Positive radial symmetric solutions for a class of elliptic problems with critical exponent and-1 growth

for u ∈ H1 0(B). It is well known that the singular term leads to the non-di erentiability of I on H1 0(B). In fact, since I(tu) → +∞ as t → 0+, I is not continuous at the point 0. Therefore, it is di cult to nd out the local minimizer and the mountain pass type solutions of problem (1.1). In order to nd rstly a local minimizer solution, we consider the following problem ∆u + μ u = 0, in B, u = 0, on ∂B. (1.2)


Introduction and main result
for u ∈ H (B). It is well known that the singular term leads to the non-di erentiability of I on H (B). In fact, since I(tu) → +∞ as t → + , I is not continuous at the point . Therefore, it is di cult to nd out the local minimizer and the mountain pass type solutions of problem (1.1). In order to nd rstly a local minimizer solution, we consider the following problem (1.2) According to Theorem 1 of [1], we know that problem (1.2) has a unique positive solution wµ ∈ C +α (B) ∩ C(B)( < α < ) with wµ ≥ cϕ (where ϕ is an eigenfunction corresponding to the smallest eigenvalue λ of the problem ∆ϕ + µϕ = , ϕ| ∂B = ). Moreover, [1] proved that the following inequality holds B ϕ t dx < +∞, (1.3) if and only if t ∈ ( , ).
The following singular elliptic problem has been extensively considered    ∆u + u p + µ u γ = , in Ω, u = , on ∂Ω, (1.4) where Ω ⊂ R N (N ≥ ) is a bounded domain with smooth boundary ∂Ω, < p ≤ * − and γ > . For examples, [2][3][4][5][6][7][8][9][10][11] studied the case of < γ < for problem (1.4). Particularly, by the variational method and Nehari method, Sun, Wu and Long investigated the multiplicity of positive solutions for the singular elliptic problem for the rst time in [8]. And, Yang [11] discussed problem (1.4) with p = * − for the rst time, and obtained two positive solutions by using the variational method and sub-supersolution method. For the case γ > , problem (1.4) is considered by [1,12,13]. To our best knowledge, problem (1.4) with γ = is only investigated by [4], and two positive solutions are obtained when < p < * − . A nature question is whether there exist positive solutions for problem (1.4) with γ = and p = * − . In the present note, we give a positive answer by the critical point theory for nonsmooth functionals, and obtain two positive solutions for problem (1.1). Moreover, based on the moving plane technique, we study the symmetry and monotonicity properties of positive solutions to problem (1.1).
In order to study problem (1.1), we de ne f : Consider the following auxiliary problem (1.5) Problem (1.5) has a variational structure given by the functional Then the functional J is only continuous in H (B). In the following, we need nd out critical points of J, and prove that they are weak solutions of problem (1.1). Now our main result is as follows.

Proof of Theorem 1.1
We divide two parts to prove Theorem 1.1. First, by using the critical point theory for nonsmooth functionals, we prove that problem (1.1) has at least two positive solutions. Then, we prove that the solution of problem (1.1) is radially symmetric and monotone decreasing about the origin by the moving plane method.

. Existence of Two Positive Solutions
By the de nition of F, the following statement is valid.
From the above information, one has By the dominated convergence theorem, (2.1) holds. The proof is complete.
We now recall some concepts adapted from critical point theory for nonsmooth functionals. Let (X, d) be a complete metric space, f : X → R be a continuous functional in X. Denote by |Df |(u) the supremun of δ in [ , ∞) such that there exist r > , a neighborhood U of u ∈ X, and a continuous map σ : (2.4) The extended real number |Df |(u) is called the weak slope of f at u, see [14,15]. A sequence {un} of X is called Palais-Smale sequence of the functional f , if |Df |(un) → as n → ∞ and f (un) is bounded. We say that u ∈ X is a critical point of f if |Df |(u) = . Since u → |Df |(u) is lower semicontinuous, any accumulation point of a (PS) sequence is clearly a critical point of f .
Since looking for positive solutions of problem (1.1), we consider the functional J as de ned on the closed positive cone P of H (B) P is a complete metric space and J is a continuous functional on P. Then we have the following conclusion.

Lemma 2.2.
Assume that u ∈ P and |DJ|(u) < +∞, then for any v ∈ P there holds Proof. Similar to the proof of Lemma 3.1 in [15].
Consequently, we assume that there exist sequences {un} ⊂ P and {tn} ⊂ [ , +∞), such that un → u, tn → + , and where sn = tn v−un → + as n → ∞. Divided by sn in (2.6), one has Notice that where ξn ∈ (un − sn un , un + sn(v − un)), which implies that ξn → u (un → u) as sn → + . Note that F(x, t) is increasing in t, then I ,n ≥ for all n. Applying Fatou's Lemma to I ,n , we obtain for v ∈ P. For I ,n , by the di erential mean value theorem, we have From the above information, one has By the arbitrariness of the sign of φ, we can deduce that (2.7) holds. Thus, u is a weak solution of problem (1.5).
(ii) This follows from (i). The proof of Lemma 2.3 is completed.
Let S be the best Sobolev constant, namely  (2.14) Since f (x, un)un ≤ for all n, by the dominated convergence theorem, one gets Set wn = un − u and lim n→∞ wn = l > , by Brézis-Lieb's lemma, Lemma 2.1 and (2.14), we have On the other hand, for φ ∈ P, taking v = un + φ in (2.11) and letting n → ∞, by Fatou's lemma, we obtain Therefore, similar to the proof of (i) in Lemma 2.3, for all φ ∈ H (B), we have In particular, one has  We see that there exist constants ρ, r, Λ > , such that J(u)| Sρ ≥ r for every µ ∈ ( , µ ). Moreover, for u ∈ H (B)\{ }, it holds So we obatin J(tu) < for all u + ≠ and t small enough. Therefore, for u small enough, one has According to Lemma 2.1, similar to [16], we can easy obtain that d is attained at some u * ∈ Bρ . According to Lemma 2.3 and Lemma 2.5 (i), we obtain the following result. where c > is a constant. Since J(u * + tφε) → −∞ as t → ∞, J(u * ) < , according to Lemma 2.5 (i), we can assume that there exist t , t > such that sup t≥ J(u * + tφε) = sup t∈[t ,t ] J(u * + tφε).
Since u * is a positive solution of problem (1.1), we have for all t ≥ . Therefore, from (2.22), (2.23) and the following inequality such that g ′ (t) > for < t < tmax and g ′ (t) < for t > tmax . Moreover, g(tmax) = max t≥ g(t). By a standard regularity argument, one has u * ∈ C (B, R + ) and there exists a positive constant C(independent of x) such that u * < C.
for all µ ∈ ( , µ ). This leads us to the proof of Lemma 2.7.

. Radially Symmetric and Monotone Decreasing Solution
In this part, we shall prove that every positive solution u of problem (1.1) is radially symmetric and monotone decreasing about the origin. Let u = µ v, then we deduce that v satis es the following equation (2.25) Therefore, it is su cient to prove that v is radially symmetric and monotone decreasing about the origin. First, we introduce some notations. Choose any direction to be the x direction. Let T λ = {x ∈ R n |x = λ, for some λ ∈ R} be the moving plane, and the set Σ λ = {x ∈ B ( )|x < λ} be the region to the left of the plane. We use the standard notation for the re ection of x about the plane T λ . We denote v(x λ ) = v λ (x), then compare the values of v(x) and v λ (x), let Otherwise, after a direct calculation, we derive that w λ (x λ ) = −w λ (x), hence it is said to be anti-symmetric. We carry out the proof in two steps. To begin with, we show that for λ su ciently close to − , we have This provides a starting point to move the plane T λ . Next, we move the plane T λ along the x direction to the right as long as inequality (2.26) holds. The plane T λ will eventually stop at some limiting position λ , where λ = sup{λ|wρ(x) ≥ , ρ ≤ λ}, then we are able to claim that The symmetry and monotonicity of solution v about T λ follow naturally from the proof. Also, because of the arbitrariness of the x direction, we conclude that v must be radially symmetric and monotone decreasing about the origin.
Step 1. We show that for λ su ciently close to − , we have If not, we denote there exists some point x ∈ Σ λ , such that By problem (2.25) and the Mean Value Theorem, we can easy derive that −∆w λ (x ) = −∆v λ (x ) − (−∆v(x )) where ξ λ (x ), η λ (x ) are in between v λ (x ) and v(x ), and the last inequality is due to the negative point x of Then we obtain that Since v ∈ C ∞ (B) ∩ L ∞ (B) (see [4]), v(x ) is bounded. We can see that c(x ) > provided µ enough small. On the other hand, at the minmum point, we have that −∆w λ (x ) + c(x )w λ (x ) < , which contradicts with (2.29). Therefore (2.27) hods. This completes the preparation for the moving of planes.
Step 2. Inequality (2.27) provides a starting point, from which we move the plane T λ toward the right as long as (2.27) holds to its limiting position to show that v is monotone decreasing about the origin. More precisely, we de ne λ = sup{λ|wρ(x) ≥ , ρ ≤ λ}.
We will show that λ = , and w λ (x) ≡ . (2.30) If not, then λ < , we show that the plane can be moved further right to cause a contradiction with the de nition of λ . More precisely, there exists a small ε > such that for all λ ∈ (λ , λ + ε) such that w λ (x) ≥ , x ∈ Σ λ , (2.31) which contradicts the de nition of λ . Hence, (2.30) must be true.