle {\mathcal {I}}_{k}^{\prime }(u_{j}),u_{j}-u\rangle -\langle {\mathcal {I}}_{k}^{\prime }(u),u_{j}-u\rangle \\&=\int _{\Omega }(|\nabla u_{j}|^{p-2}\nabla u_{j}-|\nabla u|^{p-2}\nabla u)(\nabla u_{j}-\nabla u){\textrm{d}}x\\&\quad +\int _{\Omega }(|\nabla u_{j}|^{q-2}\nabla u_{j}-|\nabla u|^{q-2}\nabla u)(\nabla u_{j}-\nabla u){\textrm{d}}x\\&\quad -\int _{\Omega }f(x,u_{j}-k)(u_{j}-u){\textrm{d}}x+\int _{\Omega }f(x,u-k)(u_{j}-u){\textrm{d}}x\\&\quad -\int _{\Omega }|u_{j}|^{p^{*}-1}(u_{j}-u){\textrm{d}}x+\int _{\Omega }|u|^{p^{*}-1}(u_{j}-u){\textrm{d}}x. \end{aligned}$$

Then by Lemma 1 (with $k_{j}=k$), (2.4) and Brézis–Lieb lemma, we have that

$$\begin{aligned} \Vert u_{j}-u\Vert _{p}^{p}+\Vert u_{j}-u\Vert _{q}^{q}=|u_{j}-u|_{p^{*}}^{p^{*}}+o(1). \end{aligned}$$

(2.6)

Assume that

$$\begin{aligned} \Vert u_{j}-u\Vert _{p}^{p}=a+o(1),\quad \Vert u_{j}-u\Vert _{q}^{q}=b+o(1), \end{aligned}$$

then by Sobolev inequality (see (1.8)) and (2.6), we know

$$\begin{aligned} a\ge S(a+b)^{\frac{p}{p^{*}}}\ge S a^{\frac{p}{p^{*}}}. \end{aligned}$$

(2.7)

If $a=0$, then we complete the proof, and if $a>0$, then (2.7) implies that $a\ge S^{\frac{N}{p}}$.

Applying Lemma 1, (2.4) and Brézis–Lieb lemma again and (2.6), we get

$$\begin{aligned} {\mathcal {I}}_{k}(u_{j})={\mathcal {I}}_{k}(u_{j}-u)+{\mathcal {I}}_{k}(u)+o(1)\ge \frac{a}{N}+{\mathcal {I}}_{k}(u)+o(1). \end{aligned}$$

(2.8)

Finally, similar as (2.1) and (2.2), we know ${\mathcal {I}}_{k}(u)\ge 0$ by (2.5). Therefore, we obtain

$$\begin{aligned} \liminf _{j\rightarrow +\infty }{\mathcal {I}}_{k}(u_{j})\ge \frac{1}{N} S^{\frac{N}{p}}=c^{\star }, \end{aligned}$$

which contradicts the assumption $c<c^{\star }$ and thus $a=0$, that is, $\Vert u_{j}-u\Vert _{p}\rightarrow 0$ as $j\rightarrow +\infty$. $\square$

Now, we want to estimate the mountain pass value $c_{k}$ given as (1.10) follow the arguments in [10]. We first recall the following technical estimates proved in [6]. From (H), there exist a positive constant b and a non-empty open set $A\subset \Omega$ such that

$$\begin{aligned} h(x)\ge b,\quad x\in A. \end{aligned}$$

(2.9)

We assume that $0\in A$ without loss of generality, and take a cutoff function $\eta \in C_{0}^{\infty }(A)$ such that $0\le \eta \le 1$ in A, $\eta =1$ in $B_{\rho }(0)$ and $\eta =0$ in $A{\setminus } B_{2\rho }(0)$, where $\rho$ is a small positive constant. Define

$$\begin{aligned} U_{\varepsilon }(x)=\frac{\eta (x)}{\left( \varepsilon ^{\frac{p}{p-1}}+|x|^{\frac{p}{p-1}}\right) ^{\frac{N-p}{p}}},\quad u_{\varepsilon }(x)=\frac{U_{\varepsilon }(x)}{|U_{\varepsilon }|_{p^{*}}} \end{aligned}$$

for $\varepsilon >0$. Hence, we know $|u_{\varepsilon }|_{p^{*}}=1$ and one can prove that (see, for example, [6])

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\nabla u_{\varepsilon }|^{p}{\textrm{d}}x=S+O\left( \varepsilon ^{\frac{N-p}{p-1}}\right) , \end{aligned}$$

(2.10)

where S is defined by (1.8),

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\nabla u_{\varepsilon }|^{q}{\textrm{d}}x= \left\{ \begin{array}{ll} O\left( \varepsilon ^{\frac{N(p-q)}{p}}\right) \quad &{}\text{ if }\;q>\frac{N(p-1)}{N-1},\\ O\left( \varepsilon ^{\frac{N(N-p)}{(N-1)p}}|\log \varepsilon |\right) \quad &{}\text{ if }\;q=\frac{N(p-1)}{N-1},\\ O\left( \varepsilon ^{\frac{(N-p)q}{p(p-1)}}\right) \quad &{}\text{ if }\;q<\frac{N(p-1)}{N-1}, \end{array} \right. \end{aligned}$$

(2.11)

and

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|u_{\varepsilon }|^{r}{\textrm{d}}x= \left\{ \begin{array}{ll} O\left( \varepsilon ^{\frac{Np-(N-p)r}{p}}\right) \quad &{}\text{ if }\,\,r>\frac{N(p-1)}{N-p},\\ O\left( \varepsilon ^{\frac{N}{p}}|\log \varepsilon |\right) \quad &{}\text{ if }\,\,r=\frac{N(p-1)}{N-p},\\ O\left( \varepsilon ^{\frac{(N-p)r}{p(p-1)}}\right) \quad &{}\text{ if }\,\,r<\frac{N(p-1)}{N-p}. \end{array} \right. \end{aligned}$$

(2.12)

For $\varepsilon >0$ and $0<\theta \le 1$, we let

$$\begin{aligned} U_{\varepsilon ,\theta }(x)=\frac{\eta \left( \frac{x}{\theta }\right) }{\left( \varepsilon ^{\frac{p}{p-1}}+|x|^{\frac{p}{p-1}}\right) ^{\frac{N-p}{p}}},\quad u_{\varepsilon ,\theta }(x)=\frac{U_{\varepsilon ,\theta }(x)}{|U_{\varepsilon ,\theta }|_{p^{*}}}. \end{aligned}$$

Then $|u_{\varepsilon ,\theta }|_{p^{*}}=1$ and one can derive that following estimates for $u_{\varepsilon ,\theta }$.

Lemma 2

For all $k\in (0,+\infty )$ and positive constant $\tau _{0}$, as $\varepsilon \rightarrow 0$ and $\frac{\varepsilon }{\theta }\rightarrow 0$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\nabla u_{\varepsilon ,\theta }|^{p}{\textrm{d}}x=S+O\left( \left( \frac{\varepsilon }{\theta }\right) ^{\frac{N-p}{p-1}}\right) , \end{aligned}$$

(2.13)

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\nabla u_{\varepsilon ,\theta }|^{q}{\textrm{d}}x= \left\{ \begin{array}{ll} O\left( \varepsilon ^{\frac{N(p-q)}{p}}\right) \quad &{}\text{ if }\,\,q>\frac{N(p-1)}{N-1},\\ O\left( \varepsilon ^{\frac{N(N-p)}{(N-1)p}}|\log \frac{\varepsilon }{\theta }|\right) \quad &{}\text{ if }\,\,q=\frac{N(p-1)}{N-1},\\ O\left( \varepsilon ^{\frac{(N-p)q}{p(p-1)}}\cdot \theta ^{\frac{N(p-1)-(N-1)q}{p-1}}\right) \quad &{}\text{ if }\,\,q<\frac{N(p-1)}{N-1}, \end{array} \right. \end{aligned}$$

(2.14)

and

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}h(x)\left[ \left( u_{\varepsilon ,\theta }-\frac{k}{\tau _{0}}\right) ^{+}\right] ^{r}{\textrm{d}}x\ge \left\{ \begin{array}{ll} O\left( \varepsilon ^{\frac{Np-(N-p)r}{p}}\right) \quad &{}\text{ if }\,\,r>\frac{N(p-1)}{N-p},\\ O\left( \varepsilon ^{\frac{N}{p}}|\log \frac{\varepsilon }{\theta }|\right) \quad &{}\text{ if }\,\,r=\frac{N(p-1)}{N-p},\\ O\left( \varepsilon ^{\frac{(N-p)r}{p(p-1)}}\cdot \theta ^{\frac{N(p-1)-(N-p)r}{p-1}}\right) \quad &{}\text{ if }\,\,r<\frac{N(p-1)}{N-p}. \end{array} \right. \end{aligned}$$

(2.15)

Proof

The proofs are depending on the observation

$$\begin{aligned} U_{\varepsilon ,\theta }(x)=\theta ^{-\frac{N-p}{p-1}}U_{\frac{\varepsilon }{\theta }}\left( \frac{x}{\theta }\right) \end{aligned}$$

and for all $k\in (0,+\infty )$, there exists $C_{k}>0$ such that for $\varepsilon$ small

$$\begin{aligned} u_{\varepsilon ,\theta }(x)-\frac{k}{\tau _{0}}\ge \frac{1}{2}u_{\varepsilon ,\theta }(x)\quad \text{ if }\quad |x|<C_{k}\varepsilon ^{\frac{1}{p}}. \end{aligned}$$

Then, combing with (2.10)–(2.12), we can get the desired results. One also can see Lemma 3.2 in [10] and Lemma 3 in [8]. $\square$

We prove the following result.

Lemma 3

For all $k\in (0,+\infty )$, we have $c_{k}<c^{\star }$.

Proof

We want to prove that

$$\begin{aligned} \max _{t\ge 0}{\mathcal {I}}_{k}(tu_{\varepsilon _{j},\theta _{j}})<c^{\star } \end{aligned}$$

as $\varepsilon \rightarrow 0$, $0<\theta _{j}\le 1$ and $\frac{\varepsilon _{j}}{\theta _{j}}\rightarrow 0$.

For simplicity, we denote $u_{j}(x)=u_{\varepsilon _{j},\theta _{j}}(x)$. Let

$$\begin{aligned} \varphi (t)&:={\mathcal {I}}_{k}(tu_{j})\\&=\frac{t^{p}}{p}\int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x+\frac{t^{q}}{q}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(tu_{j}-k)^{+}]^{r}{\textrm{d}}x-\frac{t^{p^{*}}}{p^{*}}. \end{aligned}$$

Suppose that the conclusion of the proposition is false, then there are renamed subsequences $\{\varepsilon _{j}\}$, $\{\theta _{j}\}$ and $t_{j}>0$ such that

$$\begin{aligned} \varphi (t_{j})&=\frac{t_{j}^{p}}{p}\int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x+\frac{t_{j}^{q}}{q}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(t_{j}u_{j}-k)^{+}]^{r}{\textrm{d}}x\nonumber \\&\quad -\frac{t_{j}^{p^{*}}}{p^{*}}\ge c^{\star } \end{aligned}$$

(2.16)

and

$$\begin{aligned} \varphi ^{\prime }(t_{j})&=t_{j}^{p-1}\int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x+t_{j}^{q-1}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x-\int _{\Omega }h(x)[(t_{j}u_{j}-k)^{+}]^{r-1}u_{j}{\textrm{d}}x\nonumber \\&\quad -t_{j}^{p^{*}-1}=0. \end{aligned}$$

(2.17)

By (2.17), we have

$$\begin{aligned} t_{j}^{p-1}\int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x+t_{j}^{q-1}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x&=\int _{\Omega }h(x)[(t_{j}u_{j}-k)^{+}]^{r-1}u_{j}{\textrm{d}}x+t_{j}^{p^{*}-1}\\&\ge t_{j}^{p^{*}-1}. \end{aligned}$$

By Lemma 2,

$$\begin{aligned} \int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x\rightarrow S,\quad \int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x\rightarrow 0. \end{aligned}$$

Hence, there exists $T_{1}>0$ such that $t_{j}\le T_{1}$. On the other hand, by (2.17) again and Hölder inequality,

$$\begin{aligned} t_{j}^{p-1}\int _{\Omega }|\nabla u_{j}|^{p}{\textrm{d}}x\le t_{j}^{r-1}\int _{\Omega }h(x)u_{j}^{r}{\textrm{d}}x+t_{j}^{p^{*}-1}\le t_{j}^{r-1}|h|_{\frac{Np}{Np+(p-N)r}}+t_{j}^{p^{*}-1}. \end{aligned}$$

From (H), we know there exists $T_{2}>0$ such that $t_{j}\ge T_{2}$. Now, we consider

$$\begin{aligned} g(t)=\frac{t^{p}}{p}\Vert u_{j}\Vert _{p}^{p}-\frac{t^{p^{*}}}{p^{*}}, \end{aligned}$$

the function attains its maximum at $t_{0}=\Vert u_{j}\Vert _{p}^{\frac{p}{p^{*}-p}}$ and thus

$$\begin{aligned} \varphi (t_{j})&= g(t_{j})+\frac{t_{j}^{q}}{q}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(t_{j}u_{j}-k)^{+}]^{r}{\textrm{d}}x\nonumber \\&\le g\left( \Vert u_{j}\Vert _{p}^{\frac{p}{p^{*}-p}}\right) +\frac{t_{j}^{q}}{q}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(t_{j}u_{j}-k)^{+}]^{r}{\textrm{d}}x\nonumber \\&\le c^{\star }+O\left( \left( \frac{\varepsilon _{j}}{\theta _{j}}\right) ^{\frac{N-p}{p-1}}\right) +\frac{T_{1}^{q}}{q}\int _{\Omega }|\nabla u_{j}|^{q}{\textrm{d}}x\nonumber \\&\quad -\frac{T_{2}^{r}}{r}\int _{\Omega }h(x)\left[ \left( u_{j}-\frac{k}{T_{2}}\right) ^{+}\right] ^{r}{\textrm{d}}x, \end{aligned}$$

(2.18)

since (2.13).

Next, we divide it into two cases. If $q<\frac{N(p-1)}{N-1}$ and $r>\frac{N^{2}(p-1)}{(N-1)(N-p)}$, then

$$\begin{aligned} r>\frac{N^{2}(p-1)}{(N-1)(N-p)}>\frac{N(p-1)}{N-p}. \end{aligned}$$

We take a sequence $\varepsilon _{j}\rightarrow 0$ and set $\theta _{j}=\varepsilon _{j}^{\kappa }$, where $\kappa \in [0,1)$ is to be determined later. Submitting (2.14) and (2.15) into (2.18), we have

$$\begin{aligned} \varphi (t_{j})\le c^{\star }+O\left( \varepsilon _{j}^{(1-\kappa )\frac{N-p}{p-1}}\right) +O\left( \varepsilon _{j}^{\frac{(N-p)q}{p(p-1)}+\kappa \cdot \frac{N(p-1)-(N-1)q}{p-1}}\right) -O\left( \varepsilon _{j}^{\frac{Np-(N-p)r}{p}}\right) . \end{aligned}$$

By a direct calculation, we obtain that

$$\begin{aligned} (1-\kappa )\frac{N-p}{p-1}-\frac{Np-(N-p)r}{p}=(\bar{\kappa }-\kappa )\frac{N-p}{p-1}, \end{aligned}$$

where

$$\begin{aligned} \bar{\kappa }=\frac{(N-p)(p-1)r-(Np-2N+p)p}{(N-p)p}, \end{aligned}$$

and

$$\begin{aligned}&\frac{(N-p)q}{p(p-1)}+\kappa \cdot \frac{N(p-1)-(N-1)q}{p-1}-\frac{Np-(N-p)r}{p}\\&\quad =(\kappa -\underline{\kappa })\frac{N(p-1)-(N-1)q}{p-1}, \end{aligned}$$

where

$$\begin{aligned} \underline{\kappa }=\frac{Np(p-1)-(N-p)(p-1)r-(N-p)q}{[N(p-1)-(N-1)q]p}. \end{aligned}$$

We want to choose $\kappa \in (0,1)$ such that $\kappa >\underline{\kappa }$ and $\kappa <\bar{\kappa }$. This is possible iff $\underline{\kappa }<\bar{\kappa }$, $\bar{\kappa }<1$ and $\underline{\kappa }>0$. We can check that these inequalities are equivalent to

$$\begin{aligned} r&>\frac{N^{2}(p-1)}{(N-1)(N-p)},\\ r&>\frac{Nq}{N-p},\\ r&>\frac{N^{2}(p-1)}{(N-1)(N-p)}-\frac{N-p}{(N-1)(p-1)}, \end{aligned}$$

respectively, all of which hold under our assumption on q and r.

If $q\ge \frac{N(p-1)}{N-1}$ and $r>\frac{Nq}{N-p}$. We take a sequence $\varepsilon _{j}\rightarrow 0$ and set $\theta _{j}=1$. Since

$$\begin{aligned} r>\frac{Nq}{N-p}\ge \frac{N^{2}(p-1)}{(N-1)(N-p)}>\frac{N(p-1)}{N-p}, \end{aligned}$$

then

$$\begin{aligned} \varphi (t_{j})\le \left\{ \begin{array}{ll} c^{\star }+O\left( \varepsilon _{j}^{\frac{N-p}{p-1}}\right) +O\left( \varepsilon _{j}^{\frac{N(p-q)}{p}}\right) -O\left( \varepsilon _{j}^{\frac{Np-(N-p)r}{p}}\right) \quad &{}\text{ if }\,\,q>\frac{N(p-1)}{N-1},\\ c^{\star }+O\left( \varepsilon _{j}^{\frac{N-p}{p-1}}\right) +O\left( \varepsilon _{j}^{\frac{N(N-p)}{(N-1)p}}|\log \varepsilon _{j}|\right) -O\left( \varepsilon _{j}^{\frac{Np-(N-p)r}{p}}\right) \quad &{}\text{ if }\,\,q=\frac{N(p-1)}{N-1}. \end{array} \right. \end{aligned}$$

We finish the proofs by the facts

$$\begin{aligned} r>\frac{Nq}{N-p}&\ge \frac{N^{2}(p-1)}{(N-1)(N-p)}>\frac{(Np-2N+p)p}{(N-p)(p-1)}\Longrightarrow \frac{N-p}{p-1}\\&>\frac{Np-(N-p)r}{p} \end{aligned}$$

and

$$\begin{aligned} r>\frac{Nq}{N-p}\Longrightarrow \left\{ \begin{array}{ll} \frac{N(p-q)}{p}>\frac{Np-(N-p)r}{p}\quad &{}\text{ if }\,\,q>\frac{N(p-1)}{N-1},\\ \frac{N(N-p)}{(N-1)p}>\frac{Np-(N-p)r}{p}\quad &{}\text{ if }\,\,q=\frac{N(p-1)}{N-1}. \end{array} \right. \end{aligned}$$

We complete the proofs. $\square$

From the above results, we already know that problem ($P_{k}$) has mountain pass solutions with energy value $c_{k}$ satisfies $c_{k}<c^{\star }$. Therefore, we have

Proposition 3

For all $k\in (0,+\infty )$, problem ($P_{k}$) has a non-trivial mountain pass solution $u_{k}$.

Proof

It is easy to check that energy functional ${\mathcal {I}}_{k}$ satisfies the mountain pass geometry. In fact, we first know that

$$\begin{aligned} {\mathcal {I}}_{k}(u)&=\frac{1}{p}\Vert u\Vert _{p}^{p}+\frac{1}{q}\Vert u\Vert _{q}^{q}-\frac{1}{r}\int _{\Omega }h(x)[(u-k)^{+}]^{r}{\textrm{d}}x-\frac{1}{p^{*}}|u|_{p^{*}}^{p^{*}}\nonumber \\&\ge \frac{1}{p}\Vert u\Vert _{p}^{p}+\frac{1}{q}\Vert u\Vert _{q}^{q}-\frac{1}{r}|h|_{\frac{Np}{Np+(p-N)r}}S^{-\frac{r}{p}}\Vert u\Vert _{p}^{r}-\frac{1}{p^{*}}S^{-\frac{p^{*}}{p}}\Vert u\Vert _{p}^{p^{*}}, \end{aligned}$$

(2.19)

where the Hölder and Sobolev inequalities have been used. For $1<q<p\le r<p^{*}$, there exists two constants $\rho ,\,\beta >0$ such that ${\mathcal {I}}_{k}(u)>\beta$ for any $u\in W_{0}^{1,p}(\Omega )$ satisfies $\Vert u\Vert _{p}=\rho$. On the other hand, for $u\in W_{0}^{1,p}(\Omega )$ such that $u^{+}\not \equiv 0$, we can deduce that

$$\begin{aligned} {\mathcal {I}}_{k}(tu)\rightarrow -\infty \quad \text{ as }\quad t\rightarrow +\infty . \end{aligned}$$

So we can choose $t_{0}>0$ such that $\Vert t_{0}u\Vert _{p}>\rho$ and ${\mathcal {I}}_{k}(t_{0}u)<0$.

Hence, there exists a (PS) sequence $\{u_{j}\}$ of ${\mathcal {I}}_{k}$ of mountain pass type at level $c_{k}$ (see [23]). Since ${\mathcal {I}}_{k}(|u|)\le {\mathcal {I}}_{k}(u)$ for all $u\in W_{0}^{1,p}(\Omega )$, we can assume that $\{u_{j}\}\subset \mathcal {M}$. Lemma 2.5 ensures that $c_{k}<c^{\star }$ and Lemma 2 implies that (PS) condition holds for $\{u_{j}\}$. Therefore, we know that there exists a mountain pass solution $u_{k}\in \mathcal {M}$ of ($P_{k}$).

Suppose $u_{k}=0$. Since $\{u_{j}\}$ is a $\text{(PS) }_{c_{k}}$ sequence and $u_{j}\rightharpoonup u_{k}$ in $W_{0}^{1,p}(\Omega )$, by Lemma 1, we have

$$\begin{aligned} \frac{1}{p}\Vert u_{j}\Vert _{p}^{p}+\frac{1}{q}\Vert u_{j}\Vert _{q}^{q}-\frac{1}{p^{*}}|u_{j}|_{p^{*}}^{p^{*}}&=c_{k}+o(1), \end{aligned}$$

(2.20)

$$\begin{aligned} \Vert u_{j}\Vert _{p}^{p}+\Vert u_{j}\Vert _{q}^{q}-|u_{j}|_{p^{*}}^{p^{*}}&=o(1). \end{aligned}$$

(2.21)

Equations (2.21) together with (1.8) gives

$$\begin{aligned} \Vert u_{j}\Vert _{p}^{p}\le S^{-\frac{p^{*}}{p}}\Vert u_{j}\Vert _{p}^{p^{*}}+o(1). \end{aligned}$$

(2.22)

If $\Vert u_{j}\Vert _{p}^{p}\rightarrow 0$ for a renamed subsequence, then (2.20) implies $c_{k}=0$ which is a contradiction. Then, by (2.21), we have

$$\begin{aligned} \Vert u_{j}\Vert _{p}^{p}\ge S^{\frac{N}{p}}+o(1). \end{aligned}$$

Finally, dividing (2.21) by $p^{*}$ and subtracting from (2.20) give

$$\begin{aligned} c_{k}=\left( \frac{1}{p}-\frac{1}{p^{*}}\right) \Vert u_{j}\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{p^{*}}\right) \Vert u_{j}\Vert _{q}^{q}+o(1) \ge c^{\star }+o(1), \end{aligned}$$

that is, $c_{k}\ge c^{\star }$, contrary to assumption. $\square$

We can prove that the mountain pass solution $\{u_{k}\}$ obtained in Proposition 3 is uniformly bounded in $W^{1,p}_{0}(\Omega )$.

Lemma 4

There exists a constant $\Lambda >0$ such that for all $k\in (0,+\infty )$ and for all $u_{k}$ being a mountain pass solution of ($P_{k}$), we have $\Vert u_{k}\Vert _{p}\le \Lambda$.

Proof

This lemma is a direct conclusion of (2.1) and (2.2) since $c_{k}<c^{\star }$. $\square$

We finish this section by showing the weak limit of mountain pass solution is also a solution

Lemma 5

Let $\{k_{j}\}\subset (0,+\infty )$ be a sequence such that $k_{j}\rightarrow k\in (0+\infty ]$ and $\{u_{k_{j}}\}$ is a sequence of mountain pass solutions of $(P_{k_{j}})$. Then there exists $u\in W_{0}^{1,p}(\Omega )$ such that $u_{k_{j}}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega )$, up to a subsequence. In addition, u is a solution of problem ($P_{k}$).

Proof

It follows from Lemma 4 that $\{u_{k_{j}}\}$ is uniformly bounded in $W_{0}^{1,p}(\Omega )$ and thus there exists $u\in W_{0}^{1,p}(\Omega )$ such that $u_{k_{j}}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega )$, up to a subsequence. Moreover, since ${\mathcal {I}}_{k_{j}}^{\prime }(u_{k_{j}})=0$, for any $v\in W_{0}^{1,p}(\Omega )$, we have

$$\begin{aligned} \int _{\Omega }|\nabla u_{k_{j}}|^{p-2}\nabla u_{k_{j}}\nabla v{\textrm{d}}x+\int _{\Omega }|\nabla u_{k_{j}}|^{q-2}\nabla u_{k_{j}}\nabla v{\textrm{d}}x&=\int _{\Omega }f(x,u_{k_{j}}-k_{j})v{\textrm{d}}x\\&\quad -\int _{\Omega }u_{k_{j}}^{p^{*}-1}v{\textrm{d}}x. \end{aligned}$$

By Lemma 1 and (2.3), we know u solves equation ($P_{k}$) by passing the limit $k_{j}\rightarrow k$ as $j\rightarrow +\infty$. $\square$

3 Proofs of the Main Results

This section is devoted to prove our main results.

Proof of Theorem 1

Firstly, similar as the proof of Lemma 3, we can get

$$\begin{aligned} c_{\infty }\le c^{\star }. \end{aligned}$$

(3.1)

Next, we show that ${\mathcal {I}}_{\infty }(u)>c^{\star }$ for any non-trivial solution u of $P_{\infty }$. In fact, since u is a non-trivial solution u of $P_{\infty }$, then

$$\begin{aligned} 0=\langle {\mathcal {I}}_{\infty }^{\prime }(u),u\rangle =\Vert u\Vert _{p}^{p}+\Vert u\Vert _{q}^{q}-|u|_{p^{*}}^{p^{*}}\Longrightarrow \Vert u\Vert _{p}^{p}+\Vert u\Vert _{q}^{q}=|u|_{p^{*}}^{p^{*}}. \end{aligned}$$

(3.2)

So

$$\begin{aligned} {\mathcal {I}}_{\infty }(u)&=\frac{1}{p}\Vert u\Vert _{p}^{p}+\frac{1}{q}\Vert u\Vert _{q}^{q}-\frac{1}{p^{*}}|u|_{p^{*}}^{p^{*}}\nonumber \\&\ge \frac{1}{p}(\Vert u\Vert _{p}^{p}+\Vert u\Vert _{q}^{q})-\frac{1}{p^{*}}|u|_{p^{*}}^{p^{*}}\nonumber \\&=\left( \frac{1}{p}-\frac{1}{p^{*}}\right) |u|_{p^{*}}^{p^{*}}. \end{aligned}$$

(3.3)

By (1.8), we also know

$$\begin{aligned} S<\frac{\Vert u\Vert _{p}^{p}}{|u|_{p^{*}}^{p}}\le \frac{\Vert u\Vert _{p}^{p}+\Vert u\Vert _{q}^{q}}{|u|_{p^{*}}^{p}}. \end{aligned}$$

(3.4)

Combining (3.2) and (3.4), we have

$$\begin{aligned} |u|_{p^{*}}>S^{\frac{1}{p^{*}-p}}. \end{aligned}$$

Finally, we can obtain that ${\mathcal {I}}_{\infty }(u)>c^{\star }$ by submitting the previous estimate into (3.3). $\square$

Here, we obtain some technical results before proving the remained theorems.

Lemma 6

For all $k\in (0,+\infty )$, suppose $u_{k}$ is mountain pass solution of ($P_{k}$), then there is a positive constant $\lambda$ independent of k satisfies

$$\begin{aligned} \max _{u\in \gamma _{k}}{\mathcal {I}}_{k}(u)=c_{k} \end{aligned}$$

with $\gamma _{k}=\left[ 0,\lambda \frac{u_{k}}{\Vert u_{k}\Vert _{p}}\right]$ is segment.

Proof

By (2.19) and the definition of $c_{k}$, we know there exists constant $\tau >0$ such that

$$\begin{aligned} c_{k}\ge \tau . \end{aligned}$$

(3.5)

Indeed, there exists $\rho >0$ and $\tau >0$ small such that ${\mathcal {I}}_{k}(u)\ge \tau$ if $\Vert u\Vert _{p}=\rho$ and ${\mathcal {I}}_{k}(u)>0$ if $\Vert u\Vert _{p}<\rho$.

Then, from (3.5), Hölder inequality and the fact $\langle {\mathcal {I}}^{\prime }(u_{k}),u_{k}\rangle =0$, we get

$$\begin{aligned} \tau \le c_{k}={\mathcal {I}}(u_{k})\le \frac{1}{q}(\Vert u_{k}\Vert _{p}^{p}+\Vert u_{k}\Vert _{q}^{q})\le C|u_{k}|_{p^{*}}^{r}+\frac{1}{q}|u_{k}|_{p^{*}}^{p^{*}} \end{aligned}$$

which yields that there exists $\rho >0$ such that

$$\begin{aligned} |u_{k}|_{p^{*}}\ge \rho \end{aligned}$$

(3.6)

for all $k\in (0,+\infty )$.

Next, we claim, for all $k\in (0,+\infty )$, that

$$\begin{aligned} c_{k}=\max _{t\ge 0}{\mathcal {I}}_k(tu_{k}). \end{aligned}$$

(3.7)

We prove the claim is true. Define $\varphi _{k}(t)={\mathcal {I}}_{k}(tu_{k})$. It follows from $\langle {\mathcal {I}}^{\prime }(u_{k}),u_{k}\rangle =0$ that

$$\begin{aligned} \varphi _{k}^{\prime }(t)&=(t^{q-1}-t^{p-1})\Vert u_{k}\Vert _{q}^{q}+(t^{p-1}-t^{p^{*}-1})|u_{k}|_{p^{*}}^{p^{*}}\\&\quad +\int _{\Omega }h(x)u_{k}\left( t^{p-1}[(u_{k}-k)^{+}]^{r-1}-[(tu_{k}-k)^{+}]^{r-1}\right) {\textrm{d}}x. \end{aligned}$$

Since $q<p\le r<p^{*}$, one can observe that $\varphi _{k}(t)$ is increasing in (0, 1) and non-increasing in $(1,+\infty )$. Hence,

$$\begin{aligned} \max _{t\ge 0}\varphi _{k}(t)=\varphi _{k}(1)={\mathcal {I}}_{k}(u_{k})=c_{k}. \end{aligned}$$

By Lemma 4 and (3.6), we can choosing $T>1$ such that

$$\begin{aligned} {\mathcal {I}}_{k}(Tu_{k})\le {\mathcal {I}}_{\infty }(Tu_{k})\le 0. \end{aligned}$$

Then, let $\lambda =\Lambda T$, where $\Lambda$ given in Lemma 4, we can obtain

$$\begin{aligned} {\mathcal {I}}_{k}\left( \lambda \frac{u_{k}}{\Vert u_{k}\Vert _{p}}\right) ={\mathcal {I}}_{k}\left( \frac{\Lambda T}{\Vert u_{k}\Vert _{p}}u_{k}\right) \le {\mathcal {I}}_{k}\left( Tu_{k}\right) \le 0. \end{aligned}$$

(3.8)

Finally, the deserved result is a direct conclusion of (3.7) and (3.8). $\square$

The next lemma shows that the mountain pass value $c_{k}$ is continuous respect to parameter k.

Lemma 7

Suppose $\{k_{j}\}\subset (0,+\infty )$ is a sequence satisfies $k_{j}\rightarrow k<+\infty$ as $j\rightarrow +\infty$, then $c_{k_{j}}\rightarrow c_{k}$. In addition, for any compact intervals $J\subset (0,+\infty )$, there exists $\tau _{J}>0$ such that if $k\in J$, then

$$\begin{aligned} c_{k}\le c^{\star }-\tau _{J}. \end{aligned}$$

Proof

By the mean value theorem, there exists $\sigma _{k}\in (k_{j},k)$ (notice that we may assume $k_{j}<k$) such that

$$\begin{aligned} |h(x)([(u-k)^{+}]^{r}-[(u-k_{j})^{+}]^{r})|=r|k-k_{j}||h(x)[(u-\sigma _{k})^{+}]^{r-1}|, \end{aligned}$$

and then,

$$\begin{aligned} \left| \int _{\Omega }h(x)([(u-k)^{+}]^{r}-[(u-k_{j})^{+}]^{r}){\textrm{d}}x\right| \le C|k-k_{j}|, \end{aligned}$$

(3.9)

for $u\in \mathcal {B}_{R}(0):=\{u\in W_{0}^{1,p}(\Omega ):\Vert u\Vert _{p}\le R\}$, where C is positive constant depending on R. By (3.9), we know, for any bounded subset in $W_{0}^{1,p}(\Omega )$, that

$$\begin{aligned} {\mathcal {I}}_{k_{j}}\rightarrow {\mathcal {I}}_{k}\quad \text{ uniformly } \text{ as }\;\,j\rightarrow +\infty . \end{aligned}$$

(3.10)

Recall $\gamma _{k}$ defined in Lemma 6, and set

$$\begin{aligned} \Sigma :=\{\gamma _{j}: j=k_{1},\,k_{2},\ldots \,\,\text{ or }\,\,j=k\}. \end{aligned}$$

Then, $\Sigma \subset \mathcal {B}_{\lambda }(0)$ and by (3.10), we have

$$\begin{aligned} c_{k_{j}}=\inf _{\gamma \in \Sigma }\max _{u\in \gamma }I_{k_{j}}(u)\rightarrow \inf _{\gamma \in \Sigma }\max _{u\in \gamma }I_{k}(u)=c_{k}\quad \text{ as }\quad j\rightarrow +\infty . \end{aligned}$$

We observe that $J\subset (0,+\infty )$ is compact and $c_{k}$ is continuous in k, so the maximum $\bar{c}$ on J can be attained. By Lemma 3, we know $c_{k}<c^{\star }$ for any $k\in J$. Hence, we take $\tau _{J}=c^{\star }-\bar{c}$. $\square$

Now, we are in position to prove Theorem 2.

Proof of Theorem 1

Firstly, the existence of mountain pass solutions to problem ($P_{k}$) is proved in Proposition 3. Next, we show $x_{0}^{k}$ exists. Suppose that $|u_{k}|_{\infty }\le k$ and that $u_{k}$ is a mountain pass solution of ($P_{k}$), then $(u_{k}-k)^{+}\equiv 0$, and thus, $u_{k}$ solves $P_{\infty }$, which is impossible (see Theorem 1). So, $|u_{k}|_{\infty }> k$, and $\Omega (u_{k})$ is not empty (see (1.11)). We can choose $x_{0}^{k}\in \Omega (u_{k})$.

By Lemma 5, we know u solves ($P_{k}$). Now, we need to show u is a mountain pass solution to ($P_{k}$). From Lemma 1, we know

$$\begin{aligned} {\mathcal {I}}_{k}(u_{k_{j}})-{\mathcal {I}}_{k_{j}}(u_{k_{j}})=\frac{1}{r}\int _{\Omega }h(x)([(u_{k_{j}}-k)^{+}]^{r}-[(u_{k_{j}}-k_{j})^{+}]^{r}){\textrm{d}}x\rightarrow 0. \end{aligned}$$

By Lemma 7, we know ${\mathcal {I}}_{k_{j}}(u_{k_{j}})\rightarrow c_{k}<c^{\star }$, up to a subsequence. On the other hand, by Lemma 1 again, one can check that ${\mathcal {I}}_{k}^{\prime }(u_{k_{j}})\rightarrow 0$. Hence, $\{u_{k_{j}}\}$ is a (PS) sequence of ${\mathcal {I}}_{k}$. So by Lemma 2, $u_{k_{j}}\rightarrow u$ in $W_{0}^{1,p}(\Omega )$. Moreover, using Lemma 7 again, we have

$$\begin{aligned} {\mathcal {I}}_{k}(u)=\lim _{j\rightarrow +\infty }{\mathcal {I}}_{k_{j}}(u_{k_{j}})=\lim _{j\rightarrow +\infty }c_{k_{j}}=c_{k}. \end{aligned}$$

We complete the proofs. $\square$

Suppose $k\rightarrow +\infty$, then the subcritical perturbation term will vanish, and we will show the energy of mountain pass solutions tends to threshold value $c^{\star }$.

Lemma 8

Assume that $u_{k}$ is the mountain pass solution of ($P_{k}$) for $k\in (0,+\infty )$, then

$$\begin{aligned} \lim _{k\rightarrow +\infty }{\mathcal {I}}_{k}(u_{k})=c^{\star }. \end{aligned}$$

Proof of Theorem 2

From Lemma 5, then $u_{k}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega )$ and u solves $P_{\infty }$. By Lemma 1 and (1.8), we can get

$$\begin{aligned} {\mathcal {I}}_{k}(u_{k})&=\frac{1}{p}\Vert u_{k}\Vert _{p}^{p}+\frac{1}{q}\Vert u_{k}\Vert _{q}^{q}-\frac{1}{p^{*}}|u_{k}|^{p^{*}}_{p^{*}}+o(1)\\&\ge \frac{1}{p}\Vert u_{k}\Vert _{p}^{p}-\frac{1}{p^{*}}S^{-\frac{p^{*}}{p}}\Vert u_{k}\Vert ^{p^{*}}_{p}+o(1). \end{aligned}$$

Define $g(t)=\frac{1}{p}t^{p}-\frac{1}{p^{*}}S^{-\frac{p^{*}}{p}}t^{p^{*}}$, then it is easy to check that

$$\begin{aligned} \max _{t\ge 0}g(t)=g\left( S^{\frac{N}{p^{2}}}\right) =c^{\star }. \end{aligned}$$

Since the $W_{0}^{1,p}$-sphere of radius $S^{\frac{N}{p^{2}}}$ separates 0 and $\mathcal {N}$, by the variational characterization of $c_{k}$ we get

$$\begin{aligned} {\mathcal {I}}_{k}(u_{k})=c_{k}\ge c^{\star }+o(1). \end{aligned}$$

The converse inequality follows by applying Lemma 3 for all k. $\square$

Finally, we prove the last two theorems based on the previous lemmas.

Proof of Theorem 3

By Lemma 5, we know that there exists $u\in W_{0}^{1,p}(\Omega )$ solves $P_{\infty }$ and satisfies $u_{k}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega )$. We claim that $u\equiv 0$. In fact, inserting $\langle {\mathcal {I}}_{k}^{\prime }(u_{k}),u_{k}\rangle =0$ into ${\mathcal {I}}_{k}$, we can have

$$\begin{aligned} {\mathcal {I}}_{k}(u_{k})&=\left( \frac{1}{p}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{q}^{q}\\&\quad +\frac{1}{p^{*}}\int _{\Omega }h(x)[(u_{k}-k)^{+}]^{r-1}u_{k}{\textrm{d}}x-\frac{1}{r}\int _{\Omega }h(x)[(u_{k}-k)^{+}]^{r}{\textrm{d}}x. \end{aligned}$$

From Lemmas 1 and 8, we know

$$\begin{aligned} \left( \frac{1}{p}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{q}^{q}\rightarrow c^{\star }. \end{aligned}$$

On the other hand, by $\langle {\mathcal {I}}_{\infty }^{\prime }(u),u\rangle =0$, we can get

$$\begin{aligned} {\mathcal {I}}_{\infty }(u)&=\left( \frac{1}{p}-\frac{1}{p^{*}}\right) \Vert u\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{p^{*}}\right) \Vert u\Vert _{q}\\&\le \liminf _{k\rightarrow +\infty }\left( \left( \frac{1}{p}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{p}^{p}+\left( \frac{1}{q}-\frac{1}{p^{*}}\right) \Vert u_{k}\Vert _{q}^{q}\right) \\&=c^{\star }. \end{aligned}$$

However, this is impossible unless $u\equiv 0$, see the proof of Theorem 1.

Hence, by Proposition 1, we have

$$\begin{aligned} |\nabla u_{k_{j}}|^{p}\rightharpoonup \mu \ge \sum _{i\in I}\mu _{i}\delta _{x_{i}},\quad |u_{k_{j}}|^{p^{*}}\rightharpoonup \nu =\sum _{i\in I}\nu _{i}\delta _{x_{i}}, \end{aligned}$$

(3.11)

where set I is at most countable, $x_{i}\in \bar{\Omega }$, $\delta _{x_{i}}$ is the Dirac measure at $x_{i}$ and

$$S\nu _{i}^{\frac{p}{p^{*}}}\le \mu _{i}.$$

By a standard argument (see the proof of [12, Lemma 2.3]), one also can get

$$\begin{aligned} \nu _{i}\ge S^{\frac{p^{*}}{p^{*}-p}}, \end{aligned}$$

which infer (2) and (3) in Theorem 3.

Finally, using Lemma 8 and (3.11), we know there exists unique index $i_{0}$ such that $\mu _{i_{0}}=\nu _{i_{0}}=S^{\frac{N}{p}}$. Hence, we may choose a subsequence of $\{u_{k_{j}}\}$ such that the sequence $\{x_{0}^{k_{j}}\}$ converge to the same point $x_{0}\in \bar{\Omega }$. $\square$

Finally, we prove Theorem 4.

Proof of Theorem 4

Suppose $\{u_{k}\}$ are mountain pass solutions of ($P_{k}$) and if

$$\begin{aligned} \Omega (u_{k})\cap \text{ supp }\,h=\emptyset , \end{aligned}$$

then $h(x)[(u_{k}-k)^{+}]^{r-1}\equiv 0$ which means $u_{k}$ solves $P_{\infty }$, contradicts with Theorem 1.

Therefore, for all $k\in (0,+\infty )$, there exists

$$\begin{aligned} x_{0}^{k}\in \Omega (u_{k})\cap \text{ supp }\,h. \end{aligned}$$

Then, as the proof of Theorem 3, we know the limit $x_{0}$ of $x_{0}^{k}$ belongs to $\text{ supp }\,h$. $\square$