### Abstract

In this article, we investigate the effect of the coefficient *f*(*z*) of the sub-critical nonlinearity. For sufficiently large *λ *> 0, there are at least *k *+ 1 positive solutions of the semilinear elliptic equations

where 1 ≤ *q *< 2 < *p *< 2* = 2*N*/(*N *- 2) for *N *≥ 3.

**AMS (MOS) subject classification**: 35J20; 35J25; 35J65.

##### Keywords:

semilinear elliptic equations; concave and convex; positive solutions### 1 Introduction

For *N *≥ 3, 1 ≤ *q *< 2 < *p *< 2* = 2*N*/(*N *- 2), we consider the semilinear elliptic equations

where *λ *> 0.

Let *f *and *h *satisfy the following conditions:

(*f *1) *f *is a positive continuous function in ℝ^{N }and lim_{|z| → ∞ }*f*(*z*) = *f*_{∞ }> 0.

(*f*2) there exist *k *points *a*^{1}, *a*^{2},..., *a*^{k }in ℝ^{N }such that

and *f*_{∞ }< *f*_{max}.

(*h *1)

Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain

have been studied by Ambrosetti et al. [1] (*h *≡ 1, and 1 < *q *< 2 < *p *≤ 2* = 2*N*/(*N*- 2)) and Wu [2]
*q *< 2 < *p *< 2*). They proved that this equation has at least two positive solutions for sufficiently
small *c *> 0. More general results of Equation (*E*_{c}) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].

In this article, we consider the existence and multiplicity of positive solutions
of Equation (*E*_{λ}) in ℝ^{N}. For the case *q *= *λ *= 1 and *f*(*z*) ≡ 1 for all *z *∈ ℝ^{N}, suppose that *h *is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (*E*_{λ}) admits at least two positive solutions in ℝ^{N}. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (*E*_{λ}) admits at least two positive solutions in ℝ^{N}. For the case *q *= *λ *= 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka
[9] asserted that Equation (*E*_{λ}) admits at least four positive solutions in ℝ^{N}, where *f*(*z*) ≢ 1, *f*(*z*) ≥ 1 - *C *exp((-(2 + *δ*) |*z*|) for some *C, δ *> 0, and sufficiently small
*u *+ *u *= *f*(*z*)*v*^{p-1 }+ *λh*(*z*) *v*^{q-1 }in ℝ^{N }for sufficiently small *λ *> 0.

By the change of variables

Equation (*E*_{λ}) is transformed to

Associated with Equation (*E*_{ε}), we consider the *C*^{1}-functional *J*_{ε}, for *u *∈ *H*^{1 }(ℝ^{N}),

where
*H*^{1 }(ℝ^{N}) and *u*_{+ }= max{*u*, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (*E*_{ε}) are equivalent to the critical points of *J*_{ε}. This article is organized as follows. First of all, we use the argument of Tarantello
[11] to divide the Nehari manifold **M**_{ε }into the two parts
*E*_{ε}). Finally, in Section 4, we show that the condition (*f*2) affects the number of positive solutions of Equation (*E*_{ε}), that is, there are at least *k *critical points
*J*_{ε }such that
*i *≤ *k*.

Let

then

For the semilinear elliptic equations

we define the energy functional

where **N**_{ε }= {*u *∈ *H*^{1 }(ℝ^{N}) \ {0} | *u*_{+ }≢ 0 and

(*i*) if *f *≡ *f*_{∞}, we define

where **N**_{∞ }= {*u *∈ *H*^{1 }(ℝ^{N}) \ {0} | *u*_{+ }≢ 0 and

(*ii*) if *f *≡ *f*_{max}, we define

where **N**_{max }= {*u *∈ *H*^{1 }(ℝ^{N}) \ {0} | *u*_{+ }≢ 0 and

**Lemma 1.1**

**Proof**. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].

Our main results are as follows.

(*I*) Let Λ = *ε*^{2(p-q)/(p-2)}. Under assumptions (*f *1) and (*h*1), if

where ∥*h*∥_{# }is the norm in
*E*_{ε}) admits at least a positive ground state solution. (See Theorem 3.4)

(*II*) Under assumptions (*f*1) - (*f*2) and (*h*1), if *λ *is sufficiently large, then Equation (*E*_{λ}) admits at least *k *+ 1 positive solutions. (See Theorem 4.8)

### 2 The Nehari manifold

First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions
in *H*^{1}(ℝ^{N}) for some functional *J.*

**Definition 2.1 **(*i*) *For β *∈ ℝ, *a sequence *{*u*_{n}} *is a *(*PS*)_{β}*-sequence in H*^{1}(ℝ^{N}) *for J if J*(*u*_{n}) = *β *+ *o*_{n}(1) *and J*'(*u*_{n}) = *o*_{n}(1) *strongly in H*^{-1 }(ℝ^{N}) *as n *→ ∞, *where H*^{-1 }(ℝ^{N}) *is the dual space of H*^{1}(ℝ^{N});

(*ii*) *J satisfies the *(*PS*)_{β}*-condition in H*^{1}(ℝ^{N}) *if every (PS)*_{β}*-sequence in H*^{1}(ℝ^{N}) *for J contains a convergent subsequence.*

Next, since *J*_{ε }is not bounded from below in *H*^{1 }(ℝ^{N}), we consider the Nehari manifold

where

Note that **M**_{ε }contains all nonnegative solutions of Equation (*E*_{ε}). From the lemma below, we have that *J*_{ε }is bounded from below on **M**_{ε}.

**Lemma 2.2 ***The energy functional J*_{ε }*is coercive and bounded from below on ***M**_{ε}.

**Proof**. For *u *∈ **M**_{ε}, by (2.1), the Hölder inequality

Hence, we have that *J*_{ε }is coercive and bounded from below on **M**_{ε}.

Define

Then for *u *∈ **M**_{ε}, we get

We apply the method in Tarantello [11], let

**Lemma 2.3 ***Under assumptions *(*f*1) *and *(*h*1), *if *0 < Λ (= *ε*^{2(p-q)/(p-2)}) < Λ_{0}, *then *

**Proof**. See Hsu and Lin [[10], Lemma 5].

**Lemma 2.4 ***Suppose that u is a local minimizer for J*_{ε }*on ***M**_{ε }*and *
*Then *
*in H*^{-1 }(ℝ^{N}).

**Proof**. See Brown and Zhang [[4], Theorem 2.3].

**Lemma 2.5 ***We have the following inequalities.*

(*i*)
*for each *

(*ii*)
*for each *

(*iii*)
*for each *

(*iv*) *If *
*then J*_{ε}(*u*) > 0 *for each *

**Proof**. (*i*) It can be proved by using (2.2).

(*ii*) For any

(*iii*) For any

(*iv*) For any
*iii*), we get that

Thus, if
*J*_{ε}(*u*) ≥ *d*_{0 }> 0 for some constant *d*_{0 }= *d*_{0}(*ε, p, q, S*, ∥*h*∥_{# }, *f*_{max}).

For *u *∈ *H*^{1 }(ℝ^{N}) \ {0} and *u*_{+ }≢ 0, let

**Lemma 2.6 ***For each u *∈ *H*^{1 }(ℝ^{N})\ {0} *and u*_{+ }≢ 0, *we have that*

(*i*) *if *
*then there exists a unique positive number *
*such that *
*and J*_{ε}(*t*^{-}*u*) = sup_{t ≥ 0 }*J*_{ε}(*tu*);

(*ii*) *if *0 < Λ ( = *ε*^{2(p-q)/(p-2)}) < Λ_{0 }*and *
*then there exist unique positive numbers *
*such that *
*and*

**Proof**. See Hsu and Lin [[10], Lemma 7].

Applying Lemma 2.3

Define

**Lemma 2.7 **(*i*) *If *0 < Λ ( = *ε*^{2(p-q)/(p-2)}) < Λ_{0}, *then *

(*ii*) *If *0 < Λ < *q*Λ_{0}/2, *then *
*for some constant d*_{0 }= *d*_{0 }(*ε, p, q, S*, ∥*h*∥_{#}, *f*_{max}).

**Proof**. (*i*) Let

Then

By the definitions of *α*_{ε }and

(*ii*) See the proof of Lemma 2.5 (*iv*).

Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.

**Lemma 2.8 **(*i*) *There exists a *
*-sequence *{*u*_{n}} *in ***M**_{ε }*for J*_{ε};

(*ii*) *There exists a *
*-sequence *{*u*_{n}} *in *
*for J*_{ε};

(*iii*) *There exists a *
*-sequence *{*u*_{n}} *in *
*for J*_{ε}.

### 3 Existence of a ground state solution

In order to prove the existence of positive solutions, we claim that *J*_{ε }satisfies the (PS)_{β}-condition in *H*^{1}(ℝ^{N}*) *for
*ε*^{2(p-q)/(p-2) }and *C*_{0 }is defined in the following lemma.

**Lemma 3.1 ***Assume that h satisfies *(*h*1) *and *0 < Λ ( = *ε*^{2(p-q)/(p-2)}) < Λ_{0}. *If *{*u*_{n}} *is a *(*PS*)_{β}*-sequence in H*^{1}(ℝ^{N}) *for J*_{ε }*with u*_{n }⇀ *u weakly in H*^{1 }(ℝ^{N}), *then *
*in H*^{-1 }(ℝ^{N}) *and *
*where*

and

**Proof**. Since {*u*_{n}} is a (PS)_{β}-sequence in *H*^{1}(ℝ^{N}) for *J*_{ε }with *u*_{n }⇀ *u *weakly in *H*^{1 }(ℝ^{N}), it is easy to check that
*H*^{-1}(ℝ^{N}) and *u *≥ 0. Then we have

**Lemma 3.2 ***Assume that f and h satisfy *(*f*1) *and *(*h*1). *If *0 < Λ ( = *ε*^{2(p-q)/(p-2)}) < Λ_{0}, *then J*_{ε }*satisfies the *(*PS*)_{β}*-condition in H*^{1}(ℝ^{N}) *for *

**Proof**. Let {*u*_{n}} be a (PS)_{β}-sequence in *H*^{1}(ℝ^{N}) for *J*_{ε }such that *J*_{ε}(*u*_{n}) = *β + o*_{n}(1) and
*H*^{-1}(ℝ^{N}). Then

where *c*_{n }= *o*_{n}(1), *d*_{n }= *o*_{n}(1) as *n *→ ∞. It follows that {*u*_{n}} is bounded in *H*^{1}(ℝ^{N}). Hence, there exist a subsequence {*u*_{n}} and a nonnegative *u *∈ *H*^{1 }(ℝ^{N}) such that
*H*^{-1 }(ℝ^{N}), *u*_{n }⇀ *u *weakly in *H*^{1 }(ℝ^{N}), *u*_{n }⇀ *u *a.e. in ℝ^{N}, *u*_{n }⇀ *u *strongly in
*s *< 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.

Next, claim that

For any *σ *> 0, there exists *r *> 0 such that

Applying (*f*1) and *u*_{n }→ *u *in

Let *p*_{n }*= u*_{n }*- u*. Suppose *p*_{n }↛ 0 strongly in *H*^{1 }(ℝ^{N}). By (3.1)-(3.4), we deduce that

Then

By Theorem 4.3 in Wang [12], there exists a sequence {*s*_{n}} ⊂ ℝ^{+ }such that *s*_{n }= 1 + *o*_{n}(1), {*s*_{n }*p*_{n}} ⊂ **N**_{∞ }and *I*_{∞}(*s*_{n }*p*_{n}) = *I*_{∞}(*p*_{n}) *+ o*_{n}(1). It follows that

which is a contradiction. Hence, *u*_{n }→ *u *strongly in *H*^{1}(ℝ^{N}).

**Remark 3.3 ***By Lemma 1.1, we obtain*

*and *
*for *0 < Λ < Λ_{0}.

By Lemma 2.8 (*i*), there is a
*u*_{n}} in **M**_{ε }for *J*_{ε}. Then we prove that Equation (*E*_{ε}) admits a positive ground state solution *u*_{0 }in ℝ^{N}.

**Theorem 3.4 ***Under assumptions *(*f*1), (*h*1), *if *0 < Λ ( = *ε*^{2(p-q)/(p-2)}) < Λ_{0}, *then there exists at least one positive ground state solution u*_{0 }*of Equation *(*E*_{ε}) *in *ℝ^{N}. *Moreover, we have that *
*and*

**Proof**. By Lemma 2.8 (*i*), there is a minimizing sequence {*u*_{n}} ⊂ **M**_{ε }for *J*_{ε }such that *J*_{ε}(*u*_{n}) *= α*_{ε }+ *o*_{n}(1) and
*H*^{-1 }(ℝ^{N}). Since
*u*_{n}} and *u*_{0 }∈ *H*^{1 }(ℝ^{N}) such that *u*_{n }→ *u*_{0 }strongly in *H*^{1 }(ℝ^{N}). It is easy to see that
*E*_{ε}) in ℝ^{N }and *J*_{ε}(*u*_{0}) = *α*_{ε}. Next, we claim that

We get that

Otherwise,

It follows that

which contradicts to *α*_{ε }< 0. By Lemma 2.6 (*ii*), there exist positive numbers

which is a contradiction. Hence,

By Lemma 2.4 and the maximum principle, then *u*_{0 }is a positive solution of Equation (*E*_{ε}) in ℝ^{N}.

### 4 Existence of *k *+ 1 solutions

From now, we assume that *f *and *h *satisfy (*f*1)-(*f*2) and (*h*1). Let *w *∈ *H*^{1 }(ℝ^{N}) be the unique, radially symmetric, and positive ground state solution of Equation
(*E*0) in ℝ^{N }for *f *= *f*_{max}. Recall the facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).

(*i*)
*θ *< 1 and

(*ii*) for any *ε *> 0, there exist positive numbers *C*_{1},
*z *∈ ℝ^{N}

and

For 1 ≤ *i *≤ *k*, we define

Clearly,
*ii*), there is a unique number
*i *≤ *k.*

We need to prove that

**Lemma 4.1 **(*i*) *There exists a number t*_{0 }> 0 *such that for 0 *≤ *t *≤*t*_{0 }*and any ε *> 0, *we have that*

(*ii*) *There exist positive numbers t*_{1 }*and ε*_{1 }*such that for any t > t*_{1 }*and ε < ε*_{1}, *we have that*

**Proof**. (*i*) Since *J*_{ε }is continuous in
*H*^{1 }(ℝ^{N}) for any *ε *> 0, and *γ*_{max }> 0, there is *t*_{0 }> 0 such that for 0 ≤ *t *≤ *t*_{0 }and any *ε *> 0

(*ii*) There is an *r*_{0 }> 0 such that *f *(*z*) ≥ *f*_{max}/2 for *z *∈ *B*^{N }(*a*^{i}; *r*_{0}) uniformly in *i*. Then there exists *ε*_{1 }> 0 such that for *ε < ε*_{1}

Thus, there is *t*_{1 }>0 such that for any *t > t*_{1 }and *ε < ε*_{1}

**Lemma 4.2 ***Under assumptions *(*f*1), (*f*2), *and *(*h*1). *If 0 < Λ *( = *ε*^{2(p-q)/(p-2)}) < *q *Λ_{0}*/*2, *then*

**Proof**. By Lemma 4.1, we only need to show that

We know that sup_{t ≥0 }*I*_{max }(*tw*) = *γ*_{max}. For *t*_{0 }≤ *t *≤ *t*_{1}, we get

Since

and

then
*i.*

Applying the results of Lemmas 2.6, 2.7(*ii*), and 4.2, we can deduce that

Since *γ*_{max }< γ_{∞}, there exists *ε*_{0 }> 0 such that

Choosing 0 < *ρ*_{0 }< 1 such that

where
*f*(*a*^{i}) = *f*_{max}. Define **K **= {*a*^{i }| 1 ≤ *i *≤ *k*} and
*r*_{0 }> 0.

Let *Q*_{ε }: *H*^{1 }(ℝ^{N}) \ {0} → ℝ^{N }be given by

where *χ *: ℝ^{N }→ ℝ^{N}, *χ *(*z*) = *z *for |*z*| ≤ *r*_{0 }and *χ *(*z*) *= r*_{0}*z*/|*z*| for |*z*| > *r*_{0}.

**Lemma 4.3 ***There exists *0 < *ε*^{0 }≤ *ε*_{0 }*such that if ε < ε*^{0}, *then *
*for each *1 ≤ *i *≤ *k.*

**Proof**. Since

there exists *ε*^{0 }> 0 such that

**Lemma 4.4 ***There exists a number *
*such that if u *∈ **N**_{ε }*and *
*then *
*for any *0 < *ε < ε*^{0}.

**Proof**. On the contrary, there exist the sequences {*ε*_{n}} ⊂ ℝ^{+ }and
*n *→ ∞ and
*n *∈ ℕ. It is easy to check that {*u*_{n}} is bounded in *H*^{1 }(ℝ^{N}). Suppose *u*_{n }→ 0 strongly in *L*^{p }(ℝ^{N}). Since

and

then

which is a contradiction. Thus, *u*_{n }↛ 0 strongly in *L*^{p }(ℝ^{N}). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant *d*_{0 }> 0 and a sequence

Let
*v*_{n}} and *v *∈ *H*^{1 }(ℝ^{N}) such that *v*_{n }⇀ *v *weakly in *H*^{1 }(ℝ^{N}). Using the similar computation in Lemma 2.6, there is a sequence

We deduce that a convergent subsequence
*H*^{1 }(ℝ^{N}). By (4.2), then
*H*^{1 }(ℝ^{N}) and
*z*_{n }→ *z*_{0 }∈ **K**.

(*i*) Claim that the sequence {*z*_{n}} is bounded in ℝ^{N}. On the contrary, assume that |*z*_{n}| → ∞, then

which is a contradiction.

(*ii*) Claim that *z*_{0 }∈ **K**. On the contrary, assume that *z*_{0 }∉ **K**, that is, *f(z*_{0}) < *f*_{max}. Then using the above argument to obtain that

which is a contradiction. Since *v*_{n }→ *v *≠ 0 *in H*^{1 }(ℝ^{N}), we have that

which is a contradiction.

Hence, there exists a number
*u *∈ **N**_{ε }and
*ε < ε*^{0}.

From (4.1), choosing

For each 1 ≤ *i *≤ *k*, define

**Lemma 4.5 ***If *
*and J*_{ε }(*u*) ≤ *γ*_{max }+ *δ*_{0}/2, *then there exists a number *
*such that *
*for any *

**Proof**. We use the similar computation in Lemma 2.6 to get that there is a unique positive
number

such that
*c *> 0 (independent of *u)*. First, since

and *J*_{ε }is coercive on **M**_{ε}, then
*c*_{1 }and *c*_{2 }(independent of *u*). Next, we claim that
*c*_{3 }> 0 (independent of *u*). On the contrary, there exists a sequence

By (2.3),

which is a contradiction. Thus,
*c *> 0 (independent of *u*). Now, we get that

From the above inequality, we deduce that