A Note on Radial Solutions to the Critical Lane-Emden Equation with a Variable Coefficient

Abstract

In this note, we consider the following problem

$$\displaystyle \begin {cases} -\Delta u=(1+g(x))u^{\frac {N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end {cases} $$

where N ≥ 3 and $B\subset \mathbb {R}^N$ is the unit ball centered at the origin and g(x) is a radial Hölder continuous function such that g(0) = 0. We prove the existence and nonexistence of radial solutions by the variational method with the concentration compactness analysis and the Pohozaev identity.

1 Introduction

We study the following problem

$$\displaystyle \begin{aligned} \begin{cases} -\Delta u=(1+g(x))u^{\frac{N+2}{N-2}},\ u>0\text{ in }B\\ u=0\text{ on }\partial B, \end{cases}{} \end{aligned} $$

(1.1)

where $B\subset \mathbb {R}^N$ is the unit ball centered at the origin with N ≥ 3, g is a locally Hölder continuous function in $\overline {B}$ and radial, i.e., g(x) = g(|x|). We note that a typical case is given by g(x) = |x|^β with β ≥ 0. We will show some existence and nonexistence results for (1.1).

First let us consider the next basic problem which is extensively investigated by many authors;

$$\displaystyle \begin{aligned} \begin{cases} -\Delta u=u^{\frac{N+2}{N-2}},\ u>0\text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases}{} \end{aligned} $$

(1.2)

where Ω is a smooth bounded domain in $\mathbb {R}^N$ with N ≥ 3. Since the nonlinearity $u^{\frac {N+2}{N-2}}$ has the critical growth, as is well-known, due to the lack of the compactness of the associated Sobolev embedding $H_0^1(\Omega )\hookrightarrow L^{\frac {2N}{N-2}}(\Omega )$, the existence/nonexistence of solutions of (1.2) becomes a very delicate and interesting question. In fact, in contrast to the subcritical case, we can prove that (1.2) has no smooth solution if Ω is a star-shaped domain by the Pohozaev identity [15] (see also [6]). Hence in order to ensure the existence of solutions of (1.2), we need some “perturbation” to (1.2). A celebrated work in this direction is given by [6]. They add a lower order term λu ^q (1 ≤ q < (N + 2)∕(N − 2)) to the critical nonlinearity $u^{\frac {N+2}{N-2}}$ (i.e., replace $u^{\frac {N+2}{N-2}}$ by $u^{\frac {N+2}{N-2}}+\lambda u^q$) and successfully show the existence of solutions of (1.2). After that, [4, 5, 8] prove that the topological perturbation to the domain can also induce solutions to (1.2). See also [10, 14] for the effect of the geometric perturbation to the domain. Furthermore, another perturbation is found by Ni [13]. He considers a variable coefficient |x|^α with α > 0 on $u^{\frac {N+2}{N-2}}$. More precisely he investigates

$$\displaystyle \begin{aligned} \begin{cases} -\Delta u=|x|{}^\alpha u^{p},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end{cases}{} \end{aligned} $$

(1.3)

where α > 0 and $p\in (1,\frac {N+2+2\alpha }{N-2})$. The crucial role of the variable coefficient |x|^α appears in the following compactness lemma for radially symmetric functions in $H^1_0(B)$. Here we define H _r(B) is a subspace of $H_0^1(B)$ which consists of all radial functions.

Lemma 11 ([13])

The map u↦|x|^m u from H _r(B) to L ^p(B) is compact, for $p\in [1,\tilde {m})$ where

$$\displaystyle \begin{aligned} \tilde{m}=\begin{cases} \frac{2N}{N-2-2m}\mathit{\text{ if }}m<\frac{N-2}{2}\\ \infty\ \mathit{\text{ otherwise }}. \end{cases} \end{aligned}$$

Applying this, one successfully obtains the existence of a mountain pass solution of (1.3) for all $p\in \left (1,\frac {N+2+2\alpha }{N-2}\right )$. The exponent p can be supercritical (i.e., $p > \frac {N+2}{N-2}$) if α > 0. We here note that, for the critical or supercritical case in (1.3), the essential point to assure the existence seems that $u^{\frac {N+2}{N-2}}$ has a variable coefficient which is radial and attains 0 at the origin (see Example 2.1 in [17]). In view of this it is an interesting question that whether it is possible to ensure the existence of solutions in the case where the coefficient does not attain 0 at the origin. Very recently, Ai-Cowan [2] study another problem including our problem (1.1). Applying their dynamical system approach, which is developed in [1], the authors in [2] can confirm the existence of radially symmetric solutions of (1.1) for the case g(x) = |x|^β with β ∈ (0, N − 2). An interesting point in this case is that the coefficient (1 + g(x)) attains a local minimum at the origin that is not 0. Hence we cannot apply Lemma 11 directly. Then it is an interesting question to investigate how the coefficient can exclude the non-compactness of their nonlinearity. Motivated by this, we investigate (1.1) via the variational method. Our aim is to give a variational interpretation on the results in [2] and further, to extend their results to a more general coefficient which has a local minimum at the origin.

Now in order to explain our main results, we give a comment on the results in [2]. In the variational point of view, it seems better to write the right hand side of the equation of (1.1) as $u^{\frac {N+2}{N-2}}+g(x)u^{\frac {N+2}{N-2}}$. Then the first term is actually noncompact. On the other hand, the second one becomes compact by Lemma 11 if g(x) behaves like |x|^β with β > 0. Then we clearly expect that it would play the role of the subcritical perturbation λu ^q with 1 ≤ q < (N + 2)∕(N − 2) in [6] mentioned above.

Then, it is natural to consider the next more general problem. (See also the generalization in [2].)

$$\displaystyle \begin{aligned} \begin{cases} -\Delta u=u^{\frac{N+2}{N-2}}+\lambda k(x) f(u),\ u>0&\text{ in }B,\\ u=0&\text{ on }\partial B{} \end{cases} \end{aligned} $$

(1.4)

where λ > 0 is a parameter and $k:\overline {B}\to \mathbb {R}$ and $f:\mathbb {R}\to \mathbb {R}$ satisfy some of the next assumptions.

• (k1) k(x)≢0 is a nonnegative Hölder continuous function on $\overline {B}$ and radial, i.e., k(x) = k(|x|).

• (k2) k(x) = O(|x|^β) (|x|→ 0) for some β > 0.

• (k3) There exist constants γ ≥ β > 0 and C, δ > 0 such that k(|x|) ≥ C|x|^γ for all |x|∈ (0, δ).

• (f1) f(t) is locally Hölder continuous function on [0, ∞] and f(t) ≥ 0 for all t > 0 and f(t) = 0 for all t ≤ 0.

• (f2) $\lim _{t\to 0}\frac {f(t)}{t}=0$ and $\lim _{t\to \infty }\frac {f(t)}{t^q}=0$ for q = (N + 2 + 2β)∕(N − 2).

• (f3) There exists a constant θ > 2 such that f(t)t ≥ θF(t) for all t ≥ 0 where $F(t):=\int _0^tf(s)ds$.

Now, we give our main results.

Theorem 12

We have the following.

• (i) If k, f satisfy (k1), (k2), (k3), (f1), (f2), (f3) and further,

  • (f4) $\lim _{t\to \infty }\frac {f(t)}{t^p}=\infty$ for $p=\max \left \{1,\frac {2\gamma +6-N}{N-2}\right \}$,

  then (1.4) admits a radially symmetric solution for all λ > 0.

• (ii) If k, f verify (k1), (k2), (f1), (f2), (f3) and further,

  • (k4) there exists a point $x_0\in \overline {B}$ such that k(x ₀) > 0 and,

  • (f5) there exists a constant c > 0 such that f(t) > 0 for all t ∈ (0, c),

  then, there exists a constant λ ^∗ > 0 such that (1.4) has a radially symmetric solution for all λ > λ ^∗.

Remark 13

The hypothesis in Theorem 12 (i) permits the case where k(x) = |x|^β for β > 0 and $f(u)=u_+^{q}$ with any $q\in (\max \{1,(2\beta +6-N)/(N-2)\},(N+2+2\beta )/(N-2))$. The condition $q>\max \{1,(2\beta +6-N)/(N-2)\}$ is assumed to lower the mountain pass energy down to the level for which the local compactness of the Palais-Smale sequences is valid. See Lemmas 23 and 24 for the detail. On the other hand, (ii) is valid for $f(u)=u_+^{q}$ with any q ∈ (1, (N + 2 + 2β)∕(N − 2)).

Remark 14

A similar problem is considered in [7] and [9]. The existence and nonexistence for the linear perturbation case with k(r) = r ^β for β > 0 and f(t) = t ₊ are completed by [7]. Furthermore, the superlinear perturbation case with k(r) = r ^β for β > 0 and $f(t)=t_+^{q}$ with q ∈ (1, (N + 2 + 2β)∕(N − 2)) is treated in [9]. Our theorem gives a generalization of a part of their results.

A nonexistence result on (1.4) is given by the Pohozaev identity as follows.

Theorem 15

Let $\lambda \in \mathbb {R},$ k(x) = |x|^β with β ≥ 0, $f(u)=u_+^q$ and q ≥ 1. Then (1.4) admits no solution if one of the following is true;

• (i) q ∈ [1, (2β + N + 2)∕(N − 2)] and λ ≤ 0, or

• (ii) $q\ge \frac {2\beta +N+2}{N-2}$ and λ ≥ 0, or,

• (iii) β = 0 and q = (N + 2)∕(N − 2).

Remark 16

The same conclusion holds even if we replace the domain B by any star-shaped domain. See the argument in Sect. 3.

Now we come back to our main question on (1.1). The desired existence results are given as a corollary of (i) of Theorem 12.

Corollary 17

We assume

• (g1) g(x) is Hölder continuous and g ≥−1 on $\overline {B}$ and radial, i.e., g(x) = g(|x|),

• (g2) g(0) = 0, and

• (g3) there exist constants γ ∈ (0, N − 2), δ ∈ (0, 1] and C > 0 such that g(|x|) ≥ C|x|^γ for all |x|∈ (0, δ).

Then (1.1) admits at least one radially symmetric solution.

Remark 18

This theorem generalizes Theorem 2 in [2] for the case g(|x|, u) = g(|x|). To see this, note first that their condition (6) in [2] implies (g2) and (g3). Furthermore, since (g3) is a condition for the behavior of g only near the origin, we can easily construct an example which satisfies (g2) and (g3), but not (6). In addition, they prove Theorem 2 in [2] by dynamical system approach while we shall prove it via the variational method with the concentration compactness analysis. Hence our proof can give a variational interpretation and a generalization of their theorem.

By Corollary 17, we have the existence of solution of (1.1) if g(x) = λ|x|^β with β ∈ (0, N − 2) and λ > 0. For the case including β ≥ N − 2, we have the next corollary as a direct consequence of (ii) in Theorem 12.

Corollary 19

Let λ > 0, g(x) = λk(x) and k(x) is a nonnegative Hölder continuous function in $\overline {B}$ such that k(0) = 0 and k(x) = k(|x|). Furthermore, assume there exists a point $x_0\in \overline {B}$ such that k(x ₀) > 0. Then there exists a constant λ ^∗ > 0 such that (1.1) admits at least one radially symmetric solution for all λ > λ ^∗.

Remark 110

Corollary 19 implies that if g(x) = λ|x|^β with β > 0, a radially symmetric solution exists for all sufficiently large λ > 0. Furthermore, we remark that this generalizes Theorem 1 of [2].

The existence results above are best possible in the following sense. We have the following nonexistence result.

Theorem 111

Let g(x) = λ|x|^β with β ≥ 0 and $\lambda \in \mathbb {R}$ . Then (1.1) does not admit any radially symmetric solution if β = 0 and $\lambda \in \mathbb {R}$ , or β ≥ 0 and λ ≤ 0. In addition if β ≥ N − 2, there exists a constant λ _∗ > 0 which depends on β and N such that (1.1) has no radially symmetric solution for all λ ∈ [0, λ _∗].

Remark 112

In our computation, we can choose

$$\displaystyle \begin{aligned} \lambda_*=\begin{cases}\frac{2(N-1)}{N-2}\text{ if }\beta=N-2,\\ \frac{2(N-1)}{N-2}\left(\frac{2N-2+\beta}{\beta-N+2}\right)^{\frac{\beta-N+2}{N-2}}\text{ if }\beta>N-2. \end{cases} \end{aligned}$$

For the detail, see the proof of Theorem 111 in Sect. 3.

1.1 Organization of This Paper

This paper consists of three sections with an appendix. In Sect. 2, we give the proof of the existence results. In Sect. 3, we show the nonexistence assertions by the Pohozaev identity. Lastly in Appendix A, we give a remark on the proof for the critical case for the reader’s convenience. Throughout this paper we define H _r(B) as a subspace of $H_0^1(B)$ which consists of all the radial functions. Furthermore we put 2^∗ = 2N∕(N − 2) and define the Sobolev constant S > 0 as usual by

$$\displaystyle \begin{aligned} S:=\inf_{u\in H^1_0(B)\setminus\{0\}}\frac{\| u \|{}^2}{\int_B|u|{}^{2^*}dx} \end{aligned}$$

where ∥u∥² =∫_B|∇u|² dx. Finally we define B _s(0) as a N dimensional ball centered at the origin with radius s > 0.

2 Existence Results

In this section, we give a proof of the existence results of our main theorems and corollaries. In the following we always suppose (k1), (k2), (f1) and (f2). For the problem (1.4), we define the associated energy functional

$$\displaystyle \begin{aligned} I(u)=\frac{1}{2}\|u\|{}^2-\frac{1}{2^*}\int_Bu_+^{2^*}dx-\int_B kF(u)dx\ \ (u\in H_r(B)). \end{aligned}$$

Then noting our assumptions and Lemma 11, it is standard to see that I(u) is well-defined and continuously differentiable on H _r(B). In addition, by (k1) and (f1), the usual elliptic theory and the strong maximum principle ensure that every critical point of I is a solution of (1.4). Hence our aim becomes to look for critical points of I. We first prove the mountain pass geometry of I [3].

Lemma 21

We have

• (a) ∃ρ, a > 0 such that I(u) ≥ a for all u ∈ H _r(B) with ∥u∥ = ρ, and

• (b) for all u ∈ H _r(B) ∖{0}, I(tu) →−∞ as t →∞,

for all λ > 0.

Proof

First note that by (f1) and (f2), we have that for any ε > 0, there exists a constant C > 0 such that |f(t)|≤ εt + Ct ^p for all t ≥ 0 and some p ∈ (1, (N + 2 + 2β)∕(N − 2)). Then Lemma 11 and the Sobolev inequality give

$$\displaystyle \begin{aligned} I(u)\ge \left(\frac{1}{2}-\frac{\lambda \varepsilon}{\mu_1}\right)\|u\|{}^2-\lambda C\|u\|{}^{p+1}-C\|u\|{}^{2^*} \end{aligned}$$

for all u ∈ H _r(B). Taking ε ∈ (0, μ ₁∕(4λ)), we get (a) for all λ ∈ (0, ∞).

Next, since k(x)f(u) ≥ 0 for all $x\in \overline {B}$ and $u\in \mathbb {R}$, we have for all t > 0 and u ∈ H _r(B) ∖{0} that

$$\displaystyle \begin{aligned} I(tu)\le \frac{t^2}{2}\|u\|{}^2-\frac{t^{2^*}}{2^*}\int_Bu_+^{2^*}dx. \end{aligned}$$

Since 2 < 2^∗, we obtain I(tu) →−∞ as t →∞, which shows (b). This finishes the proof. □

Noting Lemma 21, we define

$$\displaystyle \begin{aligned} \Gamma:=\{\gamma\in C([0,1],H_r(B))\ |\ \gamma(0)=0,\ \gamma(1)=e\} \end{aligned}$$

with e ∈ H _r(B) satisfying ∥e∥ > ρ and I(e) ≤ 0. Then we put

$$\displaystyle \begin{aligned} c_\lambda:=\inf_{\gamma\in \Gamma}\max_{u\in\gamma([0,1])}I(u). \end{aligned}$$

We next show the local compactness property of the Palais-Smale sequences of I. Here, as usual, we say that (u _n) ⊂ H _r(B) is a (PS)_c sequence for I if I(u _n) → c for some $c\in \mathbb {R}$ and I′(u _n) → 0 in $H_r^{-1}(B)$ as n →∞ where $H_r^{-1}(B)$ is the dual space of H _r(B).

Lemma 22

Suppose f satisfies (f3) and λ > 0. If (u _n) ⊂ H _r(B) is a (PS) _c sequence for a value c < S ^N∕2∕N, then (u _n) contains a subsequence which strongly converges in H _r(B) as n →∞.

Proof

By (f3), we obtain that

$$\displaystyle \begin{aligned} c+o(1)&=I(u_n)-\frac{1}{\min\{2^*,\theta\}}\langle I'(u_n),u_n\rangle+o(1)\|u_n\|\\ &\ge \left(\frac{1}{2}-\frac{1}{\min\{2^*,\theta\}}\right)\|u_n\|{}^2+o(1)\|u_n\| \end{aligned}$$

This shows the boundedness of (u _n) in H _r(B). Hence noting (f1), (f2) and Lemma 11, we have that, up to a subsequence, there exists a nonnegative function u ∈ H _r(B) such that

(2.1)

as n →∞. Furthermore, since (u _n) ⊂ H _r(B), the concentration compactness lemma (Lemma I.1 in [12]) implies that there exist values ν ₀, μ ₀ ≥ 0 such that

in the measure sense where δ ₀ denotes the Dirac measure with mass 1 which concentrates at $0\in \mathbb {R}^N$ and

$$\displaystyle \begin{aligned} S\nu_0^{\frac{2}{2^*}}\le \mu_0.{} \end{aligned} $$

(2.2)

Let us show ν ₀ = 0. If not, we define a smooth test function ϕ in $\mathbb {R}^N$ such that ϕ = 1 on B(0, ε), ϕ = 0 on B(0, 2ε)^c and 0 ≤ ϕ ≤ 1 otherwise. We also assume |∇ϕ|≤ 2∕ε. Then noting (f1), (f2) and using (k1), (k2), (2.1) and Lemma 11, we get

$$\displaystyle \begin{aligned} 0&=\lim_{n\to \infty}\langle I'(u_n),u_n\phi\rangle\\ &=\lim_{n\to \infty}\left(\int_B\nabla u_n \cdot \nabla (u_n\phi)dx-\int_B (u_n)_+^{2^*}\phi dx-\lambda\int_B kf(u_n)u_n\phi dx\right)\\ &=\lim_{n\to \infty}\left(\int_B|\nabla u_n|{}^2\phi dx-\int_B (u_n)_+^{2^*}\phi dx-\lambda\int_B kf(u_n)u_n\phi dx+\int_B u_n\nabla u_n \cdot \nabla \phi dx\right)\\ &= \int_{\overline{B}}\phi d\mu-\int_{\overline{B}}\phi d\nu+o(1) \end{aligned}$$

where o(1) → 0 as ε → 0. It follows that

$$\displaystyle \begin{aligned} 0\ge \mu_0-\nu_0. \end{aligned}$$

Then by (2.2), we obtain

$$\displaystyle \begin{aligned} \nu_0\ge S^{\frac{N}{2}}. \end{aligned}$$

Using this estimate, we have by (f3) that

$$\displaystyle \begin{aligned} c&=\lim_{n\to \infty}\left(I(u_n)-\frac 12\langle I'(u_n),u_n\rangle\right)\\ &\ge\frac{1}{N}\lim_{n\to \infty}\int_{\overline{B}}d\nu\\ &\ge \frac{S^{\frac{N}{2}}}{N} \end{aligned}$$

which contradicts our assumption. It follows that

$$\displaystyle \begin{aligned} \lim_{n\to \infty}\int_B (u_n)_+^{2^*}dx= \int_B u^{2^*}dx. \end{aligned}$$

Then the usual argument proves u _n → u in H _r(B). We finish the proof. □

Next we estimate the mountain pass energy c _λ. To do this, we use the Talenti function $U_\varepsilon (x):=\frac {\varepsilon ^{\frac {N-2}{2}}}{(\varepsilon ^2+|x|{ }^2)^{\frac {N-2}{2}}}$ [16]. Moreover we define a cut off function $\psi \in C^{\infty }_0(B)$ such that ψ(x) = ψ(|x|), supp{ψ}⊂ B _δ(0) and ψ = 1 on B _η(0) for some η ∈ (0, δ). We set u _ε := ψU _ε and $v_\varepsilon :=u_\varepsilon /\|u_\varepsilon \|{ }_{L^{2^*}(B)}\in H_r(B)$. Then, if $q>\max (2\gamma +6-N)/(N-2)$, a similar calculation with that in [6] shows that

$$\displaystyle \begin{aligned} \begin{cases} \|v_\varepsilon\|{}^2=S+O(\varepsilon^{N-2})\\ \|v_\varepsilon\|{}_{L^{2^*}(B)}=1,\\ \int_Bkv_\varepsilon^{q+1}dx\ge C\int_B|x|{}^\gamma v_\varepsilon^{q+1}dx= C' \varepsilon^{a}+O(\varepsilon^{N-2}) \end{cases}{} \end{aligned} $$

(2.3)

where $a=\gamma +N-\frac {(N-2)(q+1)}{2}$ and C, C′ > 0 are constants. Let us prove the next lemma (Cf. Lemma 2.1 in [6]).

Lemma 23

Assume that k verifies (k3). Then if

$$\displaystyle \begin{aligned} \lim_{\varepsilon\to0}\varepsilon^{\gamma+2}\int_0^{\varepsilon^{-1}} F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr=\infty{} \end{aligned} $$

(2.4)

holds, we have c _λ < S ^N∕2∕N for all λ > 0.

Proof

Let v _ε ∈ H _r(B) as above. Then from Lemma 21, we find a constant t _ε > 0 such that I(t _ε v _ε) =max_t≥0 I(tv _ε). Since

$$\displaystyle \begin{aligned} 0=\frac{d}{dt}|{}_{t=t_\varepsilon}I(tv_\varepsilon)= t_\varepsilon \|v_\varepsilon\|{}^2 - t_\varepsilon^{2^*-1} - \int_B kf(t_\varepsilon v_\varepsilon) v_\varepsilon dx \end{aligned}$$

and ∫_B kf(v _ε)v _ε dx ≥ 0 by (k1) and (f1), we have

$$\displaystyle \begin{aligned} t_\varepsilon\le \|v_\varepsilon\|{}^{\frac{2}{2^*-2}}=:T_\varepsilon. \end{aligned}$$

Since $T_\varepsilon =\|v_\varepsilon \|{ }^{\frac {2}{2^*-2}}$ is the maximum point of the map $t\mapsto \frac {t^2}{2}\|v_\varepsilon \|{ }^2-\frac {t^{2^*}}{2^*}$, we get by (2.3) that for any t > 0

$$\displaystyle \begin{aligned} I(t v_\varepsilon)&\le I(t_\varepsilon v_\varepsilon)\\ &\le \frac{T_\varepsilon^2}{2}\|v_\varepsilon\|{}^2-\frac{T_\varepsilon^{2^*}}{2^*}-\int_B kF(t_\varepsilon v_\varepsilon)dx\\ &\le \frac{S^{\frac{N}{2}}}{N}-\int_B kF(t_\varepsilon v_\varepsilon)dx+O(\varepsilon^{N-2}). \end{aligned}$$

Therefore once we prove

$$\displaystyle \begin{aligned} \lim_{\varepsilon\to0}\varepsilon^{-(N-2)}\int_BkF(t_\varepsilon v_\varepsilon)dx=\infty,{} \end{aligned} $$

(2.5)

we conclude c _λ ≤ I(t _ε v _ε) < S ^N∕2∕N for all small ε > 0. This completes the proof. Lastly let us ensure (2.5). To do this, we first claim that lim_ε→0 t _ε = S ^(N−2)∕4. Indeed, using (f2), for any δ > 0, there exists a constant C _δ > 0 such that

$$\displaystyle \begin{aligned} \int_B \frac{kf(t_\varepsilon v_\varepsilon)v_\varepsilon}{t_\varepsilon}dx\le t_\varepsilon^{q-1}\delta\int_B |x|{}^\beta v_\varepsilon^{q+1}dx+C_\delta \int_B|x|{}^\beta v_\varepsilon^2dx. \end{aligned}$$

Since t _ε ≤ T _ε = O(1), $\int _B |x|{ }^\beta v_\varepsilon ^{q+1}dx=O(1)$ by q = (N + 2 + 2β)∕(N − 2) and $\int _B|x|{ }^\beta v_\varepsilon ^2dx=o(1)$ as ε → 0, we get

$$\displaystyle \begin{aligned} 0 \le \limsup_{\varepsilon\to0}\int_B \frac{kf(t_\varepsilon v_\varepsilon)v_\varepsilon}{t_\varepsilon}dx \le O(1) \delta. \end{aligned}$$

Since δ > 0 can be chosen arbitrarily small, we conclude that

$$\displaystyle \begin{aligned} \lim_{\varepsilon\to0}\int_B \frac{kf(t_\varepsilon v_\varepsilon)v_\varepsilon}{t_\varepsilon}dx=0. \end{aligned}$$

Then since 〈I′(t _ε v _ε), v _ε〉 = 0, we have

$$\displaystyle \begin{aligned} t_\varepsilon=\left(\|v_\varepsilon\|{}^2-\int_B\frac{kf(t_\varepsilon v_\varepsilon)v_\varepsilon}{t_\varepsilon} dx\right)^{\frac{1}{2^*-2}}.\end{aligned} $$

This with (2.3) proves that lim_ε→0 t _ε = S ^(N−2)∕4. In particular, t _ε converges to a positive value as ε → 0. Now we calculate by (k3) that

$$\displaystyle \begin{aligned} \varepsilon^{-(N-2)}\int_BkF(t_\varepsilon v_\varepsilon)dx&\ge C_1\varepsilon^{-(N-2)}\int_{0}^\eta F\left[t_\varepsilon\left(\frac{\varepsilon}{\varepsilon^2+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr\\ &\ge C_2\varepsilon^{\gamma+2}\int_0^{\frac{\eta}{\varepsilon}} F\left[t_\varepsilon\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr\\ &\ge C_3\varepsilon^{\gamma+2}\int_0^{\frac{D}{\varepsilon}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr \end{aligned}$$

for some constant C ₁, C ₂, C ₃, D > 0 where in the last inequality we replace $\varepsilon /t_\varepsilon ^{(N-2)/2}$ by ε which does not change the conclusion below. If D ≥ 1, we clearly get (2.5) by our assumption (2.4). If D < 1, we obtain

$$\displaystyle \begin{aligned} \varepsilon^{\gamma+2}\int_0^{\frac{D}{\varepsilon}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr= &\varepsilon^{\gamma+2}\int_0^{\frac{1}{\varepsilon}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr\\ &-\varepsilon^{\gamma+2}\int_{\frac{D}{\varepsilon}}^{\frac{1}{\varepsilon}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr. \end{aligned}$$

Finally, note that (f2) shows

$$\displaystyle \begin{aligned} \varepsilon^{\gamma+2}\int_{\frac{D}{\varepsilon}}^{\frac{1}{\varepsilon}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma+N-1}dr=o(1) \end{aligned}$$

where o(1) → 0 as ε → 0. This finishes the proof. □

The next lemma confirms that under our assumptions, f(t) satisfies (2.4).

Lemma 24

Assume (k3). Then, if f satisfies (f4), then (2.4) holds true.

Proof

By (f4), for any M > 0, there exists a constant R > 0 such that f(t) ≥ Mt ^p where $p=\max \{1,\frac {2\gamma +6-N}{N-2}\}$. Furthermore, note that if r ≤ Cε ^−1∕2 for C = (2R)^{−(N−2)∕2}, we get

$$\displaystyle \begin{aligned} \left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\ge R \end{aligned}$$

for all small ε > 0. It follows that

$$\displaystyle \begin{aligned} \varepsilon^{\gamma+2}\int_0^{\varepsilon^{-1}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right] &r^{\gamma +N-1}dr \ge \varepsilon^{\gamma+2}\int_0^{C\varepsilon^{-\frac 12}}F\left[\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{N-2}{2}}\right]r^{\gamma +N-1}dr\\ &\ge \varepsilon^{\gamma+2} \frac{M}{p+1} \int_0^{C\varepsilon^{-\frac 12}}\left(\frac{\varepsilon^{-1}}{1+r^2}\right)^{\frac{(N-2)(p+1)}{2}}r^{\gamma +N-1}dr\\ &\to \infty \end{aligned}$$

as ε → 0. This completes the proof. □

Lemma 25

If k, f satisfy (k4) and (f5), we have a constant λ ^∗ > 0 such that c _λ < S ^N∕2∕N for all λ > λ ^∗.

Proof

Since k(x ₀) > 0 by (k4), there exist constants 0 < r ₁ < |x ₀| < r ₂ < 1 such that k > 0 on $\overline {B(0,r_2)}\setminus B(0,r_1)$. Then we choose a radial function $u\in C_0^{\infty }(B)\setminus \{0\}$ such that u ≥ 0 and $\text{supp}\{u\}\subset \overline {B(0,r_2)}\setminus B(0,r_1)$. Then by Lemma 21, we have a constant t _λ > 0 such that I(t _λ u) =max_t>0 I(tu). Since $\frac {d}{dt}|{ }_{t=t_\lambda }I(tu)=0$, we get

$$\displaystyle \begin{aligned} \|u\|{}^2-t_\lambda^{2^*-2}\int_Bu_+^{2^*}dx-\lambda\int_B\frac{kf(t_\lambda u)u}{t_\lambda}dx=0 \end{aligned}$$

It follows that t _λ → 0 as λ →∞. If not, there exists a sequence (λ _n) ⊂ (0, ∞) such that λ _n →∞ and $t_{\lambda _n}\to t_0>0$ for some value t ₀ > 0 as n →∞. But this is impossible in view of the previous formula and (f5). Then it follows from (k1) and (f1) that

$$\displaystyle \begin{aligned} c_\lambda\le I(t_\lambda u)\le t_\lambda^2\|u\|{}^2\to 0 \end{aligned}$$

as λ →∞. This finishes the proof. □

Then we prove the existence assertions of main theorems.

Proof of Theorem 12

First note that under the assumption in Lemma 21 and the mountain pass theorem ([3], see also Theorem 2.2 in [6]), there exists a (PS)$_{c_\lambda }$ sequence (u _n) ⊂ H _r(B) of I. Hence our aim is to see that (u _n) has a subsequence which strongly converges in H _r(B). This fact follows from Lemmas 21, 22, 23 and 24, which proves (i). The proof of (ii) is completed by Lemmas 21, 22 and 25. This completes the proof of Theorem 12. □

Proof of Corollary 17

The proof is clear from (i) of Theorem 12. □

Remark 26

Here we remark on (g1) and (g2). We first note that non-negativity of k in (k1) is needed only to apply the maximum principle. Hence it is clear that in the present case it can be weakened to g ≥−1 in (g1). Furthermore, by (g1), the associated energy functional

$$\displaystyle \begin{aligned} I(u)=\frac 12 \|u\|{}^2-\frac{1}{2^*}\int_B(1+g)|u|{}^{2^*}dx \end{aligned}$$

is always well-defined. Hence we can weaken (k2) in Theorem 12 to the condition k(0) = 0. Finally, in the present case, since we do not assume k(|x|) = O(|x|^β) for β > 0, in principle, we cannot use Lemma 11 directly in the proof of Lemma 22. Although the modification is trivial, we will give the modified proof in Appendix A for the readers’ convenience.

Proof of Corollary 19

The proof is immediate by (ii) of Theorem 12. □

3 Nonexistence Results

In this section, we prove the nonexistence results by the Pohozaev identity. Since some results still hold true for the star-shaped domain, we first consider the problem

$$\displaystyle \begin{aligned} \begin{cases} -\Delta u=|u|{}^{2^*-2}u+g|u|{}^{q-1}u\text{ in }\Omega\\ u=0\text{ on }\partial \Omega, \end{cases}{} \end{aligned} $$

(3.1)

where $\Omega \subset \mathbb {R}^N$ with N ≥ 3 is a bounded smooth domain, q ≥ 1 and g is a $C^1(\overline {\Omega })$-function. Now, let us recall the formula

$$\displaystyle \begin{aligned} \int_\Omega\left\{\frac{x\cdot \nabla g}{q+1}+\left(\frac{N}{q+1}-\frac{N-2}{2}\right)g\right\}|u|{}^{q+1}dx=\frac 12\int_{\partial \Omega}(x\cdot \nu)|\nabla u|{}^2ds_x {} \end{aligned} $$

(3.2)

holds for any solution $u\in C^1(\overline {\Omega })$. This is the Pohozaev identity for (3.1).

Theorem 31

Let $\lambda \in \mathbb {R}$ and g(x) = λ|x|^β with β ≥ 0. Then if Ω is a strictly star-shaped domain, (3.1) has no C ¹ solution if either one of the following holds;

• (i) λ ≤ 0 and q ≤ (N + 2 + 2β)∕(N − 2) or,

• (ii) λ ≥ 0 and q ≥ (N + 2 + 2β)∕(N − 2) or otherwise,

• (iii) β = 0, $\lambda \in \mathbb {R}$ and q = (N + 2)∕(N − 2).

Proof

Let $u\in C^1(\overline {\Omega })$ be a solution of (3.1). Then under the assumption in the theorem, we get by (3.2) that

$$\displaystyle \begin{aligned} \lambda\int_\Omega\left(\frac{\beta+N}{q+1}-\frac{N-2}{2}\right)|x|{}^\beta |u|{}^{q+1}dx=\frac 12\int_{\partial \Omega}(x\cdot \nu)|\nabla u|{}^2ds_x. \end{aligned}$$

Then if one of (i)–(iii) holds, the left hand side is nonpositive. It is easy to obtain the conclusion if the left hand side is strictly negative, since u is zero outside the origin, and hence also at the origin by continuity. On the other hand, if the left hand side is zero, since x ⋅ ν>0 by our assumption, we have |∇u|≡ 0 on ∂ Ω. Then from the principle of unique continuation we must have u ≡ 0 in Ω. This shows the proof. □

Proof of Theorem 15

The proof is a direct consequence of Theorem 31. □

Lastly let us show the proof of Theorem 111. To do this, we assume q ≥ 1 and u = u(r) (r ∈ [0, 1]) is a solution of

$$\displaystyle \begin{aligned} \begin{cases} -u''-\displaystyle\frac{(N-1)}{r}u'=|u|{}^{\frac{4}{N-2}}u+g|u|{}^{q-1}u\text{ in }(0,1),\\ u'(0)=0=u(1). \end{cases}{} \end{aligned} $$

(3.3)

with a C ¹ function g(r) on [0, 1]. In addition, we suppose ψ(r) (r ∈ [0, 1]) is a smooth test function such that ψ(0) = 0. Then we have the following. (See [6] and also [11].)

Theorem 32

If u is a solution of (3.3), we get

(3.4)

Proof

Multiplying the equation in (3.3) by r ^N−1 ψu′ gives

$$\displaystyle \begin{aligned} &\psi(1)|u'(1)|{}^2-\int_0^1|u'|{}^2\left\{r^{N-1}\psi'-(N-1)r^{N-2}\psi\right\}dr\\ &=\frac{N-2}{N}\int_0^1|u|{}^{2^*}\left\{r^{N-1}\psi'+(N-1)r^{N-2}\psi\right\}dr\\ &\ \ \ +\frac{\lambda(q+1)}{2}\int_0^1|u|{}^{q+1}\left\{g' r^{N-1}\psi+r^{N-1}g\psi'+(N-1)r^{N-2}g\psi\right\}dr. {} \end{aligned} $$

(3.5)

On the other hand, we multiply the equation in (3.3) by (r ^N−1 ψ′− (N − 1)r ^N−2 ψ)u and compute

$$\displaystyle \begin{aligned} &\int_0^1|u'|{}^2\left\{r^{N-1}\psi'-(N-1)r^{N-2}\psi\right\}dr\\ &-\frac 12\int_0^1u^2\left\{r^{N-1}\psi'''+(N-1)(N-3)r^{N-4}(\psi-r\psi')\right\}dr\\&=\int_0^1 |u|{}^{2^*}\left\{r^{N-1}\psi'-(N-1)r^{N-2}\psi\right\}dr\\ &\ \ \ +\lambda\int_0^1 g(r)|u|{}^{q+1}\left\{r^{N-1}\psi'-(N-1)r^{N-2}\psi\right\}dr. \end{aligned} $$

(3.6)

Combining (3.5) and (3.6), we complete the proof. □

Proof of Theorem 111

The first assertion follows from Theorem 31. Let us prove the second assertion. To do this, assume λ > 0 and u = u(r) (r ∈ [0, 1]) is a radially symmetric solution of (1.1). Then it satisfies

$$\displaystyle \begin{aligned} \begin{cases} -u''-\displaystyle\frac{(N-1)}{r}u'=(1+g)|u|{}^{\frac{4}{N-2}}u\text{ in }(0,1),\\ u'(0)=u(1)=0, \end{cases}{} \end{aligned} $$

(3.7)

where we put g(r) = λr ^β. Again choose a smooth test function ψ such that ψ(0) = 0. Then by Theorem 32, we have

$$\displaystyle \begin{aligned} &\frac 12\int_0^1 u^2 r^{N-4}\left\{r^3\psi'''-(N-1)(N-3)r \psi'+(N-1)(N-3)\psi\right\}dr\\ &=\psi(1)|u'(1)|{}^2 \\ &+\frac 1N\int_0^1|u|{}^{2^*}\left\{-(N-2)g' r^{N-1}\psi+2(N-1)(1+g(r))(r^{N-2}\psi-r^{N-1}\psi')\right\}dr. {} \end{aligned} $$

(3.8)

We fix β ≥ N − 2 and then select ψ(r) = ar ^N−1 + br so that r ³ ψ‴ − (N − 1)(N − 3)rψ′ + (N − 1)(N − 3)ψ = 0 and ψ(0) = 0. This ODE has an explicit solution ψ(r) = ar ^N−1 + br + cr ^−(N−3) where $a,b,c\in \mathbb {R}$ are arbitrary constants. Since we assume ψ(0) = 0, we must have c = 0, i.e., ψ(r) = ar ^N−1 + br. Then we get

$$\displaystyle \begin{aligned} &\psi(1)|u'(1)|{}^2 \\ &+\frac 1N\int_0^1|u|{}^{2^*}\left\{-(N-2)g' r^{N-1}\psi+2(N-1)(1+g(r))(r^{N-2}\psi-r^{N-1}\psi')\right\}dr=0. {} \end{aligned} $$

(3.9)

Substituting ψ(r) = ar ^N−1 + br into

$$\displaystyle \begin{aligned} h(r):=-(N-2)k' r^{N-1}\psi+2(N-1)(1+k)(r^{N-2}\psi-r^{N-1}\psi'), \end{aligned}$$

we see

$$\displaystyle \begin{aligned} &h(r)=r^{2N-3} \times \\ &\left[-\lambda a (N-2)\left\{2(N-1)+\beta\right\}r^{\beta}-\lambda b\beta (N-2) r^{\beta-N+2}-2a(N-1)(N-2)\right]. \end{aligned} $$

Finally, we choose a < 0 and b = |a| > 0. In particular, we have ψ(1) = a + b = 0. Then some elementary calculations show that if we set

$$\displaystyle \begin{aligned} \lambda_*=\begin{cases}\frac{2(N-1)}{N-2}\text{ if }\beta=N-2,\\ \frac{2(N-1)}{N-2}\left(\frac{2N-2+\beta}{\beta-N+2}\right)^{\frac{\beta-N+2}{N-2}}\text{ if }\beta>N-2, \end{cases} \end{aligned}$$

we assure that h ≠ 0 and h ≥ 0 for all λ ∈ [0, λ _∗]. Therefore in view of (3.9), we reach to a contradiction if λ ∈ [0, λ _∗]. This finishes the proof. □

A Critical Case

In this appendix, we give a proof of Lemma 22 under the assumption in Corollary 17 for the readers’ convenience. Especially we will use only the condition (g2) which is weaker than (k2).

Lemma A1

Assume (g1), (g2) and (u _n) ⊂ H _r(B) is a (PS) _c sequence of

$$\displaystyle \begin{aligned} I(u)=\frac{1}{2}\|u\|{}^2-\frac{1}{2^*}\int_B(1+g)u_+^{2^*}dx. \end{aligned}$$

Then if $c<S^{\frac {N}{2}}/N$, (u _n) has a subsequence which strongly converges in H _r(B).

Proof

From the definition we have

$$\displaystyle \begin{aligned} c+o(1)&=I(u_n)-\frac{1}{2^*}\langle I'(u_n),u_n\rangle+o(1)\|u_n\|\\ &\ge \frac 1N \|u_n\|{}^2+o(1)\|u_n\|. \end{aligned}$$

This implies (u _n) is bounded in H _r(B). Then we can assume that there exists a nonnegative function u ∈ H _r(B) such that

up to a subsequence. By the concentration compactness lemma, we can suppose that there exist values μ ₀, ν ₀ ≥ 0 such that

in the measure sense, where δ ₀ denotes the Dirac delta measure concentrated at the origin with mass 1 as before. Furthermore, we have

$$\displaystyle \begin{aligned} S\nu_0^{\frac{2}{2^*}}\le\mu_0.{} \end{aligned} $$

(A.1)

We show ν ₀ = 0. To this end, we assume ν ₀ > 0 on the contrary. Then, for small ε > 0, we define a smooth test function ϕ as in the proof of Lemma 22. Since I′(u _n) → 0 in H ⁻¹(B) and (u _n) is bounded, we have

$$\displaystyle \begin{aligned} 0&=\lim_{n\to \infty}\langle I'(u_n),u_n\phi\rangle\\ &=\lim_{n\to \infty}\left(\int_B\nabla u_n \cdot \nabla (u_n\phi)dx-\int_B (1+g)(u_n)_+^{2^*}\phi dx\right)\\ &=\lim_{n\to \infty}\left(\int_B|\nabla u_n|{}^2\phi dx-\int_B (1+g)(u_n)_+^{2^*}\phi dx+\int_B u_n\nabla u_n \cdot \nabla \phi dx\right)\\ &= \int_{\overline{B}}\phi d\mu-\int_{\overline{B}}(1+g)\phi d\nu+o(1) \end{aligned}$$

where o(1) → 0 as ε → 0. Taking ε → 0 and noting g(0) = 0, we obtain

$$\displaystyle \begin{aligned} 0\ge\mu_0-\nu_0. \end{aligned}$$

Then using (A.1), we get

$$\displaystyle \begin{aligned} \nu_0\ge S^{\frac N2}.{} \end{aligned}$$

Finally, noting this estimate, we see

$$\displaystyle \begin{aligned} c&=\lim_{n\to \infty}\left(I(u_n)-\frac 12\langle I'(u_n),u_n\rangle\right)\\ &=\frac{1}{N}\lim_{n\to \infty}\int_{\overline{B}}(1+g)d\nu\\ &\ge \frac{S^{\frac{N}{2}}}{N} \end{aligned}$$

since g(0) = 0, which is a contradiction. It follows that

$$\displaystyle \begin{aligned} \lim_{n\to \infty}\int_B (1+g)(u_n)_+^{2^*}dx=\int_B (1+g)u_+^{2^*}dx. \end{aligned}$$

Then a standard argument shows that u _n → u in H _r(B). This completes the proof. □