We deal with the variational study of some type of nonlinear inhomogeneous elliptic problems arising in models of solar flares on the halfplane $\mathbb{R}^n_+$

Calahorrano, M.

Mena, H.

Electronic Journal of Qualitative Theory of Differential Equations

M. Calahorrano

H. Mena

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Electronic journal of qualitative theory of differential equations

Remarks on inhomogeneous elliptic problems arising in astrophysics

Marco Calahorrano

Hermann Mena

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Electronic Journal of Qualitative Theory of Differential Equations2005, No. 19, 1-12; http://www.math.u-szeged.hu/ejqtde/REMARKS ON INHOMOGENEOUS ELLIPTICPROBLEMS ARISING IN ASTROPHYSICSMARCO CALAHORRANO & HERMANN MENAAbstract. We deal with the variational study of some type ofnonlinear inhomogeneous elliptic problems arising in models of so-lar flares on the halfplane Rn+.1. IntroductionIn this paper we study a boundary value problem of type(1.1){−∆u + c(x)u = λm(y)f(u) Rn+u(z, 0) = h(z) ∀z ∈ Rn−1where x = (z, y) ∈ Rn−1 × R+ ≡ Rn+ with R+ = {y ∈ R : y > 0} andn ≥ 2, f :]−∞, +∞[→ R is a function satisfying:(f-1) There exists s0 > 0 such that f(s) > 0 for all s ∈]0, s0[.(f-2) f(s) = 0 for s ≤ 0 o s ≥ s0.(f-3) f(s) ≤ asσ, a is a positive constant and 1 < σ < n+2n−2if n > 2or σ > 1 if n = 2.(f-4) There exists l > 0 such that |f(s1)− f(s2)| ≤ l|s1 − s2|, for alls1, s2 ∈ R.h is a non-negative bounded smooth function, h 6= 0, min h < s0, c ≥ 0,c ∈ L∞(Ω)⋂C(Ω) and mes{x ∈ Ω : c(x) = 0} = 0.The problem (1.1) is a generalization of an astrophysical gravity modelof solar flares in the half plane R2+, given in [1], namely:(1.2){−∆u = λe−βyf(u) R2+u(x, 0) = h(x) ∀x ∈ Rbesides 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 C1 functionsuch that ∫ +∞0ym(y)dy < +∞more general than e−βy.1991 Mathematics Subject Classification. 35J65, 85A30.Key words and phrases. Solar flares, variational methods, inhomogeneous semi-linear elliptic problems.EJQTDE, 2005 No. 19, p. 1We 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 casec(x) = 0. In fact, the result presented here follows from the one ob-tained by the authors.First of all we note that problem (1.1) is equivalent to(1.3){−∆ω + c(x)ω = λm(y)f(ω + τ) Rn+ω(z, 0) = 0 ∀z ∈ Rn−1where ω = u− τ and τ is solution of the problem(1.4){−∆τ + c(x)τ = 0 Rn+τ(z, 0) = h(z) ∀z ∈ Rn−1We will study (1.3) instead of (1.1).The problem (1.1), or equivalently (1.3), is interesting not only onwhole Rn+, but also in an arbitrary big but finite domain in Rn+, for ex-ample for semidisks DR = {(x, y) ∈ Rn−1×R+ : |x|2+y2 < R2, y > 0},with R big enough.Motivated by this observation in section 2, we will study the followingapproximate problem(1.5){−∆ω + c(x)ω = λm(y)f(ω + τ) DRω = 0 ∂DRwhose solutions are related to those of (1.3).Using variational techniques we will prove the existence of an inter-val Λ ⊂ R+ such that for all λ ∈ Λ there exists at least three positivesolutions of (1.5), with R large enough.Finally in section 3 we prove the existence of solutions of (1.3) aslimit of a special family of solutions of (1.5) obtained in theorem 5 andits uniqueness to λ small enough.2. Problem in DRLetting Ω be either DR or Rn+, we denote by Lpm(Ω) the usual weightedLp space on Ω for a suitable weight m and 1 ≤ p < ∞, and by V 1,2m (Ω),V 1,2c (Ω) the completion of C∞0 (Ω) in the norm‖u‖2V1,2m (Ω)=∫Ωu2(z, y)m(y)dzdy +∫Ω|∇u|2dzdyEJQTDE, 2005 No. 19, p. 2and‖u‖2V1,2c (Ω)=∫Ωu2(x)c(x)dx +∫Ω|∇u|2dxLet m : R+ → R+ be such that(2.1) 0 < M ≡∫ +∞0ym(y)dy < +∞it is easy to prove for all functions u ∈ C∞0 (Ω) the following inequalityholds, see [8].(2.2)∫Ωu2(x, y)m(y)dxdy ≤ M∫Ω|∇u|2dxdythen V 1,2m (DR) ∼ H10 (DR) ∼ V1,2c (DR) and V1,2m (Rn+) ∼ D1,2(Rn+) whereH10 (DR) is the usual Sobolev space with the norm ‖∇(.)‖L2(DR) andD1,2(Rn+) is the completion of C∞0 (Rn+) for the norm ‖∇(.)‖L2( n+).On the other hand if R′≥ R, then(2.3) V 1,2c (DR) ⊂ V1,2c (DR′ ) ⊂ V1,2c (Rn+) ⊂ V1,2m (Rn+)There exists many results about immersion of weighted Sobolevspaces into weighted Lebesgue spaces. Here we will take into accountone suitable result for our problem.Let m : R+ → R+ be a bounded C1 function such that there existsk > 0 such that(2.4) |(log m)′| ≤ kthen the identity map is an immersion from V 1,2m (Ω) into Lpmp2(Ω) for1 < p < 2nn−2if n ≥ 31 < p if n=2More precisely, there exists a constant K = K(k, sup m) such that(2.5) ‖u‖Lpmp2(Ω) ≤ CsK‖u‖V 1,2m (Ω)where Cs is the usual Sobolev immersion constant. The immersion iscompact if Ω = DR.Now we will begin to study (1.3) by variational methods. For thispurpose, for all λ ≥ 0 and for all non negative function τ such that‖τ‖Lσ+1m < +∞ we associate the functional Ψλ,τ : V1,2c (Rn+) → R(2.6) Ψλ,τ (u) =12∫ n+{|∇u|2 + c(x)u2} − λ∫ n+mF (u + τ)EJQTDE, 2005 No. 19, p. 3where F (t) =∫ t0f(s)ds, m ∈ C1(R+) and m̂ ≡ m2σ+1 satisfying (2.1)and (2.4).Ψλ,τ is a C1 functional, so if u ∈ V 1,2c (Rn+) is a critical point of Ψλ,τthen u is a weak, and by regularity a classical solution of (1.3).Remark 1. i. If we consider Ψλ,τ,R : V1,2c (DR) → R,Ψλ,τ,R(u) =12∫DR{|∇u|2 + c(x)u2} − λ∫DRmF (u + τ)its critical points are weak, and by regularity, strong solutions of (1.5).Furthermore if R ≤ R′≤ +∞, then for all u ∈ V 1,2c (DR)Ψλ,τ,R′ (u) ≤ Ψλ,τ,R(u) ≤ Ψλ,0,R(u)more preciselyΨλ,τ,R′ (u) = Ψλ,τ,R(u)− λ∫DR′−DRmF (τ) ≤ Ψλ,τ,R(u)Here DR′ with R′= +∞ means Rn+.ii. Since f is bounded, Ψλ,τ,R is coercive, bounded from below andverifies Palais-Smale condition for all λ non negative.Lemma 2. For each R > 0 denote θR : Rn → Rn the mapθR(z, y) ≡(zR, y)and ΘR the scaling η → ηR ≡ ηoθR. Theni. ∀r > 0, ΘR(V1,2c (Dr)) ⊂ V1,2c (θ−1R Dr) and if R ≥ 1, V1,2c (θ−1R Dr) ⊂V 1,2c (DRr).ii. If η ∈ C∞0 (Rn+), is non identically 0, then(2.7) ‖∇ηR‖L2(   n+) → +∞ as R → +∞iii. Let f be defined before and m such that verifies (2.1). Then thereexists 0 < λ < ∞ such that if λ > λ, η ∈ C∞0 (Rn+), η ≥ 0, nonidentically 0 and(2.8) λ ≤ Q(η) ≡12∫ n+{|∇η|2 + ‖c‖L∞η2}∫ n+m(y)F (η)< λthen there exists rn > 0 : ηR ∈ V1,2c (DR′ ), ∀R′, R: R′≥ Rrn ≥ rn andfor all non negative function τ .a. Ψλ,τ,R′ (ηR) < 0, ∀R′, R: R′≥ Rrn ≥ rn.b. Ψλ,τ,Rrn(ηR) → −∞ as R → +∞EJQTDE, 2005 No. 19, p. 4Proof.- This proof follows almost directly from lemma 6 in [8]. How-ever, by completeness we present all the proof.i. It is immediate from the definition of ΘR.ii. We observe|∇ηR|2(z, y) =1R2|∇η|2θR +(1−1R2)|∂yη|2θR(z,y)thus, changing variables(2.9) ‖∇ηR‖2L2( n+)= Rn−1[1R2∫ n+|∇η|2 +(1−1R2)∫ n+|∂yη|2]so, since∫ n+|∂yη|2 > 0, (2.9) implies (2.7).iii. Set(2.10) λ ≡ inf{Q(η) : η ∈ C∞0 (Rn+), η ≥ 0, η 6= 0}by (f-3) and since F is bounded(2.11)b2≡ sups>0F (s)s2< +∞so, by (2.2) and since c(x) ≥ 0∫ n+m(y)F (η) ≤bM2∫ n+|∇η|2 + ‖c‖L∞η2hence0 <1bM≤ λ < ∞Let λ > Q(η) be, since η ∈ C∞0 (Rn+), there exists rn > 0 such that suppη ⊂ DRrn , for all R ≥ 1. Then by i. and (2.3) ηR ∈ V1,2c (θ−1R Drn) ⊂V 1,2c (DRrn) ⊂ V1,2c (DR′ ) for all R′≥ Rrn ≥ rn.For simplicity from now on we call Rrn ≡ Rn, where R ≥ 1.Then, by remark 1(2.12) Ψλ,τ,R′ (ηR) ≤ Ψλ,τ,Rn(ηR) ≤ Ψλ,0,Rn(ηR)On the other hand, if we define the function ξ : R+ → Rξ(R) ≡1Rn−1‖∇ηR‖2L2( n+)=1R2∫ n+|∇η|2 +(1−1R2) ∫ n+|∂yη|2=[ ∫ n+|∇zη|2R2∫ n+|∂yη|2+ 1]∫ n+|∂yη|2is non increasing. So applying ξ(R) ≤ ξ(1) to (2.9)∫DRn|∇ηR|2 =∫ n+|∇ηR|2 ≤ Rn−1∫ n+|∇η|2EJQTDE, 2005 No. 19, p. 5furthermore∫DRnc(x)η2R =∫ n+c(x)η2R ≤ Rn−1‖c‖L∞∫ n+η2and ∫DRnm(y)F (ηR) =∫ n+m(y)F (ηR) = Rn−1∫ n+m(y)F (η)soΨλ,0,Rn ≤ Rn−1[12∫ n+|∇η|2 + ‖c‖L∞η2 − λ∫ n+m(y)F (η)]then(2.13) Ψλ,0,Rn ≤Rn−12∫ n+|∇η|2 + ‖c‖L∞η2(1−λQ(η))thus, from (2.12) and (2.13) we obtain immediately a and b.Remark 3. i. Let m : R+ → R+ be a bounded C1 function and letm̂ ≡ m2σ+1 . It is easy to prove that m verifies (2.4) if and only if m̂does it. Furthermore, given a positive constant k, |(log m)′| ≤ k if andonly if |(log m̂)′| ≤ 2σ+1k.ii. If there exists a non negative value m1 ≥ 0 such that {m > 1} ⊂[0, m1] and0 < M̂ ≡∫ +∞0ym̂(y)dy < +∞then0 < M ≡∫ +∞0ym(y)dy < +∞Indeed, since m̂ > 1 if and only if m > 1 and 0 < 2σ+1< 1M =∫ m>1ym(y)dy +∫ m≤1ym(y)dy ≤(sup mm12)+ M̂ < +∞Lemma 4. There exists a positive constant C = C(a, σ, k, sup m, M̂)such that for all λ < λ(‖τ‖Lσ+1m ( n+)) and for all u : ‖u‖V1,2c ( n+)=‖τ‖Lσ+1m ( n+), Ψλ,τ (u) > 0 where λ(‖τ‖Lσ+1m ( n+)) ≡ C‖τ‖1−σLσ+1m ( n+).Moreover λ(‖τ‖Lσ+1m ( n+)) → +∞ as ‖τ‖Lσ+1m ( n+)→ 0EJQTDE, 2005 No. 19, p. 6Proof.- Let u ∈ V 1,2c (Rn+) be, using (f-3) and Minkowsky inequalitywith respect to measure m(y)dxdy and (2.2), (2.5) we obtain0 ≤∫ n+mF (u + τ) =∫ n+m∫ u+τ0f(t)dt ≤aσ + 1∫ n+m(u + τ)σ+1≤aσ + 1(‖u‖Lσ+1 mσ+12+ ‖τ‖Lσ+1m )σ+1≤aσ + 1(CsK(1 + M̂)12 ‖∇u‖L2(  n+) + ‖τ‖Lσ+1m )σ+1≤aσ + 1(CsK(1 + M̂)12 ‖u‖V1,2c ( n+)+ ‖τ‖Lσ+1m )σ+1then(2.14)Ψλ,τ (u) ≥12‖u‖2V1,2c ( n+)−λaσ + 1(CsK(1+M̂)12 ‖u‖V 1,2c ( n+)+‖τ‖Lσ+1m )σ+1then, if we defineC ≡σ + 12a(CsK(k, sup m)(1 + M̂)12 + 1)−σ−1then Ψλ,τ(u) > 0 for all λ < λ ≡ C‖τ‖1−σLσ+1m ( n+), and since σ > 1. Thelemma is proved.Theorem 5. Let us assume (f-1-2-3-4), let m:R+ → R+ be a C1function such that m and m̂ ≡ m2σ+1 verify (2.1) and (2.4), and letτ : Rn+ → R+ be a C1 function, non identically 0. So there existspositive constants C = C(a, σ, k, sup m, M̂) and λ such that if(2.15) ‖τ‖Lσ+1m ( n+)<(cλ) 1σ−1then∀λ : λ < λ < λ ≡ C‖τ‖1−σLσ+1m ( n+)there exists a positive R0 = R0(λ) such that for all R ≥ R0, (1.5) hasat least three strictly positive solutions.Proof.- Let C = C(a, σ, k, sup m, M̂) and λ be the positive constantdefined in lemmas 4 and 2 respectively . Since τ verifies (2.15), bylemma 4 and remark 1, for all λ ∈]λ, λ[ and for all R ≥ 1(2.16) Ψλ,τ,R(u) > 0 ∀u ∈ V1,2c (DR) : ‖u‖V 1,2c (DR) = ‖τ‖Lσ+1m (Rn+)On the other hand, fixed λ ∈]λ, λ[, η ∈ C∞0 (Rn+), and letting rn > 0, theradius of any semidisk Drn such that supp η ⊂ Drn, by lemma 2 thereEJQTDE, 2005 No. 19, p. 7exists R1 ≥ 1 such that for all R ≥ R1rn, we have ηR1 ∈ V1,2c (DR),furthermore(2.17) ‖τ‖Lσ+1m ( n+)< ‖∇ηR1‖L2(DR) = ‖∇ηR1‖L2( n+)< ‖ηR1‖V 1,2c ( n+)and(2.18) Ψλ,τ,R(ηR1) < µ < 0where µ ∈ R defined asµ ≡ min0≤t≤‖τ‖Lσ+1m (  n+)12t2 − λaσ + 1(CsK(1 + M̂)12 t + ‖τ‖Lσ+1m )σ+1Let R ≥ R1, we divide the proof in three steps.1. Local minimum.- LetνR ≡ infBΓΨλ,τ,R(u)where BΓ = {u ∈ V1,2c (DR) : ‖u‖V 1,2c (DR) < Γ ≡ ‖τ‖Lσ+1m ( n+)}.Since Ψλ,τ,R(0) < 0, νR < 0. Furthermore µ ≤ νR < 0, by (2.14) andremark 1 . Therefore inf∂BΓ Ψλ,τ,R > νR.Now we will prove that νR is achieved in BΓ. Using a modification inthe proof of proposition 5 and corollaries 6 and 7 in [3], we can obtaina sequence (un)n in BΓ such thatΨλ,τ,R(un) → νRΨ′λ,τ,R(un) → 0since Ψλ,τ,R verifies Palais-Smale condition, there exists a subsequence(unk)k such that unk → u1,R in V1,2c (DR) and u1,R 6= 0 because 0 it isnot a critical point of Ψλ,τ,R.2. Absolute minimum.- LetuR ≡ infV1,2c (DR)Ψλ,τ,RThen uR < µ, by (2.17). Now using similar arguments to local min-imum, but without any modification, we have that uR is achieved inV 1,2c (DR) at a function u2,R.3.Mountain pass.- LetcR ≡ infδ∈ΛRsupu∈δΨλ,τ,R(u)where ΛR is the set of pathsΛR = {γ : γ ∈ C([0, 1], V1,2c (DR)), γ(0) = 0, γ(1) = ηR1}Since Ψλ,τ,R(0) < 0, by (2.15), (2.16) and (2.17), cR > 0.Then by the mountain pass theorem, see [4], cR is achieved in V1,2c (DR)at a function u3,R.EJQTDE, 2005 No. 19, p. 8On the other hand it is clear that u1,R, u2,R and u3,R are different,indeedΨλ,τ,R(u2,R) = uR < µ ≤ νR = Ψλ,τ,R(u1,R) < 0 < cR = Ψλ,τ,R(u3,R)Remark 6. When λ is small enough it is easy to prove uniqueness for(1.5), so u1,R = u2,R, and the local minimum in BΓ od Ψλ,τ,R is theabsolute in V 1,2c (DR).3. Problem in Rn+Ψλ,τ does not verifies Palais-Samale condition, furthermore by lemma2 and remark 1 Ψλ,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 supposem verifies (2.1). Theni. For all λ < 1bM, Ψλ,τ is coercive and bounded from below.ii. For all λ < 1lM, (1.3) has at most one solution in V 1,2c (Rn+).λ < λ holds in both cases.Proof.- i. By (2.11), (2.2) and Cauchy-Schwartz for the measuremdxdyΨλ,τ (u) ≥12‖u‖2V1,2c ( n+)−λb2∫ n+m(u + τ)2≥12‖u‖2V1,2c ( n+)−λb2(M12‖∇u‖L2(   n+) + ‖τ‖L2m( n+))2≥12(1− λbM)‖u‖2V1,2c ( n+)− (λbM12 ‖τ‖L2m( n+))‖u‖V 1,2c ( n+)−−(λb2‖τ‖2L2m( n+))so, i. is proved.ii. Uniqueness is proved as in [1] using the inequality (2.2) and (f-4).Indeed: if u1 and u2 are two solutions of (1.3) then∫ n+(u1−u2)2m ≤ M∫ n+|∇(u1−u2)|2+c(x)(u1−u2)2 ≤ Mlλ∫ n+(u1−u2)2mNow we will prove a sufficient condition to approximate solutions of(1.3) with solutions of (1.5) with R large enough.EJQTDE, 2005 No. 19, p. 9Lemma 8. Let f and τ be as above and λ ∈ R+. Suppose (Rn)nis a sequence R+ such that Rn → +∞ and (un)n is a sequence ofpositive solutions of (1.5) with Rn instead of R, such that for all n,un ∈ V1,2c (DRn) and (un)n is bounded in V1,2c (Rn+), i.e. there existsΓ′> 0 such that for all n, ‖un‖V 1,2c (DRn) < Γ′. Then, there exists asubsequence (called again (un)n)) and a function u ∈ V1,2c (Rn+) suchthat un → u weakly in V1,2c (Rn+) and u is a classical solution (1.3).Proof.- By the Caldero´n-Zygmund1 inequality for all n, un ∈ H10 (DRn)⋂H2,p(DRn) and fixed R′> 0, for any Ω′⊂⊂ DR′(3.1) ‖un‖H2,p(Ω′ ) ≤ C(‖un‖Lp(DR′ ) + ‖λm(y)f(un + τ)‖Lp(DR′ ))for all n such that Rn > R′. The constant C depends on DR′ , n, p andΩ′. Since m is decreasing and strictly positive, and (un)n is boundedin V 1,2c (Rn+), by (2.2), (2.5), (3.1) and the hypothesis of f and m, weobtain‖un‖H2,p(Ω′ ) ≤ C(m(R′)−12 CsK(1 + M)12 Γ′+ λ sup m sup f |DR′ |1p )for p such that1 < p < 2nn−2if n ≥ 31 < p if n=2and for all n such that Rn > R′.For this and the Sobolev embedding theorem for Ω′, there exists asubsequence (un)n such that if n=2,3 un → u in C1,α(Ω′) and if n ≥ 4and 1 < p < min(n2, 2nn−2)is fixed, un → u in Lq(Ω′), 1 ≤ q < npn−2p.Since Ω′is an arbitrary and relatively compact such that Ω′⊂⊂ DRnand Rn → +∞, we obtain that the above convergences are in C1,αloc (Rn+)and Lqloc(Rn+) respectively. In particular(3.2) un → u en L1loc(Rn+)On the other hand, since (un)n is bounded in V1,2c (Rn+), by (2.3), (2.5)and reflexivityun → u weakly in V1,2c (Rn+)(3.3)un → u weakly in Lpmp2(Rn+)(3.4)where1 < p < 2nn−2if n ≥ 31 < p if n=21see theorems 9.9 y 9.11 in [9]EJQTDE, 2005 No. 19, p. 10So, if we prove that for all v ∈ C∞0 (Rn+)∫ n+mf(un + τ)v →∫ n+mf(u + τ)vour lemma will follow. Based on this and for a fixed v ∈ C∞0 (Rn+) weconsider the functionw = vf(u + τ)u + τm2−p2It is easy to prove that w ∈ Lp′mp2(Rn+), where1p+ 1p′ = 1. Now∫ n+mf(un + τ)v =∫ n+m[f(un + τ)− (un + τ)f(u + τ)u + τ]v ++∫ n+mp2 (un + τ)w(3.5)by (3.4), the last term of right hand side of (3.5) tends to∫ n+mf(u +τ)v. On the other hand, by (f-4)(3.6)∣∣∣∣∫ n+m[f(un+τ)−(un+τ)f(u + τ)u + τ]v∣∣∣∣ ≤ 2l∫supp(v)m|u−un||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‖Lσ+1m ( n+).Then for all λ, 0 < λ < λ the local minima u1,R of Ψλ,τ,R, approximatethe local minima of Ψλ,τ on the ball BΓ of center 0 and radius Γ inV 1,2c (Rn+).As a consequence ν∞ ≡ infBΓ Ψλ,τ , is a minimum and by proposition 7it is the unique critical point of Ψλ,τ , if λ small enough(i.e. 0 < λ <1lM).Proof.- We only need to prove that νR → ν∞ as R → ∞. Withthis aim we consider (uR)R in C∞0 (Rn+) such that uR ∈ V1,2c (DR) andΨλ,τ,R(uR) → ν∞ as R →∞. By remark 1 λ∫ n+−DRmF (τ)dx → 0 asR →∞, because Γ < +∞.Sinceν∞ ≤ νR = Ψλ,τ,R(u1,R) ≤ Ψλ,τ,R(uR) = Ψλ,τ(uR)− λ∫ n+−DRmF (τ)then νR → ν∞ as R →∞.EJQTDE, 2005 No. 19, p. 11References[1] J.J. Aly, T. Amari,Two-dimensional Isothermal Magnetostatic Equilibria ina Gravitational Field I, Unsheared Equilibria, Astron & Astrophys. 208, pp.361-373.[2] A. 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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 SecondOrder, Second Edition, Springer Verlag, Berlin, 1983.[10] J. Heyvaerts, J. M. Lasry, M. Schatzman and P. Witomski, Solar Flares: ANonlinear Eigenvalue Problem in an Unbounded Domain. In Bifurcation andNonlinear 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 Two-dimentional Magnetohydrostatic Equilibra by an Increase of Electric Currentor 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 Applica-tions to Differential Equations, CBMS, Regional Conference Series in Mathe-matics, 65, vii, 100 p. (1986).(Received October 23, 2003)Marco CalahorranoEscuela Polite´cnica Nacional, Departamento de Matema´tica, Apartado17-01-2759, Quito, EcuadorE-mail address : calahor@server.epn.edu.ecURL: www.math.epn.edu.ec/miembros/calahorrano.htmHermann MenaEscuela Polite´cnica Nacional, Departamento de Matema´tica, Apartado17-01-2759, Quito, EcuadorE-mail address : hmena@server.epn.edu.ecURL: www.math.epn.edu.ec/~ hmenaEJQTDE, 2005 No. 19, p. 12

Remarks on inhomogeneous elliptic problems arising in astrophysics

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