数理解析研究所講究録 1204 巻 2001 年 50-57

50
Positive solutions for nonhomogeneous elliptic equations(足達慎二)

Shinji Adachi

Department of Mathematics, School of Science and Engineering, Waseda University
3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, JAPANThis paper is based on the joint work [AT1], [AT2] with K. Tanaka.

0. Introduction
In this paper, we study the existence of positive solutions for anonhomogeneous elliptic:
problem in$-\Delta u+u=g(x, u)+f(x)$ in
,
$u>0$
,
(0.1)
in
,
$\mathrm{R}^{N}$$\mathrm{R}^{N}$$\mathrm{R}^{N}$$\{$$u\in H^{1}(\mathrm{R}^{N})$

where$g(x, s)\in C(\mathrm{R}^{N}\cross \mathrm{R})$is afunction of superlinear growth, i.e.,$\lim_{sarrow\infty}\frac{g(x,s)}{s}=\infty$,. Heredenotes the usual Sobolev space
and $f(x)\in H^{-1}(\mathrm{R}^{N}),$ $f(x)\geq 0,$. We denote the duality productis the dual space of
and
over
by
and for $f(x)\in H^{-1}(\mathrm{R}^{N})$ ,andbetweenholds for any non-negative functionif
we say$f(x)\not\equiv 0$$H^{-1}(\mathrm{R}^{N})$$f(x)\geq \mathrm{O}$$H^{1}(\mathrm{R}^{N})$$H^{1}(\mathrm{R}^{N})$$H^{-1}(\mathrm{R}^{N})$$\mathrm{R}^{N}$$H^{1}(\mathrm{R}^{N})$$H^{1}(\mathrm{R}^{N})$$\langle \cdot, \cdot\rangle_{H(\mathrm{R}^{N}),H^{1}(\mathrm{R}^{N})}-1$$\langle f, \varphi\rangle_{H^{-1}(\mathrm{R}^{N}),H^{1}(\mathrm{R}^{N})}\geq 0$$\varphi\in$.

Our main aim is to study the effects of the shape of $g(x, u)$ and $f(x)$ on the existence
and multiplicity of solutions of (0.1). We first consider the existence and multiplicity ofsolutions of (0.1) for the general nonlinearity $g(x, u)$ and next consider for $g(x, u)=a(x)u^{p}$in particular.
1. Existence of two positive solutions for general nonlinearity $g(x,$u)

In this section, we will show the existence of at least two positive solutions of (0.1) undersuitable conditions. In some cases we prove the existence of two positive solutions of (0.1),
even if the existence of apositive solution of (0.1) with $f(x)\equiv 0$ is not known.
1.1. Assumption on $g(x,$u)

51
We assume that
(A1)
(A2) There exist constants$g(x, s)\in C^{1}(\mathrm{R}^{N}\cross \mathrm{R}, \mathrm{R})$.

$\delta_{0}\in[0,1)$

and $7n_{0}>0$ such thatfor all$0<g(x, s)\leq\delta_{0}s+m_{0}s^{p}$

and $s>0$ ,$x\in \mathrm{R}^{N}$

where $1<p< \frac{N+2}{N-2}$ if $N\geq 3,1<p<\infty$ if $N=1,2$ .
(A3) There exists aconstant $\theta>2$ such that$0<\theta G(x, s)\leq g(x, s)s$

for all$x\in \mathrm{R}^{N}$

and $s>0$ ,where $G(x, s)= \int_{0}^{s}g(x, \tau)d\tau$ .
(A4)$\frac{g(x,s)}{s}$is strictly increasing in$s>\mathrm{O}$

uniformly in$s \in[\mathrm{r},r_{2}],x\in \mathrm{R}^{N}\inf_{1}\frac{d}{ds}(\frac{g(x,s)}{s})>0$$x\in \mathrm{R}^{N}$in the following sense:

for all $0<r_{1}<r_{2}$ .

Moreover, we consider the situation that $g(x, s)$ approaches to some limit function$C^{1}(\mathrm{R}, \mathrm{R})$as$|x|arrow\infty$$g^{\infty}(s)\in$:(A5) $g(x, s)arrow g^{\infty}(s)$ as $|x|arrow\infty$ uniformly on any compact subset of$[0, \infty)$.

Moreover we assume(A6) There exists aconstant $\lambda>2$ such that for any
which satisfies$g(x, s)-g^{\infty}(s)\geq-e^{-\lambda|x|}(\epsilon s+C_{\epsilon}s^{p})$$\epsilon>0$

we can find aconstantfor all$x\in \mathrm{R}^{N}$$C_{\epsilon}>0$

and $s\geq 0$ .

Here the constant Ais corresponding to aconvergent rate (from below) and the condition
plays an important role in our existence result.
satisfiesthat the limit function
We remark that it follows from$\lambda>2$$g^{\infty}(s)$$(\mathrm{A}1)-(\mathrm{A}5)$

similar conditions to$(\mathrm{A}1)-(\mathrm{A}4)$$(\mathrm{A}1’)g^{\infty}(s)\in C^{1}(\mathrm{R}, \mathrm{R})$:.

(A2’) $0<g^{\infty}(s)\leq\delta_{0}s+m_{0}s^{p}$ for all $s>0$ .
$(\mathrm{A}3’)0<\theta G^{\infty}(s)\leq g^{\infty}(s)s$$( \mathrm{A}4’)\frac{d}{ds}(\frac{g^{\infty}(s)}{s})>0$

for all $s>0$ , where

for all $s>0$ .

$G^{\infty}(s)= \int_{0}^{s}g^{\infty}(\tau)d\tau$.

521.2. Known results

CaO-Zhou [CZ], Jeanjean [J] ( . Hirano [H], Zhu [Z]) studied the problem (0.1) as aperturbation from the following homogeneous equation:$\mathrm{c}.\mathrm{f}$inin$-\Delta u+u=g(x, u)$
$\{$

$u>0$
$u\in H^{1}(\mathrm{R}^{N})$In addition to similar assumptions to$(\mathrm{A}1)-(\mathrm{A}5)$$g(x, s)\geq g^{\infty}(s)$for all$\mathrm{R}^{N}$$\mathrm{R}^{N}$

,
,

(1.1)

.

, they needed$x\in \mathrm{R}^{N}$

and $s>0$(1.2)

and they succeeded to show that there exists aconstantsuch that if $f\geq 0,$,, then (0.1) has at least two positive solutions. Here the constant $M>0$
was chosen so that the corresponding functional:$M>\mathrm{O}$$f\not\equiv \mathrm{O}$$||f||_{H^{-1}(\mathrm{R}^{N})}\leq M$$I(u)=1$$j_{N}fudx$:$H^{1}(\mathrm{R}^{N})arrow \mathrm{R}$

,

where
$||u||_{H^{1}}=( \int_{\mathrm{R}^{N}}|\nabla u|^{2}+|u|^{2}dx)\frac{1}{2}$possesses the mountain pass geometry. That is, if,$||f||_{H^{-1}(\mathrm{R}^{N})}\leq M$, then $I(u)$ satisfies(i) there exists aconstant $\rho_{0}>0$ such that
$I(u)\geq 0$(ii) {(iii)

for all$u\in H^{1}(\mathrm{R}^{N});||u||_{H^{1}(\mathrm{R}^{N})}>\rho_{0}$inf$I(u)<0$ .

$u\in H^{1}(\mathrm{R}^{N})$withand $I(u)<0$ }$\neq\emptyset$$||u||_{H^{1}(\mathrm{R}^{N})}=\rho_{0}$,,$||\mathrm{u}||_{H^{1}(\mathrm{R}^{N})}<\rho 0$

To see the role of the condition (1.2), we consider here the homogeneous problem(1.1). The corresponding functional is$J(u)= \frac{1}{2}||u||_{H^{1}(\mathrm{R}^{N})}^{2}-$$\mathrm{I}\mathrm{t}^{N}G(x, u)dx$.

It is well-Jcnown that the mountain pass critical value for $J(u)$ is attained at some criticalpoint
under condition (1.2). However, without the condition (1.2), themountain pass value is not attained in general. For example, it is not under condition:$g(s, x)<g^{\infty}(s)$ for all$a>0$ . See Lions [PLLI], [PLL2] for similar arguments.
$u\in H^{1}(\mathrm{R}^{N})$$a\in \mathrm{R}^{N},$53
We also remark that is seems that the existence of positive solution for (1.1) is not known
without (1.2) in general. As far as we know, it is obtainedjust for the case $g(x, s)=a(x)s^{p}$ .
See Bahri-Li [BaYL], Bahri-Lions [BaPLL] for details. Thus the aim of our paper is toshow the existence of positive solutions of (0.1) without (1.2). Even the existence of apositive solution for homogeneous problem (1.1) is not known, we can show the existenceof at least two positive solutions for nonhomongeneous problem (0.1).
1.3. Main results

Our main result is as follows.
Theorem 1.1. Assume that$f\geq 0,$$f\not\equiv \mathrm{O},$$(A\mathit{1})-(A\mathit{6})$$||f||_{H^{-1}(\mathrm{R}^{N})}\leq M$. Then there exists aconstant$M>\mathrm{O}$such that if, then (0.1) has at least two positive solutions.

We will prove Theorem 1.1 via variational methods. We find positive solutions of (0.1)
as critical points of $I(u)$ . First we find one positive solution $u_{0}(x)$ as alocal minimum of, then 0is not asolution of our problem and the$I(u)$ near 0. We remark that if
first positive solution is obtained as aperturbation of 0. Next we find apositive solutionof (0.1) different from $u_{0}(x)$ by using the Mountain Pass Theorem. When we seek criticalpoints of $I(u)$ , we need to pay attention to the breaking down of Palais-Smale condition$f\not\equiv \mathrm{O}$for $I(u)$ .
2. Existence of four positive solutions in the case $g(x, u)=a(x)u^{p}$

In this section, we consider the equation (0.1) with $g(x, u)=a(x)u^{p}$ , that is:$-\Delta u+u=a(x)u^{p}+f(x)$

$u>0$

$\{$

$u\in H^{1}(\mathrm{R}^{N})$inin$\mathrm{R}^{N}$$\mathrm{R}^{N}$

,
,

(2.1)

,

where 1 $<p< \frac{N+2}{N-2}(N\geq 3),$ $1<p<\infty(N=1,2)$ . We also assume that for$a(x)\in C(\mathrm{R}^{N})$(H1) $a(x)>0$ for all$x\in \mathrm{R}^{N}$,(H2) $a(x)arrow 1$ as $|x|arrow\infty$ ,(H3) there exist $\delta>0$ and$C>\mathrm{O}$such that$a(x)-1\geq-Ce^{-(2+\delta)|x|}$

for all$x\in \mathrm{R}^{N}$

54
By Theorem 1.1, we see that if
is not so large, then (2.1) has at least two
positive solutions without order relation between $a(x)$ and 1. We remark that the equation(2.1) with $f(x)\equiv 0$ :$-\Delta u+u=a(x)u^{p}$
,
in
$u>0$
,
in
(2.2)
,
$||f||_{H(\mathrm{R}^{N})}-1$$\mathrm{R}^{N}$$\{$$\mathrm{R}^{N}$$u\in H^{1}(\mathrm{R}^{N})$possesses at least one positive solution only under condition. See Bahri-Li[BaYL]. ( . Bahri-Lions [BaPLL]). We also remark that Kwong [K] showed that thelimit equation:$(\mathrm{H}1)-(\mathrm{H}3)$$\mathrm{c}.\mathrm{f}$(2.3)$\{\begin{array}{l}-\Delta u+u=u^{p}\mathrm{i}\mathrm{n}\mathrm{R}^{N}u>0\mathrm{i}\mathrm{n}\mathrm{R}^{N}u\in H^{1}(\mathrm{R}^{N})\end{array}$possesses aunique positive radial solution $\omega(x)=\omega(|x|)>0$ and any positive solution$u(x)$ of (2.3) can be written as
for some$u(x)=\omega(x-x_{0})$($x_{0}\in \mathrm{R}^{N}$. Kabeya-Tanaka [KT]).

$\mathrm{c}.\mathrm{f}$In this section, we consider (2.1) under(H4) $a(x)\in(0,1]$ for all$x\in \mathrm{R}^{N},$$a(x)\not\equiv 1$.

in addition to. We will show the existence of more positive solutions under(H4). The uniqueness of positive solution of the limit equation (2.3) plays an importantrole in our existence results. Our main results are the following$(\mathrm{H}1)-(\mathrm{H}3)$$(\mathrm{H}1)-$

Theorem 2.1 ([AT1]). We assume
non-negative function $f(x)$ satisfying
positive solutions.

$(H1)-(H\mathit{4})$. Then there exists a$0<||f||_{H^{-1}(\mathrm{R}^{N})}\leq\delta_{0},$such that for$(2.1)$ possesses at least four$\delta_{0}>0$

As to an asymptotic behavior of solutions obtained in Theorem 2.1 as
we have$||f||_{H^{-1}(\mathrm{R}^{N})}arrow 0$

Theorem 2.2 ([AT1]). Assume that asequence of non-negative functions
satisfiesand$H^{-1}(\mathrm{R}^{N})$,$(f_{j}(x))_{j=1}^{\infty}\subset$$f_{j}(x)\not\equiv \mathrm{O}$$||f_{j}||_{H^{-1}(\mathrm{R}^{N})}arrow 0$

as j$arrow\infty$.

Then there exist asubsequence of–still denoted by–and four
sequences $(u_{j}^{(k)}(x))_{j\in \mathrm{N}}(k=1,2,3,4)$ of positive solutions of (2.1) with $f(x)=f_{j}(x)$ such$(f_{j}(x))_{j=1}^{\infty}$$(f_{j}(x))_{j=1}^{\infty}$

55

as $jarrow\infty$ .
(i)(ii) There exist sequences$||u_{j}^{(1)}||_{H^{1}(\mathrm{R}^{N})}arrow 0$$(y_{j}^{(2)})_{j=1}^{\infty},$$|y_{j}^{(k)}|arrow\infty$,

such that$||u_{j}^{(k)}(x)-\omega(x-y_{j}^{(k)})||_{H^{1}(\mathrm{R}^{N})}arrow 0$

for $k=2,3$ .
(iii) There exists apositive solutionas$(y_{j}^{(3)})_{j=1}^{\infty}\subset \mathrm{R}^{N}$$jarrow\infty$$v_{0}(x)$of (2.2) such thatas$||u_{j}^{(4)}(x)-v_{0}(x)||_{H^{1}(\mathrm{R}^{N})}arrow 0$$jarrow\infty$.

We use variational methods to find positive solutions of (2.1). We define for given$a(x)$ and $f(x)$$I_{a,f}\langle u$)$= \frac{1}{2}||u||_{H^{1}(\mathrm{R}^{N})}^{2}-\frac{1}{p+1}\int_{\mathrm{R}^{N}}a(x)u_{+}^{p+1}dx-\int_{\mathrm{R}^{N}}fudx$$J_{a,f}(v)= \max_{t>0}I_{a,f}(tv)$:$\Sigma_{+}arrow \mathrm{R}$:$H^{1}(\mathrm{R}^{N})arrow \mathrm{R}$

,

,

where
$\Sigma=\{v\in H^{1}(\mathrm{R}^{N});||v||_{H^{1}(\mathrm{R}^{N})}=1\}$$\Sigma_{+}=\{v\in\Sigma;v_{+}\not\equiv 0\}$

,

.

are
or $J_{a,f}(v)$ :We will see that critical points of $I_{a,f}(u)$ :corresponding to positive solutions of (2.1).
We will find critical point of $I_{a,f}(u),$ $J_{a,f}(v)$ in the following way. First we find one
positive solution $u^{(1)}(a, f;x)=u_{loc\min}(a, f;x)$ as alocal minimum of $I_{a,f}(u)$ near 0. Next
we see that the Palais-Smale compactness condition for $I_{a,f}(u)$ and $J_{a,f}(v)$ breaks down$H^{1}(\mathrm{R}^{N})arrow \mathrm{R}$$\Sigma_{+}arrow \mathrm{R}$only at levels$I_{a,f}(u_{0}(x))+\ell I_{1,0}(\omega)$$\ell=1,2,$$\ldots$

is aunique
where $I_{1,0}(u)$ is afunctional corresponding to the limit equation (2.3),
positive radial solution of (2.3) and $u_{0}(x)$ is acritical point of $I_{a,f}(u)$ . In particular, we will
see that the Palais-Smale condition holds under the level $I_{a,f}(u_{loc\min}(a, f;x))+I_{1,0}(\omega)$ .
Next we find two critical points different from $u_{loc\min}$ under the first level of breaking
down of Palais-Smale condition, that is, under the level $I_{a,f}(u_{loc\min}(a, f;x))+I_{1,0}(\omega)$ .
$\omega(x)$We use notation:$[J_{a,f}\leq c]=\{u\in\Sigma_{+} ; J_{a,f}(u)\leq c\}$for

$c\in \mathrm{R}$. We will observe that for sufficiently small$\epsilon>0$$[J_{a,f}\leq I_{a,f}(u_{loc\min}(a, f;x))+I_{1,0}(\omega)-\epsilon]$

56
is not empty and(2.4)$\mathrm{c}\mathrm{a}\mathrm{t}([J_{a,f}\leq I_{a,f}(u_{l_{oCm}:n}(a, f;x))+I_{1,0}(\omega)-\epsilon])\geq 2$

provided $f(x)\geq 0,$ $f(x)\not\equiv 0$ andis sufficiently small. Herestandsfor the Lusternik-Schnirelman category. We find two positive solutions $u^{(2)}(a, f;x)$ and$u^{(3)}(a, f;x)$ satisfying$||f||_{H^{-1}(\mathrm{R}^{N})}$$I_{a,f}(u^{(k)}(a, f;x))<I_{a,f}(u_{l_{o\mathrm{C}m}:n}(a, f :$\mathrm{c}\mathrm{a}\mathrm{t}(\cdot)$x))+I_{1,0}(\omega)$

for $k=2,3$ .

(2.5)

We remark that for $f\equiv 0$ , we see that$u_{l_{ocm}:n}(a, 0;x)\equiv 0$

and
(2.6)

$[J_{a,0}\leq I_{a,0}(u_{l_{o\mathrm{C}m}:n}(a, 0;x)+I_{1,0}(\omega)]=\emptyset$and (2.4) is the key of our proof. To get (2.4), we use the following interaction phenomenon, Bahri-Li [BaYL], Bahri-Loins [BaPLL], Taubes
as in [AT2] ( . Bahri-Coron[T] :$\mathrm{c}.\mathrm{f}$$[\mathrm{B}\mathrm{a}\mathrm{C}]$$)$$I_{a,f}(u_{l_{o\mathrm{C}m}:n}(a, f;x)+\omega(x-y))<I_{a,f}(u_{l_{o\mathrm{C}m}:n}(a, f;x))+I_{1,0}(\omega)$

for sufficiently large $|y|\geq 1$ .
To find the fourth positive solution, we adapt the minimax method of Bahri-Li [BaYL]
to our functional $J_{a,f}(v)$ . More precisely, we define$b_{a,f}= \inf_{\gamma\in}\sup_{y\in \mathrm{R}^{N}}J_{a,f}(\gamma(y))$

,

where
$\Gamma=${$\gamma\in C(\mathrm{R}^{N},$$\Sigma_{+});\gamma(y)=\frac{\omega(\cdot-y)}{||\omega||_{H^{1}(\mathrm{R}^{N})}}$

for large

$|y|$

}.

Then we will find apositive solution $u^{(4)}(a, f;x)$ corresponding to the minimax value
which satisfies$b_{a,f}$$I_{a,f}(u^{(4)}(a, f;x))>I_{a,f}(u_{l_{ocm}:n}(a, f;x))+I_{1,0}(\omega)$

for sufficiently small
essential way.

$||f||_{H^{-1}(\mathrm{R}^{N})}$. To show Theorem 2.2, we also use (2.5) and (2.6) in an

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$\mathrm{R}^{N}$$[\mathrm{B}\mathrm{a}\mathrm{C}]$$\mathrm{R}^{N}$$\mathrm{R}^{N}$$126\mathrm{A}$$\mathrm{R}^{n}$