p -Laplacian Equations in R N + with Critical Boundary Nonlinearity

: In this paper, we consider the following p -Laplacian equation in R N + with critical boundary nonlinearity. The existence of inﬁnitely many solutions of the equation is proved via the truncation method.


Introduction
In this paper, we consider the following p-Laplacian equation in R N + with critical boundary nonlinearity    − ∆ p u = 0, in R N + , |∇u| p−2 ∂u ∂n + |u| p−2 u = |u| p−2 u + µ|u| q−2 u, on R N−1 = ∂R N + , where 1 < p < N, max{p, p − 1} < q < p = (N−1)p N−p , µ > 0 and ∆ p is the p-Laplacian operator, ∆ p u = div(|∇u| p−2 ∇u). We are looking for axial solutions of the Equation (1) that are solutions of the form u(x) = u(|y|, s), where we denote x ∈ R N + by x = (y, s) ∈ R N−1 × [0, ∞) and we identify R N−1 = ∂R N + , y = (y, 0) for y ∈ R N−1 if there is no confusion. Introduce in C ∞ 0 (R N + ) a norm by Let W be the completion of C ∞ 0 (R N + ) with respect to this norm and W r be the subspace of W of axial functions, that is, W r = {u | u ∈ W, u(x) = u(|y|, s), x = (y, s) ∈ R N + } .
The problem (1) has a variational structure given by the functional Notice that p = (N−1)p N−p is the critical exponent for the Sobolev imbedding from W 1,p (R N + ) to L q (R N−1 ), p ≤ q ≤ p. Moreover, the imbedding from W r to L q (R N−1 ) is continuous for p ≤ q ≤ p and compact only for p < q < p due to the dilations. Therefore, the Palais-Smale condition is not satisfied by the functional I and the problem (1) lacks the necessary compactness property. Since the pioneering work of Brezis and Nirenberg [1], significant progress has been made in recent decades for these kinds of problems lacking compactness. In particular, the authors of [2] dealt with the Laplacian equation with critical growth in the bounded domain − ∆u = |u| 2 * −2 u + λu, in Ω, where Ω ⊂ R N , N ≥ 3 is a regular bounded domain, and 2 * = 2N N−2 . While the authors of [3] considered the Laplacian equation with subcritical nonlinear term in the whole space R N − ∆u + a(x)u = |u| q−2 u, in R N , where 2 < q < 2 * and a(x) is the potential function. As to the p-Laplacian equation, there is a lot of significant work, whether in the field of ordinary differential equations [4][5][6] or partial differential equations [7][8][9], the authors of [7] considered − ∆ p u + a(x)u = |u| p * −2 u + λu, in Ω, where p * = N p N−p . All of these authors found the solutions as limits of approximated equations with subcritical growth in bounded domains. The lack of compactness due to dilations (in the case (2) and (4)) and shifts (in the case (3)) does not allow for deducing that a sequence of approximate solutions must have a convergent subsequence, but the fact that they solve the approximated problems gives, with use of a local Pohožaev identity, some extra estimates which lead to a proof of desired compactness.
In the Existing literature, some researchers considered the existence of finite multiple solutions [10,11]. While the subcritical problems in bounded domains have infinitely many solutions. In order to show the existence of multiple solutions of the original problems, we need to check that multiple solutions of approximated problems do not converge to the same solution of the limit problems. This is hard work. In both [2,3], some estimates on the Morse index are employed, which has been used as one of the possible devices to distinguish the limit of the multiple approximate solutions by their original variational characterization. For general p-Laplacian equations, we have no information on the Morse index; therefore, the approach in this last step in [2,3] can not be extended in a straightforward way to problems involving the p-Laplacian operator. Here, we will use the truncation method, as we did in [8,9]. First, we consider some truncated problems, the solutions of which will be used as approximate solutions. By a concentration-compactness analysis, similar to that in [2,3,7], in particular with use of a local Pohožaev identity, the theorem of convergence of approximate solutions is proved. We show that, by a careful choice of the approximate nonlinear terms, the approximated problems and the original problem share more and more solutions, as the approximation parameter tends to zero. For more references, we refer the readers to [12][13][14][15][16][17][18].
Let us describe the truncation method in more details. Let ψ ∈ C ∞ 0 (R, [0, 1]) be an even function such that ψ(t) = 1 for |t| ≤ 1, ψ(t) = 0 for |t| ≥ 2 and ψ is decreasing in [1,2]. Define the auxiliary functions for λ ∈ (0, 1], Instead of the original problem (1), we consider the truncated problem In addition, the problem (6) has a variational structure given by the functional Notice that the functional I λ , λ > 0 is subcritical at the infinity and the imbedding from W r to L q (R N−1 ), p < q < p is compact. Therefore, the functional I λ , λ > 0 satisfies the Palais-Smale condition.
Here are our main results.
Consequently, if λ < ν, then u is a solution of the problem (1).
Throughout the paper, we use the following notations: we use · and | · | q to denote the norms of W and L q (R N−1 ), respectively, and → to denote the weak and the strong convergence, respectively. In addition, we use the notations . The paper is organized as follows. In Section 2, we do the concentration-compactness analysis of the approximate solution sequence and prove the convergence Theorem 1. In Section 3, we construct a sequence of critical values of the truncated functionals by the symmetric mountain pass lemma. Finally, we prove the existence Theorem 2 by showing that approximated solutions are also solutions of the original problem for a sufficiently small parameter.

The Profile Decomposition
In this section, we analyze the concentration behavior for the solutions of the problem (6) as λ → 0 and prove Theorem 1 . First, we list the properties of the auxiliary functions, defined in (5) in the following lemma.
Proof. The proof is straightforward. We verify (e)-(g). By the definition of F λ and f λ , we have ( f ) and (g) follow from (7), and (e) follows from (7) and (a), (d) of this lemma.

Lemma 2.
Let λ n ≥ 0, u n ∈ W r such that DI λ n (u n ) = 0, I λ n (u n ) ≤ L. Then, {u n } is bounded in W r .
Proof. By Lemma 1 ( f ), we have Hence, {u n } is bounded in W r .
and D r be the subspace of D of axial functions, Let D be the dilation group Notice that the operator g σ of D is an isometry in both D and L p (R N−1 ). The imbedding from D r to L p (R N−1 ) is compact with respect to the group D that is a sequence {u n } of D r , satisfying g σ n u n 0 in D r for any sequence {g σ n } of D, denoted by u n D 0 in D r , must converge to zero in L p (R N−1 ) . Now, let u n be a bounded sequence of W r . By [19,20], we have the following profile decomposition: where u ∈ W r , U k ∈ D r , r n ∈ D r , σ n,k ∈ (0, ∞) and Λ is an index set, satisfying (d) r n 0 in D r as n → ∞, consequently r n → 0 in L p (R N−1 ) as n → ∞ .
We refer to [19,20] for general concepts of compactness and the profile decomposition and relevant results. For reader's convenience, we consider the compactness of the imbedding from D r to L p (R N−1 ) with respect to the dilation group D. Lemma 3. Assume λ n > 0, λ n → 0 as n → ∞, u n ∈ W r , DI λ n (u n ) = 0 and I λ n (u n ) ≤ L . Assume that the profile decomposition (9) holds. Then, for ϕ ≥ 0, ϕ ∈ W r . Consequently, for some c > 1, Proof. We prove the conclusion for the function V k . u n satisfies the equation in the weak form Denote u n = g −1 σ n,k u n . For ϕ ∈ W r , take g σ n,k ϕ as a test function in (13). By a variable change, we obtain where λ n = λ n σ N−p p n,k . In the above, we have used the fact that which can be proved by the very definition of the function f λ . Since R N−1 | u n | p dy = R N−1 |u n | p dy is bounded and u n is axial, for any where S p is the Sobolev constant of the imbedding D → L p (R N−1 ). By Lemma A4, u n is uniformly bounded in D 2δ (y). Consequently, by Equation (14) and the following elementary inequality (15), u n converges in W 1,p (B δ (y)) and in W 1,p loc (R N + ). The following inequality (15) is useful for problems involving the p-Laplacian operator [21]. There exists a constant c p such that, for ξ, η ∈ R N , Let v n = | u n |, v n converge to V k = |U k | in W 1,p loc (R N + ) and satisfy the inequality By a density argument, (17) holds for ϕ ∈ D, ϕ ≥ 0 .

Lemma 4.
The index set Λ in the profile decomposition (9) is finite.
Proof. By Lemma 3, V k satisfies the inequality (12). Choose ϕ = V k in (12). By the Sobolev imbedding theorem By the property (3) of the decomposition (9), Λ is a finite set.

Safe Regions
Assume the profile decomposition (9) with a finite index set Λ. Denote and define the so-called safe regions [2] For these regions, we have a good estimate.

Proposition 1.
There exists a constant c, independent of n, such that

Corollary 1.
There exists a constant c, independent of n, such that In order to prove these estimates, we start with the following definition.
Proof. By Lemma A6 for γ < 1, we have By Proposition 2, we have |u n | p 1 ,p 2 ,σ n ≤ c for any p 1 , p 2 such that 1 − 1 By Lemma A4 and Lemma 5, Hence, We complete the proof of Proposition 1. To prove Corollary 1, we choose a function ϕ ∈ C ∞ 0 (R N ) such that ϕ(x) = 1 for x ∈ A 3 n ∪ T 3 n and ϕ(x) = 0 for x ∈ A 2 n ∪ T 3 n and |∇ϕ| ≤ 2σ 1 p n . Testing the Equation (13) by ϕ p u n , we obtain Hence,

Pohožaev Identity
In the remainder of this section, following the idea of [2,3], we apply the local Pohožaev identity to prove the convergence Theorem 1.
Let ϕ ∈ C ∞ 0 (R N ), then Proof. Multiplying (34) by (x, ∇u)ϕ and integration by parts, we obtain Multiplying (34) by uϕ and integration by parts, we obtain Eliminating the term R N + |∇u| p ϕ dx, we obtain the local Pohožaev identity.
Proof of the convergence Theorem 1. We apply the local Pohožaev identity to the function u n . Let Choose ϕ ∈ C ∞ 0 (R N , [0, 1]) such that ϕ(x) = 1 for |x| ≤ 3σ We estimate (38). Notice that the integrals of the right-hand side of (38) are taken over the domains A 3 n and T 3 n . By Proposition 1 and Corollary 1, we know RHS of (38) ≤ c On the other hand, by Lemma 1 (7), we have Without loss of generality, assume σ n,1 = σ n = min{σ n,k | k ∈ Λ}. Choose L large enough such that we arrive at a contradiction for σ n large enough, since q + 1 > p = (N−1)p N−p . The index set Λ in the profile decomposition (9) must be empty, and (9) reduces to u n = u + r n , and r n → 0 in L p (R N−1 ) as n → ∞ .
That is, u n → u in L p (R N−1 ). By Lemma A4, u n is uniformly bounded, and there exists ν = ν(L) such that |u n (y)| ≤ 1 ν for y ∈ R N−1 .

Existence of Multiple Solutions
We define a sequence of critical values of the truncated functional I λ , λ > 0 by the symmetric mountain pass lemma due to Ambrosetti and Rabinowitz, and prove that the corresponding critical points are solutions of the original problem (1) for sufficiently small parameter λ .

Definition 2. Define the critical values of I
ρ is chosen as a suitable positive constant such that In fact, for u ∈ W r , u = ρ, we have
Proof. Let u n be a Palais-Smale sequence of I λ , and we have hence u n is bounded in W r . Since the imbedding from W r to L q (R N−1 ) is compact, we assume u n u in W r , u n → u in L q (R N−1 ). By Lemma 1, we have By the elementary inequalities (15), u n is a Cauchy sequence.
For more details and background material, we refer the readers to the Appendices A-C of this paper.

Results
The main results of this paper are Theorem 1 and Theorem 2.
Then, there exists a constant c = c(p, q) such that
Then, there exists a unique function u ∈ D satisfying the equation Proof. Consider the functional J defined on D J is lower semi-continuous and bounded from below. Therefore, J assumes its infimum at a function u ∈ D, which solves the equation. By the elementary inequalities (15), the solution is unique.
Step 2. Assume 0 < r < R ≤ R 0 < 1. Then, there exists c R 0 > 0 such that Taking ϕ as a test function in (a 15 ), we have and In the above, we have used p < p * d. Assume d , x j = χ j , r j = r + 1 2 j−1 (R − r), j = 1, 2, · · · . By Moser's iteration, for some t > 0, we obtain Step 3. By (A26), there exists t , c such that By iteration, we obtain In particular We also have inner estimate Lemma A5. Let u ≥ 0, u ∈ D and satisfy Then, for any γ > 0, there exists c = c(p, γ) such that