On isolated singularities of Kirchho equations

where p > 1, θ ∈ R,Mθ(u) = θ + ∫ Ω |∇u|dx, Ω is a bounded smooth domain containing the origin in R N with N ≥ 2. In the subcritical case: 1 < p < N N−2 if N ≥ 3, 1 < p < +∞ if N = 2, we employee the Schauder xed point theorem to derive a sequence of positive isolated singular solutions for the above equation such thatMθ(u) > 0. To estimate Mθ(u), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that Mθ(u) < 0, by analyzing relationship between the parameter λ and the unique solution uλ of −∆u + λup = kδ0 in B1(0), u = 0 on ∂B1(0).


Introduction and main results
A model with small variation of tension due to the changes of the length of a string is described by D'Alembert wave equation, it is also well-known as the Kirchho equation, see [15], which states as follows where τ is the tension, L = β−α is the length of the string at rest, m is the mass density, κ is the Young's modulus. The Kirchho -type problems have been attracted great attentions in the analysis of di erent nonlinear term due to the gradient term, see [9,11,24,37].
Observe that in the prototype of Kirchho model, the tension, for small deformations of the string, takes the linear form as follows: where a > , b > . When the displacement gradient is small, i.e. |∇u| , M(u) a + b|Ω| + b Ω |∇u| dx. The advantage for this approximation makes the problem have variational structure and the approximating solution could be constructed by variational methods. For example, the stationary analogue and qualitative properties of solutions to the Kirchho - (x, u) in Ω has been extensively studied in [9,10,14,16,19,30] and extended into the fractional setting in [27,28,33] and the references therein. In this case, M(u) = a + b Ω |∇u| dx is often called Kirchho function. In fact, the Kirchho function has been greatly extended for recent years. For example, the case a = , b > , which is called degenerate, has been intensely investigated recently, we refer to [34] for a physical explanation and [8,25,26,40,41] for related results in this direction.
Our interest of this paper is to study a new Kirchho -type problem by taking into account that |∇u| is not small in a bounded smooth domain Ω and the tension could be vector in a proper coordinate axis. In this situation, the Kirchho function (1.1) may be taken as M θ (u) = θ + Ω |∇u|dx, (1.2) where θ is assumed to be real number. Given a sequence of extra pressures {σm} with the support in B m ( ) and the total force F = Ω σm dx = keeps invariant. The limit of {σm} as m → +∞ in the distributional sense is Dirac mass. As we know that the corresponding solutions may blow up at the origin or blow up in the whole domain. Our aim is to clarify this limit phenomena of the solutions to some elliptic problems involving the Kirchho -type function (1.2).
More precisely, in this article we are interested in nonnegative singular solutions of the following Kirchho -type equation: where p > , M θ is de ned by (1.2) with θ ∈ R and Ω is a bounded, smooth domain containing the origin in R N with N ≥ . The following parameter plays an important role in obtaining the solutions of (1.3): G Ω is Green operator de ned as here G Ω is the Green kernel of −∆ in Ω × Ω with zero Dirichlet boundary condition. Note that ap is well-de ned when p is subcritical, that is, p < p * , where (1.5) Our rst existence result about isolated singular solutions with M θ (u) > is stated as follows.  (1.2) with θ ∈ R, ap is given by (1.4), p * is given by (1.5) and Ω is a bounded smooth domain containing the origin such that B ( ) ⊂ Ω and |Br ( )| = |Ω| where ≤ r < +∞. Let k > r θ− with θ− := min{ , θ} be such that Then for p ∈ ( , p * ), problem (1.3) has a nonnegative solution u k satisfying that and u k has following asymptotic behaviors at the origin 8) where c N > is the normalized constant and where δ is Dirac mass concentrated at the origin.
Remark 1.1. Note that ap depends on p and Ω and the value p = is critical for assumption (1.6) for N = , . Indeed, p * > occurs only for N = and N = . Due to the parameter θ, (1.6) gives a rich structure of isolated singular solutions for problem (1.3). Moreover, a discussion is put in Proposition 2.2 in Section 2.
Involving Kirhcho function M θ (u), the classical method of Lions' iteration argument in [17] does not work due to the lack of monotonicity of nonlinearity M − θ (u)u p , and also the variational method in [29] fails, since (1.3) has no variational structure. Furthermore, it is di cult to calculate precise value for Ω |∇u k | to express M θ (u k ), especially, when Ω is a general bounded domain. To overcome these di culties, we make use of the rearrangement argument to estimate the value of M θ (u) and employ the Schauder xed-point theorem to obtain the existence of isolated singular solutions in the class set of M θ (u) > .
When θ < , we can derive a branch of singular solutions such that M θ (u) < . Véron in [39] gave a survey on the isolated singularities of (1.10), in which B ( ) is replaced by general bounded domain containing the origin. With a general Radon measure and a more general absorption nonlinearity g : R → R satis es the subcritical assumption: problem (1.11) has been studied by Benilan-Brézis [1], Brézis [3], by approximating the measure by a sequence of regular functions, and nd classical solutions which converges to a weak solution. For this approach to work, uniform bounds for the sequence of classical solutions are necessary to be established. The uniqueness is then derived by Kato's inequality. Such a method has been applied to solve equations with boundary measure data in [13,[20][21][22] and other extensions in [2,5].
In the case λ = −M − θ (u), depending on the unknown function u, a di erent approach has to be taken into account to study problem (1.11). A branch of solutions such that M θ (u) < are derived from the observations that the function F(λ) = −M − θ (u) − λ is continuous and it has a zero, because we will nd two values λ , λ such that F(λ )F(λ ) < , where v λ is the unique solution of problem (1.11). This zero indicates a solution of problem (1.3).
For the singularity as |x| − /(p− ) , the di usion and the nonlinear terms play the predominant roles in (1.3), so we just consider λup, where up is the solution of −∆u + u p = in Ω \ { }. By scaling λ to meet the Kirchho function and then a solution with this type singularity is derived in Theorem 1.2. This scaling technique could be extended to obtain solutions in the supercritical case in Theorem 5.1 in Section 5.
It is worth pointing out that the method of searching solutions with the weak singularities as Φ in Theorem 1.1 could be extended into dealing with general nonlinearity f (u) when ≤ f (u) ≤ c|u| p with p ∈ ( , p * ). This method to prove Theorem 1.2 is based on the homogeneous property of nonlinearity and when the nonlinearity is not a power function, it is open but challenging to obtain solutions with such isolated singularity.
The rest of this paper is organized as follows. In Section 2, we introduce the very weak solution of equation (1.3) involving Dirac mass and give a discussion of (1.6). Section 3 is devoted to show the existence of a solution to (1.3) with M θ (u) > in Theorem 1.1. In Section 4, we search the solutions of (1.3) with M θ (u) < in Theorem 1.2. The supercritical case: N/(N − ) ≤ p < (N + )/(N − ) with N ≥ , is considered in Section 5, and we obtain there multiple isolated singular solutions of (1.3) such that M θ (u i ) > .

Preliminary . Kirchho -type problem with Dirac mass
In order to drive solutions of (1.3) with singularity (1.8), it is always transformed into nding solutions of (1.9). A function u is said to be a super (resp. sub) distributional solution of (1.9), if u ∈ L (Ω), |∇u| ∈ L(Ω), u p ∈ L (Ω, ρdx) and where ρ(x) = dist(x, ∂Ω). A function u is a distributional solution of (1.9) if u is both super and sub distributional solutions of of (1.9). Next we build the connection between the singular solutions of (1.3) and the distributional solutions of (1.9). Theorem 2.1. Assume that N ≥ , p > and u ∈ L (Ω) is a nonnegative classical solution of problem (1.3) satisfying that M θ (u) ≠ and u p ∈ L (Ω, ρdx). Then u is a very weak solution of problem (1.9) for some k ≥ . Furthermore, 3) only has zero solution and θ < .
(ii) Assume more that < p < p * . If k = , then u is removable at the origin, and if k > , then u satis es (1.8).
In order to prove Theorem 2.1, we need the following lemmas.
Proof. We follow the idea of Lemma 2.3 in [6]. In fact, from [2, Propsition 2.1] it follows that the Green kernel veri es that (2.8) Since u p ∈ L (Ω, ρdx) and u ∈ L (Ω), we may de ne the operator L by the following First we claim that for any ξ ∈ C ∞ c (Ω) with the support in Ω \ { }, In fact, since ξ ∈ C ∞ c (Ω) has the support in Ω \ { }, then there exists r ∈ ( , ) such that ξ = in Br( ) and then From Theorem 1.1 in [4], it implies that that is, Then u is a weak solution of (1.9) for some k ≥ . Case 1: M θ (u) < . We observe that Therefore, when p ≥ p * , there is no nontrivial nonnegative solution (1.3) such that M θ (u) < . Case 2: M θ (u) > . We refer to [17] for the proof. For the reader's convenience, we give the details. When p ∈ ( , N/(N − )) and k = , then We infer from u p ∈ L t (Ω) with t = ( + p N N− ) > and Proposition 2.1 that u ∈ L t p (Ω) and u p ∈ L t (Ω) with If t > Np/ , by Proposition 2.1, u ∈ L ∞ (Ω) and then it could be improved that u is a classical solution of −∆u = M θ (u) u p in Ω. (2.12) If t < Np/ , we proceed as above. By Proposition 2.1, u ∈ L t p (Ω), where Inductively, let us de ne Then there exists m ∈ N such that tm > Np and by part (i) in Proposition 2.1, u ∈ L ∞ (Ω).
It then follows that u is a classical solution of (2.12).
Then there exists n ≥ such that µ n − > and µ n − ≤ and where Γ i ≤ c|x| µ i and un ≤ p (G Ω [u p n ] + ).
Next, we claim that un ∈ L ∞ (Ω). Since un ∈ L s (Ω) for s ∈ [ , N/(N − )), letting Inductively, it implies by un ∈ L tn− (Ω) that un ∈ L tn (Ω) with Then there exists n ∈ N such that This ends the proof.

Proof.
When Ω = B ( ), we have that r = . Let Note that When p ≠ , h (k ) = implies that When p = , The rest of the proof is simple and hence we omit it.
When p = , note that is a critical value for (1.6) and we show that a < when Ω is a ball.

Proof.
When Ω = B ( ), take ξ (x) = − |x| as a test function, we derive Since w is radial symmetric and decreasing, then So for N = , Then w (r) w (r) = − r (ln r) + r ln r + −r − π ln r , We see that Therefore, we have that α < / . The proof is thus complete.

Solutions with M θ (u) >
In order to do estimates on M θ (u), we introduce the following lemma.
From (3.1), we have that This nishes the proof.
Proof of Theorem 1.1. We search for distributional solutions of by using the Schauder xed-point theorem. Let w , w be the solutions of (2.17) and denote w t = tk p w + kw , (3.3) where the parameter t > .
We claim that there exists kp > independent of θ such that for k ∈ ( , kp], if θ + r − k > there exists tp > such that We observe that if (ap tk p + k) p θ + r − k ≤ tk p , (3.5) then w t veri es (3.4), since Now we discuss what condition on k guarantee that (3.5) holds for some t > . In fact, (3.5) is equivalent to (ap tk p− + ) p ≤ t(θ + r − k) (3.6) or in the form In fact, (1.6) implies (3.7). Therefore, for k > r θ− satisfying (1.6) and taking tp = (θ + k) − p p− p , function w tp veri es (3.4). Let We claim that For u ∈ D k , we may let vn ∈ C (Ω) be a sequence of nonnegative functions converging to u in W , (Ω). Let un = vn +kw , and by the fact that w ∈ C (Ω\{ })∩W , (Ω), then un ∈ C (Ω\{ })∩W , (Ω), un ≥ kw in Ω \ { } and un converge to u + kw in W , (Ω). By the symmetric decreasing arrangement, we may denote u * n , the symmetric decreasing rearranged function of un in Br ( ), where r ≥ such that |Br ( )| = |Ω|. Observe that lim inf and Ω un dx ≥ k Ω w dx.
By Pólya-Szegő inequality, we have that Thus, Therefore, passing to the limit as n → +∞ in the above inequality we get that which implies (3.8). Therefore, from (3.4) it follows that Note that for u ∈ D k , one has that (u + kw ) p ∈ L σ (Ω) with σ ∈ ( , p N N− ), then TD k ⊂ W ,σ (Ω), where σ ∈ ( , p N N− ). Since the embeddings W ,σ (Ω) → W , (Ω), L (Ω) are compact and then T is a compact operator.
Observing that D k is a closed and convex set in L (Ω), we may apply the Schauder xed-point theorem to derive that there exists v k ∈ D k such that Tv k = v k .
Since ≤ v k ≤ tp k p w , so v k is locally bounded in Ω \ { }, then u k := v k + kw satis es (1.8), and by interior regularity results, u k is a positive classical solution of (1.3). From Theorem 2.1 we deduce that u k is a distributional solution of (1.9). has a unique positive weak solution u λ,k verifying that

Solutions with M θ (u) <
Furthermore, u λ,k is radially symmetric and decreasing with to |x| and the map λ → u λ,k is decreasing.
Proof. The existence could be seen [38, theorem 3.7] and uniqueness follows by Kato's inequality [38, theorem 2.4]. The radial symmetry of u λ,k and decreasing monotonicity with to |x| could be derived by the method of moving plane, see [12,35] for the details. It follows from Kato's inequality that the map λ → u λ,k is decreasing. This ends the proof.
So it follows by Lemma 3.1 that We claim that the map λ ∈ [λ , λ ] → M θ (u λ,k ) is continuous.
At this moment, we assume that the above argument is true. Let where v λ is the solution of (4.2) with λ ∈ [λ , λ ]. Since F is continuous in [λ , λ ], by (4.4), (4.5) and the mean value theorem, there exists λ ∈ (λ , λ ) such that F(λ ) = , that is, (4.1) has a solution u k with −M θ (u k ) = λ . From standard regularity, we have that u k is a classical solution of (1.3) and veri es the corresponding properties in the lemma. Now we prove that the map λ ∈ [λ , λ ] → M θ (u λ,k ) is continuous. Let λ ≤ λ < λ ≤ λ and u λ ,k and u λ ,k be the solutions of (4.1) with λ = λ and λ = λ respectively. Then u λ ,k < u λ ,k and M θ (u λ ,k ) < M θ (u λ ,k ). (4.6) Therefore Kato's inequality implies that which yields that This together with (4.6), give (ii) It is well known that for p ∈ ( , p * ), the problem has a positive solution vp verifying that in the set of functions satisfying (4.9). For p ∈ ( N+ N− , p * ), we have that Ω |∇up|dx < +∞, so that where m = Ω |∇up|dx and λ = (m /(−θ)) p− . We de ne Observe that F is continuous, increasing and Hence there exists a uniqueλ such that Meaning that −M − θ (vλ) =λ. We then conclude that (1.3) has a solution up := vλ with M θ (up) < . From (4.8) and the de nition of v λ , we know that up is not a weak solution of problem (1.9).
In the above three cases, there exists a uniqueλ i such that Remark 5.1. Our method to prove Theorem 5.1 is based on the homogeneous property of the nonlinearity. When the nonlinearity is not a power function, this scaling method fails and so it is challenging to provide the existence results of isolated singular solutions.