Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 19, 1-12; http://www.math.u-szeged.hu/ejqtde/

REMARKS ON INHOMOGENEOUS ELLIPTIC PROBLEMS ARISING IN ASTROPHYSICS

MARCO CALAHORRANO & HERMANN MENA

Abstract. We deal with the variational study of some type of nonlinear inhomogeneous elliptic problems arising in models of solar flares on the halfplane ℝ_{+}^{n}.

1. Introduction

In this paper we study a boundary value problem of type

(1.1)  \{ \begin{matrix}−Δu + c(x)u = λm(y)f(u) ℝ_{+}^{n} \\ u(z, 0) = h(z) ∀z ∈ ℝ^{n−1} \end{matrix}

where x = (z, y) ∈ ℝ^{n−1} × ℝ_{+} ≡ ℝ_{+}^{n} with ℝ_{+} = {y ∈ ℝ : y > 0} and n ≥ 2, f :] − ∞, +∞[→ ℝ is a function satisfying:

(f-1) There exists s_{0} > 0 such that f(s) > 0 for all s ∈]0, s_{0}[.

(f-2) f(s) = 0 for s ≤ 0 o s ≥ s_{0}.

(f-3) f(s) ≤ as^{σ}, a is a positive constant and 1 < σ < \frac{n+2}{n−2} if n > 2

or σ > 1 if n = 2. (f-4) There exists l > 0 such that |f(s_{1}) − f(s_{2})| ≤ l|s_{1} − s_{2}|, for all

s_{1}, s_{2} ∈ ℝ. h is a non-negative bounded smooth function, h ≠ 0, min h < s_{0}, c ≥ 0, c ∈ L^{∞}(Ω) ⋂ C(\overline{Ω}) and mes\{x ∈ Ω : c(x) = 0\} = 0. The problem (1.1) is a generalization of an astrophysical gravity model of solar flares in the half plane ℝ_{+}^{2}, given in [1], namely:

(1.2)  \{ \begin{matrix}−Δu & = & λe^{−βy}f(u) ℝ_{+}^{2} \\ u(x, 0) & = & h(x) ∀x ∈ ℝ \end{matrix}

besides the above mentioned conditions for f, h and β > 0. See [1], [8] and [6] for a detailed description and related problems. By this, we study the problem (1.1) with m : ℝ_{+} → ℝ_{+} a C^{1} function such that

∫_{0}^{+∞} ym(y)dy < +∞

more general than e^{−βy}.

EJQTDE, 2005 No. 19, p. 1

1991 Mathematics Subject Classification. 35J65, 85A30.

Key words and phrases. Solar flares, variational methods, inhomogeneous semilinear elliptic problems.

We shall follow the ideas of F. Dobarro and E. Lami Dozo in [8]. The authors prove the existence of solutions of (1.1) in the special case c(x) = 0. In fact, the result presented here follows from the one obtained by the authors.

First of all we note that problem (1.1) is equivalent to

(1.3)  \{ \begin{matrix}−Δω + c(x)ω = λm(y)f(ω + τ) ℝ_{+}^{n} \\ ω(z, 0) = 0  ∀z ∈ ℝ^{n−1} \end{matrix}

where ω = u − τ and τ is solution of the problem

(1.4)  \{ \begin{matrix}−Δτ + c(x)τ = 0 ℝ_{+}^{n} \\ τ(z, 0) = h(z) ∀z ∈ ℝ^{n−1} \end{matrix}

We will study (1.3) instead of (1.1).

The problem (1.1), or equivalently (1.3), is interesting not only on whole ℝ_{+}^{n}, but also in an arbitrary big but finite domain in ℝ_{+}^{n}, for example for semidisks D_{R} = {(x, y) ∈ ℝ^{n−1}×ℝ_{+} : |x|^{2}+y^{2} < R^{2}, y > 0}, with R big enough.

Motivated by this observation in section 2, we will study the following approximate problem

(1.5)  \{ \begin{matrix}−Δω + c(x)ω = λm(y)f(ω + τ) D_{R} \\ ω = 0  ∂D_{R} \end{matrix}

whose solutions are related to those of (1.3).

Using variational techniques we will prove the existence of an interval Λ ⊂ R_{+} such that for all λ ∈ Λ there exists at least three positive solutions of (1.5), with R large enough.

Finally in section 3 we prove the existence of solutions of (1.3) as limit of a special family of solutions of (1.5) obtained in theorem 5 and its uniqueness to λ small enough.

2. Problem in D_{R}

Letting Ω be either D_{R} or ℝ_{+}^{n}, we denote by L_{m}^{p}(Ω) the usual weighted L^{p} space on Ω for a suitable weight m and 1 ≤ p < ∞, and by V_{m}^{1,2}(Ω), V_{c}^{1,2}(Ω) the completion of C_{0}^{∞}(Ω) in the norm

∥u∥_{V_{m}^{1,2}(Ω)}^{2} = ∫_{Ω} u^{2}(z, y)m(y)dzdy + ∫_{Ω} |∇u|^{2}dzdy

and

∥u∥_{V_{c}^{1,2}(Ω)}^{2} = ∫_{Ω} u^{2}(x)c(x)dx + ∫_{Ω} |∇u|^{2}dx

Let m : R_{+} → R_{+} be such that

(2.1)  0 < M ≡ ∫_{0}^{+∞} ym(y)dy < +∞

it is easy to prove for all functions u ∈ C_{0}^{∞}(Ω) the following inequality holds, see [8].

(2.2)  ∫_{Ω} u^{2}(x, y)m(y)dxdy ≤ M ∫_{Ω} |∇u|^{2}dxdy

then V_{m}^{1,2}(D_{R}) ∼ H_{0}^{1}(D_{R}) ∼ V_{c}^{1,2}(D_{R}) and V_{m}^{1,2}(ℝ_{+}^{n}) ∼ D^{1,2}(ℝ_{+}^{n}) where H_{0}^{1}(D_{R}) is the usual Sobolev space with the norm ∥∇(.)∥_{L^{2}(D_{R})} and D^{1,2}(ℝ_{+}^{n}) is the completion of C_{0}^{∞}(ℝ_{+}^{n}) for the norm ∥∇(.)∥_{L^{2}(ℝ_{+}^{n})}. On the other hand if R′ ≥ R, then

(2.3)  V_{c}^{1,2}(D_{R}) ⊂ V_{c}^{1,2}(D_{R′}) ⊂ V_{c}^{1,2}(ℝ_{+}^{n}) ⊂ V_{m}^{1,2}(ℝ_{+}^{n})

There exists many results about immersion of weighted Sobolev spaces into weighted Lebesgue spaces. Here we will take into account one suitable result for our problem.

Let m : R_{+} → R_{+} be a bounded C^{1} function such that there exists k > 0 such that

(2.4)  |(log m)′| ≤ k

then the identity map is an immersion from V_{m}^{1,2}(Ω) into L_{m^{\frac{p}{2}}}^{p} (Ω) for

\begin{matrix}1 < p < \frac{2n}{n−2} & if n ≥ 3 \\ 1 < p & if n=2 \end{matrix}

More precisely, there exists a constant K = K(k, sup m) such that

(2.5)  ∥u∥_{L_{m \frac{p}{2}}^{p} (Ω)} ≤ C_{s}K∥u∥_{V_{m}^{1,2}(Ω)}

where C_{s} is the usual Sobolev immersion constant. The immersion is compact if Ω = D_{R}.

Now we will begin to study (1.3) by variational methods. For this purpose, for all λ ≥ 0 and for all non negative function τ such that ∥τ∥_{L_{m}^{σ+1}} < +∞ we associate the functional Ψ_{λ,τ} : V_{c}^{1,2}(ℝ_{+}^{n}) → R

(2.6)  Ψ_{λ,τ}(u) = \frac{1}{2} ∫_{ℝ_{+}^{n}}\{|∇u|^{2} + c(x)u^{2}\} − λ ∫_{ℝ_{+}^{n}} mF (u + τ)

where F (t) = ∫_{0}^{t} f(s)ds, m ∈ C^{1}(R_{+}) and m̂ ≡ m^{\frac{2}{σ+1}} satisfying (2.1) and (2.4).

Ψ_{λ,τ} is a C^{1} functional, so if u ∈ V_{c}^{1,2}(ℝ_{+}^{n}) is a critical point of Ψ_{λ,τ} then u is a weak, and by regularity a classical solution of (1.3).

Remark 1. i. If we consider Ψ_{λ,τ,R} : V_{c}^{1,2}(D_{R}) → R,

Ψ_{λ,τ,R}(u) = \frac{1}{2} ∫_{D_{R}}\{|∇u|^{2} + c(x)u^{2}\} − λ ∫_{D_{R}} mF (u + τ)

its critical points are weak, and by regularity, strong solutions of (1.5). Furthermore if R ≤ R′ ≤ +∞, then for all u ∈ V_{c}^{1,2}(D_{R})

Ψ_{λ,τ,R′}(u) ≤ Ψ_{λ,τ,R}(u) ≤ Ψ_{λ,0,R}(u)

more precisely

Ψ_{λ,τ,R′}(u) = Ψ_{λ,τ,R}(u) − λ ∫_{D_{R′} −D_{R}} mF (τ) ≤ Ψ_{λ,τ,R}(u)

Here D_{R′} with R′ = +∞ means ℝ_{+}^{n}. ii. Since f is bounded, Ψ_{λ,τ,R} is coercive, bounded from below and verifies Palais-Smale condition for all λ non negative.

Lemma 2. For each R > 0 denote θ_{R} : R^{n} → R^{n} the map

θ_{R}(z, y) ≡(\frac{z}{R}, y)

and Θ_{R} the scaling η → η_{R} ≡ ηoθ_{R}. Then i. ∀r > 0, Θ_{R}(V_{c}^{1,2}(D_{r})) ⊂ V_{c}^{1,2}(θ_{R}^{−1}D_{r}) and if R ≥ 1, V_{c}^{1,2}(θ_{R}^{−1}D_{r}) ⊂ V_{c}^{1,2}(D_{Rr}). ii. If η ∈ C_{0}^{∞}(ℝ_{+}^{n}), is non identically 0, then

(2.7)  ∥∇η_{R}∥_{L^{2}(ℝ_{+}^{n})} → +∞  as  R → +∞

iii. Let f be defined before and m such that verifies (2.1). Then there exists 0 < \underline{λ} < ∞ such that if λ > \underline{λ}, η ∈ C_{0}^{∞}(ℝ_{+}^{n}), η ≥ 0, non identically 0 and

(2.8)  \underline{λ} ≤ Q(η) ≡ \frac{\frac{1}{2} ∫_{ℝ_{+}^{n}}\{|∇η|^{2} + ∥c∥_{L^{∞}}η^{2}\}}{∫_{ℝ_{+}^{n}} m(y)F (η)} < λ

then there exists r_{n} > 0 : η_{R} ∈ V_{c}^{1,2}(D_{R′}), ∀R′, R: R′ ≥ Rr_{n} ≥ r_{n} and for all non negative function τ. a. Ψ_{λ,τ,R′}(η_{R}) < 0, ∀R′, R: R′ ≥ Rr_{n} ≥ r_{n}. b. Ψ_{λ,τ,Rr_{n}}(η_{R}) → −∞ as R → +∞

EJQTDE, 2005 No. 19, p. 4

Proof.- This proof follows almost directly from lemma 6 in [8]. However, by completeness we present all the proof. i. It is immediate from the definition of Θ_{R}. ii. We observe

|∇η_{R}|^{2}(z, y) = \frac{1}{R^{2}} |∇η|_{θ_{R}}^{2} + (1 − \frac{1}{R^{2}} )|∂_{y}η|_{θ_{R}(z,y)}^{2}

thus, changing variables

(2.9) ∥∇η_{R}∥_{L^{2}(ℝ_{+}^{n})}^{2} = R^{n−1}[\frac{1}{R^{2}} ∫_{ℝ_{+}^{n}} |∇η|^{2} + (1 − \frac{1}{R^{2}} ) ∫_{ℝ_{+}^{n}} |∂_{y}η|^{2}]

so, since ∫_{ℝ_{+}^{n}} |∂_{y}η|^{2} > 0, (2.9) implies (2.7). iii. Set

(2.10)  \underline{λ} ≡ inf\{Q(η) : η ∈ C_{0}^{∞}(ℝ_{+}^{n}), η ≥ 0, η ≠ 0\}

by (f-3) and since F is bounded

(2.11)  \frac{b}{2} ≡ sup_{s>0} \frac{F (s)}{s^{2}} < +∞

so, by (2.2) and since c(x) ≥ 0

∫_{ℝ_{+}^{n}} m(y)F (η) ≤ \frac{bM}{2} ∫_{ℝ_{+}^{n}} |∇η|^{2} + ∥c∥_{L^{∞}}η^{2}

hence

0 < \frac{1}{bM} ≤ \underline{λ} < ∞

Let λ > Q(η) be, since η ∈ C_{0}^{∞}(ℝ_{+}^{n}), there exists r_{n} > 0 such that supp η ⊂ D_{Rr_{n}}, for all R ≥ 1. Then by i. and (2.3) η_{R} ∈ V_{c}^{1,2}(θ_{R}^{−1}D_{r_{n}}) ⊂ V_{c}^{1,2}(D_{Rr_{n}}) ⊂ V_{c}^{1,2}(D_{R′}) for all R′ ≥ Rr_{n} ≥ r_{n}. For simplicity from now on we call Rr_{n} ≡ R_{n}, where R ≥ 1. Then, by remark 1

(2.12)  Ψ_{λ,τ,R′}(η_{R}) ≤ Ψ_{λ,τ,R_{n}}(η_{R}) ≤ Ψ_{λ,0,R_{n}}(η_{R})

On the other hand, if we define the function ξ : R_{+} → R

ξ(R) ≡ \frac{1}{R^{n−1}} ∥∇η_{R}∥_{L^{2}(ℝ_{+}^{n})}^{2} = \frac{1}{R^{2}} ∫_{ℝ_{+}^{n}} |∇η|^{2} + (1 − \frac{1}{R^{2}} ) ∫_{ℝ_{+}^{n}} |∂_{y}η|^{2}

= [\frac{∫_{ℝ_{+}^{n}} |∇_{z}η|^{2}}{R^{2} ∫_{ℝ_{+}^{n}} |∂_{y}η|^{2}} + 1] ∫_{ℝ_{+}^{n}} |∂_{y}η|^{2}

is non increasing. So applying ξ(R) ≤ ξ(1) to (2.9)

∫_{D_{Rn}} |∇η_{R}|^{2} = ∫_{ℝ_{+}^{n}} |∇η_{R}|^{2} ≤ R^{n−1} ∫_{ℝ_{+}^{n}} |∇η|^{2}

furthermore

∫_{D_{Rn}} c(x)η_{R}^{2} = ∫_{ℝ_{+}^{n}} c(x)η_{R}^{2} ≤ R^{n−1}∥c∥_{L^{∞}} ∫_{ℝ_{+}^{n}} η^{2}

and

∫_{D_{Rn}} m(y)F (η_{R}) = ∫_{ℝ_{+}^{n}} m(y)F (η_{R}) = R^{n−1} ∫_{ℝ_{+}^{n}} m(y)F (η)

so

Ψ_{λ,0,R_{n}} ≤ R^{n−1}[\frac{1}{2} ∫_{ℝ_{+}^{n}} |∇η|^{2} + ∥c∥_{L^{∞}}η^{2} − λ ∫_{ℝ_{+}^{n}} m(y)F (η)]

then

(2.13)  Ψ_{λ,0,R_{n}} ≤ \frac{R^{n−1}}{2} ∫_{ℝ_{+}^{n}} |∇η|^{2} + ∥c∥_{L^{∞}}η^{2}(1 − \frac{λ}{Q(η)})

thus, from (2.12) and (2.13) we obtain immediately a and b.

□

Remark 3. i. Let m : ℝ_{+} → ℝ_{+} be a bounded C^{1} function and let m̂ ≡ m^{\frac{2}{σ+1}} . It is easy to prove that m verifies (2.4) if and only if m̂ does it. Furthermore, given a positive constant k, |(log m)′| ≤ k if and only if |(log m̂)′| ≤ \frac{2}{σ+1}k. ii. If there exists a non negative value m_{1} ≥ 0 such that {m > 1} ⊂ [0, m_{1}] and

0 < M̂ ≡ ∫_{0}^{+∞} ym̂(y)dy < +∞

then

0 < M ≡ ∫_{0}^{+∞} ym(y)dy < +∞

Indeed, since m̂ > 1 if and only if m > 1 and 0 < \frac{2}{σ+1} < 1

M = ∫_{m̂>1} ym(y)dy + ∫_{m̂≤1} ym(y)dy ≤ (sup m\frac{m_{1}}{2} ) + M̂ < +∞

Lemma 4. There exists a positive constant C = C(a, σ, k, sup m, M̂) such that for all λ < \overline{λ}(∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}) and for all u : ∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})} = ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}, Ψ_{λ,τ}(u) > 0 where \overline{λ}(∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}) ≡ C∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}^{1−σ}.

Moreover \overline{λ}(∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}) → +∞ as ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})} → 0 EJQTDE, 2005 No. 19, p. 6 Proof.- Let u ∈ V_{c}^{1,2}(ℝ_{+}^{n}) be, using (f-3) and Minkowsky inequality with respect to measure m(y)dxdy and (2.2), (2.5) we obtain

0 ≤ ∫_{ℝ_{+}^{n}} mF (u + τ) = ∫_{ℝ_{+}^{n}} m ∫_{0}^{u+τ} f(t)dt ≤ \frac{a}{σ + 1} ∫_{ℝ_{+}^{n}} m(u + τ)^{σ+1}
≤ \frac{a}{σ + 1}(∥u∥_{L_{m \frac{σ+1}{2}}^{σ+1}} + ∥τ∥_{L_{m}^{σ+1}})^{σ+1}
≤ \frac{a}{σ + 1}(C_{s}K(1 + M̂)^{\frac{1}{2}} ∥∇u∥_{L^{2}(ℝ_{+}^{n})} + ∥τ∥_{L_{m}^{σ+1}})^{σ+1}
≤ \frac{a}{σ + 1}(C_{s}K(1 + M̂)^{\frac{1}{2}} ∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})} + ∥τ∥_{L_{m}^{σ+1}})^{σ+1}

then (2.14)

Ψ_{λ,τ}(u) ≥ \frac{1}{2}∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})}^{2}−λ\frac{a}{σ + 1}(C_{s}K(1+M̂)^{\frac{1}{2}} ∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})}+∥τ∥_{L_{m}^{σ+1}})^{σ+1}

then, if we define

C ≡ \frac{σ + 1}{2a}(C_{s}K(k, sup m)(1 + M̂)^{\frac{1}{2}} + 1)^{−σ−1}

then Ψ_{λ,τ}(u) > 0 for all λ < \overline{λ} ≡ C∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}^{1−σ}, and since σ > 1. The lemma is proved.

□

Theorem 5. Let us assume (f-1-2-3-4), let m:ℝ_{+} → ℝ_{+} be a C^{1} function such that m and m̂ ≡ m^{\frac{2}{σ+1}} verify (2.1) and (2.4), and let τ : ℝ_{+}^{n} → ℝ_{+} be a C^{1} function, non identically 0. So there exists positive constants C = C(a, σ, k, sup m, M̂) and \underline{λ} such that if

(2.15)  ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})} < (\frac{c}{\underline{λ}})^{\frac{1}{σ−1}}

then

∀λ : \underline{λ} < λ < \overline{λ} ≡ C∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}^{1−σ}

there exists a positive R_{0} = R_{0}(λ) such that for all R ≥ R_{0}, (1.5) has at least three strictly positive solutions.

Proof.- Let C = C(a, σ, k, sup m, M̂) and \underline{λ} be the positive constant defined in lemmas 4 and 2 respectively . Since τ verifies (2.15), by lemma 4 and remark 1, for all λ ∈]\underline{λ}, \overline{λ}[ and for all R ≥ 1

(2.16) Ψ_{λ,τ,R}(u) > 0 ∀u ∈ V_{c}^{1,2}(D_{R}) : ∥u∥_{V_{c}^{1,2}(D_{R})} = ∥τ∥_{L_{m}^{σ+1}(R_{+}^{n})}

On the other hand, fixed λ ∈]\underline{λ}, \overline{λ}[, η ∈ C_{0}^{∞}(ℝ_{+}^{n}), and letting r_{n} > 0, the radius of any semidisk D_{r_{n}} such that supp η ⊂ D_{r_{n}}, by lemma 2 there

EJQTDE, 2005 No. 19, p. 7 exists R_{1} ≥ 1 such that for all R ≥ R_{1}r_{n}, we have η_{R_{1}} ∈ V_{c}^{1,2}(D_{R}), furthermore

(2.17) ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})} < ∥∇η_{R_{1}}∥_{L^{2}(D_{R})} = ∥∇η_{R_{1}}∥_{L^{2}(ℝ_{+}^{n})} < ∥η_{R_{1}}∥_{V_{c}^{1,2}(ℝ_{+}^{n})}

and

(2.18)  Ψ_{λ,τ,R}(η_{R_{1}}) < μ < 0

where μ ∈ R defined as

μ ≡ min_{0≤t≤∥τ∥_{L{σ+1 \atop m}(ℝ^{n}_{+})}} \frac{1}{2}t^{2} − λ\frac{a}{σ + 1}(C_{s}K(1 + M̂)^{\frac{1}{2}} t + ∥τ∥_{L_{m}^{σ+1}})^{σ+1}

Let R ≥ R_{1}, we divide the proof in three steps.

1. Local minimum.- Let

ν_{R} ≡ inf_{B_{Γ}} Ψ_{λ,τ,R}(u)

where B_{Γ} = {u ∈ V_{c}^{1,2}(D_{R}) : ∥u∥_{V_{c}^{1,2}(D_{R})} < Γ ≡ ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}}. Since Ψ_{λ,τ,R}(0) < 0, ν_{R} < 0. Furthermore μ ≤ ν_{R} < 0, by (2.14) and remark 1 . Therefore inf_{∂B_{Γ}} Ψ_{λ,τ,R} > ν_{R}. Now we will prove that ν_{R} is achieved in B_{Γ}. Using a modification in the proof of proposition 5 and corollaries 6 and 7 in [3], we can obtain a sequence (u_{n})_{n} in B_{Γ} such that

Ψ_{λ,τ,R}(u_{n}) → ν_{R}

Ψλ,τ,R^{′}(u_{n}) → 0

since Ψ_{λ,τ,R} verifies Palais-Smale condition, there exists a subsequence (u_{n_{k}})_{k} such that u_{n_{k}} → u_{1,R} in V_{c}^{1,2}(D_{R}) and u_{1,R} ≠ 0 because 0 it is not a critical point of Ψ_{λ,τ,R}.

2. Absolute minimum.- Let

u_{R} ≡ inf_{V_{c}^{1,2}(D_{R})} Ψ_{λ,τ,R}

Then u_{R} < μ, by (2.17). Now using similar arguments to local minimum, but without any modification, we have that u_{R} is achieved in V_{c}^{1,2}(D_{R}) at a function u_{2,R}. 3.Mountain pass.- Let

c_{R} ≡ inf_{δ∈Λ_{R}} sup_{u∈δ} Ψ_{λ,τ,R}(u)

where Λ_{R} is the set of paths

Λ_{R} = {γ : γ ∈ C([0, 1], V_{c}^{1,2}(D_{R})), γ(0) = 0, γ(1) = η_{R_{1}}}

Since Ψ_{λ,τ,R}(0) < 0, by (2.15), (2.16) and (2.17), c_{R} > 0. Then by the mountain pass theorem, see [4], c_{R} is achieved in V_{c}^{1,2}(D_{R}) at a function u_{3,R}.

EJQTDE, 2005 No. 19, p. 8 On the other hand it is clear that u_{1,R}, u_{2,R} and u_{3,R} are different, indeed

Ψ_{λ,τ,R}(u_{2,R}) = u_{R} < μ ≤ ν_{R} = Ψ_{λ,τ,R}(u_{1,R}) < 0 < c_{R} = Ψ_{λ,τ,R}(u_{3,R})

□

Remark 6. When λ is small enough it is easy to prove uniqueness for (1.5), so u_{1,R} = u_{2,R}, and the local minimum in B_{Γ} od Ψ_{λ,τ,R} is the absolute in V_{c}^{1,2}(D_{R}).

3. Problem in R_{+}^{n}

Ψ_{λ,τ} does not verifies Palais-Samale condition, furthermore by lemma 2 and remark 1 Ψ_{λ,τ} is not coercive and not bounded from below. However for λ small enough:

Proposition 7. Let f be as above, let b be given by (2.11) and suppose m verifies (2.1). Then i. For all λ < \frac{1}{bM}, Ψ_{λ,τ} is coercive and bounded from below. ii. For all λ < \frac{1}{lM}, (1.3) has at most one solution in V_{c}^{1,2}(ℝ_{+}^{n}). λ < \underline{λ} holds in both cases.

Proof.- i. By (2.11), (2.2) and Cauchy-Schwartz for the measure mdxdy

Ψ_{λ,τ}(u) ≥ \frac{1}{2}∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})}^{2} − \frac{λb}{2} ∫_{ℝ_{+}^{n}} m(u + τ)^{2}

≥ \frac{1}{2}∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})}^{2} − \frac{λb}{2}(M^{\frac{1}{2}} ∥∇u∥_{L^{2}(ℝ_{+}^{n})} + ∥τ∥_{L_{m}^{2}(ℝ_{+}^{n})})^{2}
≥ \frac{1}{2}(1 − λbM)∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})}^{2} − (λbM^{\frac{1}{2}} ∥τ∥_{L_{m}^{2}(ℝ_{+}^{n})})∥u∥_{V_{c}^{1,2}(ℝ_{+}^{n})} −

− (\frac{λb}{2}∥τ∥_{L_{m}^{2}(ℝ_{+}^{n})}^{2})

so, i. is proved.

□

ii. Uniqueness is proved as in [1] using the inequality (2.2) and (f-4). Indeed: if u_{1} and u_{2} are two solutions of (1.3) then

∫_{ℝ_{+}^{n}}(u_{1}−u_{2})^{2}m ≤ M ∫_{ℝ_{+}^{n}} |∇(u_{1}−u_{2})|^{2}+c(x)(u_{1}−u_{2})^{2} ≤ Mlλ ∫_{ℝ_{+}^{n}}(u_{1}−u_{2})^{2}m

□

Now we will prove a sufficient condition to approximate solutions of (1.3) with solutions of (1.5) with R large enough.

Lemma 8. Let f and τ be as above and λ ∈ ℝ_{+}. Suppose (R_{n})_{n} is a sequence ℝ_{+} such that R_{n} → +∞ and (u_{n})_{n} is a sequence of positive solutions of (1.5) with R_{n} instead of R, such that for all n, u_{n} ∈ V_{c}^{1,2}(D_{Rn}) and (u_{n})_{n} is bounded in V_{c}^{1,2}(ℝ_{+}^{n}), i.e. there exists Γ′ > 0 such that for all n, ∥u_{n}∥_{V_{c}^{1,2}(D_{Rn})} < Γ′. Then, there exists a subsequence (called again (u_{n})_{n})) and a function u ∈ V_{c}^{1,2}(ℝ_{+}^{n}) such that u_{n} → u weakly in V_{c}^{1,2}(ℝ_{+}^{n}) and u is a classical solution (1.3).

Proof.- By the Calderón-Zygmund^{1} inequality for all n, u_{n} ∈ H_{0}^{1}(D_{R_{n}}) ⋂ H^{2,p}(D_{R_{n}}) and fixed R′ > 0, for any Ω′ ⊂⊂ D_{R′}

(3.1) ∥u_{n}∥_{H^{2,p}(Ω′)} ≤ C(∥u_{n}∥_{L^{p}(D_{R′})} + ∥λm(y)f(u_{n} + τ)∥_{L^{p}(D_{R′})})

for all n such that R_{n} > R′. The constant C depends on D_{R′}, n, p and Ω′. Since m is decreasing and strictly positive, and (u_{n})_{n} is bounded in V_{c}^{1,2}(ℝ_{+}^{n}), by (2.2), (2.5), (3.1) and the hypothesis of f and m, we obtain

∥u_{n}∥_{H^{2,p}(Ω′)} ≤ C(m(R′)^{− \frac{1}{2}} C_{s}K(1 + M)^{\frac{1}{2}} Γ′ + λ sup m sup f|D_{R′}|^{\frac{1}{p}})

for p such that

\begin{matrix}1 < p < \frac{2n}{n−2} & if n ≥ 3 \\ 1 < p & if n=2 \end{matrix}

and for all n such that R_{n} > R′. For this and the Sobolev embedding theorem for Ω′, there exists a subsequence (u_{n})_{n} such that if n=2,3 u_{n} → u in C^{1,α}(\overline{Ω′}) and if n ≥ 4

and 1 < p < min(\frac{n}{2}, \frac{2n}{n−2}) is fixed, u_{n} → u in L^{q}(Ω′), 1 ≤ q < \frac{np}{n−2p}.

Since Ω′ is an arbitrary and relatively compact such that Ω′ ⊂⊂ D_{R_{n}} and R_{n} → +∞, we obtain that the above convergences are in C_{loc}^{1,α}(ℝ_{+}^{n}) and L_{loc}^{q}(ℝ_{+}^{n}) respectively. In particular

(3.2)  u_{n} → u en L_{loc}^{1}(ℝ_{+}^{n})

On the other hand, since (u_{n})_{n} is bounded in V_{c}^{1,2}(ℝ_{+}^{n}), by (2.3), (2.5) and reflexivity

u_{n} → u weakly in V_{c}^{1,2}(ℝ_{+}^{n}) (3.3) u_{n} → u weakly in L_{m^{\frac{p}{2}}}^{p} (ℝ_{+}^{n}) (3.4)

where

\begin{matrix}1 < p < \frac{2n}{n−2} & if n ≥ 3 \\ 1 < p & if n=2 \end{matrix}

EJQTDE, 2005 No. 19, p. 10 So, if we prove that for all v ∈ C_{0}^{∞}(ℝ_{+}^{n})

1see theorems 9.9 y 9.11 in [9]

∫_{ℝ_{+}^{n}} mf(u_{n} + τ)v → ∫_{ℝ_{+}^{n}} mf(u + τ)v

our lemma will follow. Based on this and for a fixed v ∈ C_{0}^{∞}(R_{+}^{n}) we consider the function

w = v \frac{f(u + τ)}{u + τ}m^{\frac{2−p}{2}}

It is easy to prove that w ∈ L_{m^{\frac{p}{2}}}^{p′} (ℝ_{+}^{n}), where \frac{1}{p} + \frac{1}{p′} = 1. Now

∫_{ℝ_{+}^{n}} mf(u_{n} + τ)v = ∫_{ℝ_{+}^{n}} m[f(u_{n} + τ) − (u_{n} + τ)\frac{f(u + τ)}{u + τ}]v +

+ ∫_{ℝ_{+}^{n}} m^{\frac{p}{2}} (u_{n} + τ)w (3.5)

by (3.4), the last term of right hand side of (3.5) tends to ∫_{ℝ_{+}^{n}} mf(u + τ)v. On the other hand, by (f-4)

(3.6) |∫_{ℝ_{+}^{n}} m[f(u_{n}+τ)−(u_{n}+τ)\frac{f(u + τ)}{u + τ}]v| ≤ 2l ∫_{supp(v)} m|u−u_{n}||v|

So, by (3.2) the last term of the right hand side of (3.5) tends to 0. □

Theorem 9. Let f, m, and τ as in lemma 8 and let Γ ≡ ∥τ∥_{L_{m}^{σ+1}(ℝ_{+}^{n})}. Then for all λ, 0 < λ < \overline{λ} the local minima u_{1,R} of Ψ_{λ,τ,R}, approximate the local minima of Ψ_{λ,τ} on the ball B_{Γ} of center 0 and radius Γ in V_{c}^{1,2}(ℝ_{+}^{n}). As a consequence ν_{∞} ≡ inf_{B_{Γ}} Ψ_{λ,τ}, is a minimum and by proposition 7 it is the unique critical point of Ψ_{λ,τ}, if λ small enough(i.e. 0 < λ < \frac{1}{lM}).

Proof.- We only need to prove that ν_{R} → ν_{∞} as R → ∞. With this aim we consider (u_{R})_{R} in C_{0}^{∞}(ℝ_{+}^{n}) such that u_{R} ∈ V_{c}^{1,2}(D_{R}) and Ψ_{λ,τ,R}(u_{R}) → ν_{∞} as R → ∞. By remark 1 λ ∫_{ℝ_{+}^{n}−D_{R}} mF (τ)dx → 0 as R → ∞, because Γ < +∞. Since

ν_{∞} ≤ ν_{R} = Ψ_{λ,τ,R}(u_{1,R}) ≤ Ψ_{λ,τ,R}(u_{R}) = Ψ_{λ,τ}(u_{R}) − λ ∫_{ℝ_{+}^{n}−D_{R}} mF (τ)

then ν_{R} → ν_{∞} as R → ∞.

□

EJQTDE, 2005 No. 19, p. 11

References

[1] J.J. Aly, T. Amari,Two-dimensional Isothermal Magnetostatic Equilibria in a Gravitational Field I, Unsheared Equilibria, Astron & Astrophys. 208, pp. 361-373.

[2] A. Ambrosetti, Critical Points and Nonlinear Variational Problems, Cours de la Chaire Lagrange, Mémoire (nouvelle série) N 49, Supplément au Bulletin de la Société Mathématique de France, Tome 120, 1992.

[3] J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis (Pure and Applied Mathematics) J.Wiley & Sons, 1984.

[4] A. Ambrosetti, P.H. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications, J. Funct Anal. 14,pp. 349-381, 1973.

[5] M. Calahorrano, F. Dobarro, Multiple solutions for Inhomogeneous Elliptic Problems Arising in Astrophysics, Math. Mod. and Methods in Applied Sciences, 3, pp. 217-230, 1993.

[6] M. Calahorrano, H. Mena, Multiple solutions for inhomogeneous nonlinear elliptic problems arising in astrophysics, Electron. J. Differential Equations 2004, No. 49, pp. 1-10.

[7] A. Castro, Métodos de reducción via minimax, Notas del primer simposio colombiano de análisis funcional, Colciencias-ICFES, 1981.

[8] F. Dobarro, E. Lami Dozo, Variational Solutions in Solar Flares With Gravity, Partial Differential Equations (Han-Sur-Lesse, 1993), pp. 120-143, Math. Res; 82, Akademie-Verlag, Berlin, 1994.

[9] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer Verlag, Berlin, 1983.

[10] J. Heyvaerts, J. M. Lasry, M. Schatzman and P. Witomski, Solar Flares: A Nonlinear Eigenvalue Problem in an Unbounded Domain. In Bifurcation and Nonlinear Eigenvalue problems, Lecture Notes in Mathematics 782, Springer, pp. 160-191, 1980.

[11] J. Heyvaerts, J. M. Lasry, M. Schatzman and P. Witomski, Blowing up of Twodimentional Magnetohydrostatic Equilibra by an Increase of Electric Current or Pressure, Astron & Astroph. 111, pp. 104-112, 1982.

[12] J. Heyvaerts, J. M. Lasry, M. Schatzman, and P. Witomski, Quart. Appl. Math. XLI, 1, 1983.

[13] P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Applications to Differential Equations, CBMS, Regional Conference Series in Mathematics, 65, vii, 100 p. (1986).

(Received October 23, 2003)

Marco Calahorrano Escuela Politécnica Nacional, Departamento de Matemática, Apartado 17-01-2759, Quito, Ecuador

E-mail address: calahor@server.epn.edu.ec
URL: www.math.epn.edu.ec/miembros/calahorrano.htm

Hermann Mena Escuela Politécnica Nacional, Departamento de Matemática, Apartado 17-01-2759, Quito, Ecuador

E-mail address: hmena@server.epn.edu.ec
URL: www.math.epn.edu.ec/~ hmena

EJQTDE, 2005 No. 19, p. 12