REMARKS ON INHOMOGENEOUS ELLIPTIC PROBLEMS ARISING IN ASTROPHYSICS

We deal with the variational study of some type of nonlinear inhomogeneous elliptic problems arising in models of so- lar ares on the halfplane R n.

By this, we study the problem ( 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 where ω = u − τ and τ is solution of the problem 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 Motivated by this observation in section 2, we will study the following approximate problem (1.5) 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.

Problem in D R
Letting Ω be either D R or R n + , we denote by L p m (Ω) the usual weighted L p space on Ω for a suitable weight m and 1 ≤ p < ∞, and by Let m : R + → R + be such that it is easy to prove for all functions u ∈ C ∞ 0 (Ω) the following inequality holds, see [8]. (2.2) 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 More precisely, there exists a constant 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 where ) is a critical point of Ψ λ,τ then u is a weak, and by regularity a classical solution of (1.3).
its critical points are weak, and by regularity, strong solutions of (1.5).
is coercive, bounded from below and verifies Palais-Smale condition for all λ non negative.
iii.Let f be defined before and m such that verifies (2.1).Then there Proof.-Thisproof 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 thus, changing variables , there exists r n > 0 such that supp η ⊂ D Rrn , for all R ≥ 1.Then by i. and For simplicity from now on we call Rr n ≡ R n , where R ≥ 1.Then, by remark 1 On the other hand, if we define the function ξ : is non increasing.So applying ξ(R) ≤ ξ(1) to (2.9) thus, from (2.12) and (2.13) we obtain immediately a and b. ) and for all u : c (R n + ) be, using (f-3) and Minkowsky inequality with respect to measure m(y)dxdy and (2.2), (2.5) we obtain , and since σ > 1.The lemma is proved.
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.
where (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 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 0) < 0, by (2.15), (2.16) and (2.17), c R > 0. Then by the mountain pass theorem, see [4] On the other hand it is clear that u 1,R , u 2,R and u 3,R are different, indeed 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 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 < 1 bM , Ψ λ,τ is coercive and bounded from below.ii.For all λ < 1 lM , (1.3) has at most one solution in V 1,2 c (R n + ).λ < λ holds in both cases.