Positive Solution to Singular Elliptic Problems with Subcritical nonlinearities


               <jats:p>In this paper, we study the existence of a non-trivial weak solution to the following singular elliptic equations with subcritical nonlinearities:</jats:p>
               <jats:p>
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                        <jats:tex-math>\left\{ {\matrix{   { - div\left( {{{\left| x \right|}^{ - 2\beta }}\nabla u} \right) - \mu {{f(x)u} \over {{{\left| x \right|}^{2(\beta  + 1)}}}} = {{\lambda g(x)} \over {{u^\theta }}} + h(x){u^p}\,\,\,\,in\,\,\,\Omega ,} \hfill  \cr    {u > 0\,\,\,in\,\,\Omega ,} \hfill  \cr    {u = 0\,\,on\,\,\partial \Omega ,} \hfill  \cr  } } \right.</jats:tex-math>
                     </jats:alternatives>
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               </jats:p>
               <jats:p>where Ω ⊂ℝ<jats:italic>
                     <jats:sup>N</jats:sup>
                  </jats:italic> is an open bounded domain with <jats:italic>C</jats:italic>
                  <jats:sup>1</jats:sup> boundary, θ, λ > 0, <jats:inline-formula>
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                        <jats:tex-math>0 < \beta  < {{N - 2} \over 2}</jats:tex-math>
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                        <jats:tex-math>0 < \mu  < {\left( {{{N - 2(\beta  + 1)} \over 2}} \right)^2}</jats:tex-math>
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                  <jats:sup>∞</jats:sup> (Ω). We show that there exists a solution <jats:inline-formula>
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                        <jats:tex-math>u \in H_0^1\left( {\Omega ,{{\left| x \right|}^{ - 2\beta }}} \right) \cap {L^\infty }(\Omega )</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> to this problem.</jats:p>

When h(x, s) that does not depend on x, Crandall, Rabinowitz and Tartar in [7], showed the existence and continuity properties of the solution of (1.4). In [12], it has been shown that the existence of the solution to where f is a continuous function. They showed that the solution u ∈ H (Ω) if and only if γ < and that, if γ > , solution u ∉ C (Ω). For, h(x, u) = f (x)g(u), Lair and Shaker [10,11] studied the problem (1.4) , and by Zhang and Cheng, see [19], using the rst eigenfunction of the Laplacian in Ω.
The present work is more close to results in [1,8], although we generalised the results for sublinearities, i.e. for the case < p < [1, Theorem 2.4]. The aim of this paper is to show the existence to (1.1) for sublinearities. To obtain a solution to (1.1), we use the similar techniques as used in [1,2]. The organisation of this paper is as follows. Section deals with some preliminaries, basic facts which are used in the main result. Section is devoted to the proof of the main theorem. The main result of this paper is the following theorem, which we will prove in the next section.

Auxiliary results
Let n ∈ N, let gn(x) = min{g(x), n}. To prove the existence results we consider the following approximate problem: (2.1) Let us recall the following Ca arelli-Kohn-Nirenberg inequality and a compact embedding theorem (which is an extension of the classical Rellich-Kondrachov compactness theorem). [4]: There is a constant C γ,β > such that

Proposition 2.1. Ca arelli-Kohn-Nirenberg inequality
Let H (Ω, |x| − β ) be the completion of C ∞ (R N ), with respect to the following weighted norm · de ned by From the boundedness of Ω and the standard approximation arguments, it can be shown that (2.2) holds for any u ∈ H (Ω, |x| − β ) in the sense: for ≤ q ≤ * .

Proposition 2.2. General Hardy-Sobolev inequality: There is a constant
Moreover, C − β,N, is optimal and it is not achieved.
Let L q (Ω, |x| −βq ) be the weighted L q space with weighted norm de ned by then (2.5) can be written as where C > denotes a universal constant, which may change its value from line to line.

Proposition 2.3. Compact imbedding theorem[18]: Suppose that Ω ⊂ R N is an open bounded domain with C boundary and
Proof. Let n ∈ N be a xed natural no and ω be a function in L (Ω, |x| − β ). By the Lax-Milgram Theorem, the following problem has a unique solution u in H (Ω, |x| − β ) ∩ L ∞ (Ω): because the operator is coercive due to the assumptions on µ and Proposition 2.2. Now, for any u ∈ L (Ω, |x| − (β+ ) ), we de ne the mapping as follows: Let us take u as a test function, we have and so The General Hardy -Sobolev inequality on the left-hand side and Hölder inequality on the right-hand side implies that for some constant C independent of u.
This is equivalent to (2.17) Since gn(x) (ω+ n ) θ ≥ a.e., so the maximum principle implies that un ≥ and hence un be a solution of (2.1). By the result of [15], un belongs to L ∞ (Ω), because of the right-hand side of (2.10) belongs to L ∞ (Ω).

Proof of the theorem 1.1
Proof. Consider the following Dirichlet problems To prove the existence of a solution un of (3.1), we apply Sattinger monotone iteration.
We rst obtain a subsolution: By Lemma(2.4) there exists a weak solution un of the following Dirichlet problem so un solves: i.e un is a subsolution to the problem (3.1). Now, we construct a supersolution of (3.1): Let t ≥ Y λ (yet have to determine) and consider the following Dirichlet problems u n,t : u n,t is the solution of u n,t ∈ H (Ω, |x| − β ) : −div(|x| − β ∇u n,t ) − µ f (x)u n,t |x| (β+ ) + n = t (u n,t + n ) θ , (3.5) then (see [2]) ∃ c > such that u n,t L ∞ (Ω) ≤ c t θ+ . (3.6) Then the maximum principle implies that u n,t ≥ so that u n,t solves −div(|x| − β ∇u n,t ) − µ f (x)u n,t |x| (β+ ) + n = t (u n,t + n ) θ . (3.7) We show that for some t > , t n + u n,t θ ≥ The above inequality is true if t satisfy the following Suppose Y λ be such that That is there exists Λ > such that for < λ < Λ, there exists the solution of the equation (3.11). Therefore, we only have to prove that if n is large enough, we have (3.13) this will hold if n ≤ c Y θ+ λ and this latter inequality true for n large. Since Y λ independent of n. Thus we proved that for Y ≥ Y λ , u n,t is a super solution to the problem (3.1).
We show that un ≤ u n,t : De ne (3.14) Next aim to show un ≤ u n,t . Indeed  − u n,t ). (3.19) Note that, in the last line, the rst integral is negative because the real function n +l θ is decreasing if l > and the second integral is negative because t ≥ Y λ and by the de nition (3.11), Y λ ≥ λ g ∞ . Next we apply Sattinger Method: It can be shown that function w(s) is increasing. Now the classical Amann-Sattinger method will give the existence of un solution of and such that un ≤ un ≤ ω n,t ≤ c t θ+ .
(3.23) also using [2], it can be shown that the sequence {un} is increasing with respect to n, un > in Ω and for every Ω ′ ⊂ ⊂ Ω, ∃ c Ω ′ ( independent of n) such that (3.24) We will use un as a test function in the following equation:  where Ω ′ is an arbitrary open subset of Ω, such that Ω ′ ⊂ Ω. Moreover if < γ < , then u ∈ H (Ω) ∩ L ∞ (Ω). For the sake of brevity, we omit the proof.

Remark 3.3. [8]
The similar proof as in the Theorem 1.1, will work to treat the following semilinear elliptic problem: in Ω,