Search

In this paper we are concerned with the Lane-Emden-Fowler equation... where Ω ⊂ Rn (n ≥ 3) is a nonconvex polygonal domain and ε > 0. We study the asymptotic behavior of minimal energy solutions as ε > 0 goes to zero. A main part is to show that the solution is uniformly bounded near the boundary with respect to ε > 0. The moving plane method is difficult to apply for the nonconvex polygonal domain. To get around this difficulty, we derive a contradiction after assuming that the solution blows up near the boundary by using the Pohozaev identity and the Green's function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Neumann problem (P ε) : − Δ u + V (x) u = K (x) u (n + 2) / (n − 2) − ε , u > 0 in Ω , ∂ u / ∂ ν = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n ≥ 4 , ε is a small positive real, and V and K are non-constant smooth positive functions on Ω ¯ . First, we study the asymptotic behavior of solutions for (P ε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K ≡ 1 , problem (P ε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (P ε) which allow us to provide multiplicity results for (P ε) . The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a singular super-critical Trudinger-Moser inequality on a unit ball of Rn, n ≥ 3. For any p > 1, we set λp(B) = inf u∈W0 1,n (B),u̸≡0 ∫ B |∇u| n dx/ ( ∫ B |u| pdx) n/pas an eigenvalue related to the n-Laplacian. Let S be a set of radially symmetric functions. Precisely, if β ≥ 0 and α < (1 + p/n β)n−1+n/pλp(B), then there exists a positive constant ϵ0 such that when λ ≤ 1 + ϵ0,supu∈W 01,n (B)∩S, ∫ B|∇u|ndx−α( ∫ B |u|p|x|pβdx)n/p≤1∫ B |x| pβ(e αn(1+ p/n β)|u| n/n−1− λ ∑k=0 m |αn(1 + p/n β)u n/n−1 |k . The proof is based on the method of blow-up analysis. The case λ = 0 was studied by Yang-Zhu in [38]. de Figueiredo [11] considered the case p = 2, β ≥ 0, and α = 0 in two dimension. The case λ = 0, p = n, −1 < β < 0, and α = 0 was considered by Adimurthi-Sandeep [1]. Our results extend those of the above cases. [ABSTRACT FROM AUTHOR]1/(n−1)n−1, ωn−1 is the surface area of the unit ball in Rn . The proof is based on the method of blow-up analysis. The case λ = 0 was studied by Yang-Zhu in [38]. de Figueiredo [11] considered the case p = 2, β ≥ 0, and α = 0 in two dimension. The case λ = 0, p = n, −1 < β < 0, and α = 0 was considered by Adimurthi-Sandeep [1]. Our results extend those of the above cases. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Moser-Trudinger type inequality with a special degenerate potential V. Specifically, we demonstrate that for any potentials V (x) satisfying the conditions (V 1) and (V 2) , the following inequality MT (V , α , p) ≔ sup u ∈ E , ‖ u ‖ ≤ 1 ⁡ ∫ R n Φ (α n (1 + α ‖ u ‖ p n) 1 n − 1 u n n − 1 ) d x < ∞ holds if α < λ 1 (V) ; where E is the usual weighted Sobolev space associated to the potential V , the norm ‖ u ‖ n : = ∫ R n | ∇ u | n + V (x) | u | n d x , and λ 1 (V) is the first Dirichlet eigenvalue of the corresponding operator − Δ + V. The potential conditions are as follows. (V 1) 0 ≤ m = inf x ∈ R n ⁡ V (x) < sup x ∈ R n ⁡ V (x) = lim | x | → ∞ ⁡ V (x) = M ≤ ∞. (V 2) V is radially symmetric, non-decreasing. Besides, we also prove that for any potentials V (x) satisfying the condition (V 1) merely and for p = n , the previous inequality is still valid. In addition, this inequality is sharp in the sense that if p > n and α ≥ λ 1 (V) , there holds MT (V , α , p) = ∞. Meanwhile, if α > λ 1 (V) , then MT (V , α , n) = ∞. Furthermore, via a subtle blow-up analysis, we also prove the existence of an extremal function for the inequality above when α is sufficiently small. [ABSTRACT FROM AUTHOR]

We consider the Brezis-Nirenberg problem: -Δu = |u|p-1u ± εu in Ω, with u = 0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n ≤ 4, p + 1 = 2n/(n - 2) is the critical Sobolev exponent, and ε > 0 is a positive parameter. The main result of this paper shows that if n ≤ 4 there are no sign-changing solutions uε of (P-ε) with two positive and one negative blow up points. [ABSTRACT FROM AUTHOR]

Let B be the unit disc in R², H be the completion of C∞0 (B) under the norm ... We prove that the supremum in the following inequality ... can not be achieved by any functions in the function space H when α is sufficiently close to λ-1, i.e., 0 < λ1 - α ... 1, where ... Evidently, this conclusion is complementary to that of [12, Theorem 1.1 (ii)]. [ABSTRACT FROM AUTHOR]

In this paper, we establish a sharp Trudinger-Moser type inequality for a class of Schrödinger operators in ℝ². We obtain a result related to the compactness of the embedding of a subspace of W1,2(ℝ²) into the Orlicz space Lf(ℝ²) determined by f(t) = eßt² -1. Our result is similar to one obtained by Adimurthi and Druet for smooth bounded domains in ℝ², which is closely related to a compactness result proved by Lions. Furthermore, similarly to what has been done by Carleson and Chang, we prove the existence of an extremal function for this Trudinger-Moser inequality by performing a blow-up analysis. Trudinger-Moser type inequalities have a wide variety of applications to the study of nonlinear elliptic partial differential equations involving the limiting case of Sobolev inequalities and have received considerable attention in recent years. [ABSTRACT FROM AUTHOR]

The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves L n norm. Let λ 1 (Ω) = inf u ∈ W 0 1 , n (Ω) , u ≢ 0 ⁡ | | F (∇ u) | | L n (Ω) n / | | u | | L n (Ω) n be the first eigenvalue associated with n -Finsler-Laplacian. Using blow-up analysis, we obtain that sup u ∈ W 0 1 , n (Ω) , | | F (∇ u) | | L n (Ω) = 1 ⁡ ∫ Ω e λ n (1 + α | | u | | L n (Ω) n) 1 n − 1 | u | n n − 1 d x is finite for any 0 ≤ α < λ 1 (Ω) , and the supremum is infinite for any α ≥ λ 1 (Ω) , where λ n = n n n − 1 κ n 1 n − 1 (κ n is the volume of the unit wulff ball) and the function F is positive, convex and homogeneous of degree 1, and its polar F o represents a Finsler metric on R n. Furthermore, the supremum is attained for any 0 ≤ α < λ 1 (Ω). [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Abstract In this paper, we provide a new regularity criterion of smooth solutions via the one component of the velocity field in various scaling invariant spaces with a natural growth condition of the L ∞ norm near a possible blow-up time. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to the problem of existence of conformal metrics with prescribed Q -curvature on closed Riemannian manifolds of even dimension n ≥ 4 , when the kernel of the associated GJMS operator is trivial and the total integral of the corresponding Q -curvature is a positive integer multiple of the one of the n -dimensional round sphere. Indeed, exploiting the variational structure of the problem, we develop a full Morse theory and algebraic topological arguments for existence for this non-compact geometric variational problem of high order, extending the works Ahmedou and Ndiaye (2019) and Ndiaye (2015) to arbitrary even dimensions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution. [ABSTRACT FROM AUTHOR]

This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ⁡ ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ⁡ ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]

In a previous work (Adimurthi and Yang, 2010 [2] ), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space R N ( N ≥ 2 ) . Precisely, if 0 ≤ β < 1 and 0 < γ ≤ 1 − β , then there holds for any τ > 0 , sup u ∈ W 1 , N ( R N ) , ∫ R N ( | ∇ u | N + τ | u | N ) d x ≤ 1 ⁡ ∫ R N 1 | x | N β ( e α N γ | u | N N − 1 − ∑ k = 0 N − 2 α N k γ k | u | k N N − 1 k ! ) d x < ∞ , where α N = N ω N − 1 1 / ( N − 1 ) and ω N − 1 is the area of the unit sphere in R N . The above inequality is sharp in the sense that if γ > 1 − β , all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β = 0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11] . We shall investigate the singular case 0 < β < 1 and prove that for all τ > 0 , 0 < β < 1 and 0 < γ ≤ 1 − β , extremal functions for the above inequalities exist. The proof is based on blow-up analysis. [ABSTRACT FROM AUTHOR]

Let ( X n + 1 , g + ) be an ( n + 1 ) -dimensional asymptotically hyperbolic manifold with conformal infinity ( M n , [ h ˆ ] ) . The fractional Yamabe problem addresses to solve P γ [ g + , h ˆ ] ( u ) = c u n + 2 γ n − 2 γ , u > 0 on M where c ∈ R and P γ [ g + , h ˆ ] is the fractional conformal Laplacian whose principal symbol is the Laplace–Beltrami operator ( − Δ ) γ on M . In this paper, we construct a metric on the half space X = R + n + 1 , which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that n ≥ 24 for γ ∈ ( 0 , γ ⁎ ) and n ≥ 25 for γ ∈ [ γ ⁎ , 1 ) where γ ⁎ ∈ ( 0 , 1 ) is a certain transition exponent. The value of γ ⁎ turns out to be approximately 0.940197. [ABSTRACT FROM AUTHOR]

We consider the Brezis–Niremberg problem: ( P ε ) { − △ u = | u | p − 1 u + ε u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n = 4 , 5 , 6 , p + 1 = 2 n n − 2 is the critical Sobolev exponent and ε is a positive parameter. The main result of the paper generalizes the result of A. Iacopetti and F. Pacella [10] . Precisely we show that there are no low energy sign-changing solutions u ε with max ⁡ u ε / min ⁡ u ε → 0 or −∞ as ε goes to zero. [ABSTRACT FROM AUTHOR]

In this paper, we derive a sharp version of the singular Trudinger–Moser inequality, which was originally established by Adimurthi and Sandeep [2] . Moreover, extremal functions for those singular Trudinger–Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work ( [32] ), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behavior of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li [6,7] to get the desired bubble. [ABSTRACT FROM AUTHOR]

In this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the “uniformization” of the Riemann sphere with conical singularities. [ABSTRACT FROM AUTHOR]

Let W1,n(ℝn) be the standard Sobolev space. For any τ > 0, 0 < β < 1, Li and Yang [16] proved the existence of extremals for a singular Trudinger-Moser inequality. Namely, the supremum ... can be attained by some function uβ ∈ W1,n(ℝn) with ... (|∇u|n+ τ|u|n)dx=1. Here Φ(n,t)=et ...!, and αn=nωn-11/(n-1) with ωn-1 being the surface area of the (n--1) -dimensional unit sphere. In this note, we consider the compactness of the function family {uβ}0<β<1 and prove that up to a subsequence, uβ converges to some function u0 in C¹(ℝn) when β→ 0. Moreover, u0 is an extremal function of the supremum ... Let us explain the result in geometry. Denote ω0(x) = ∑j=1ndxj² and ωβ(x) = |x|-2β ω0(x) as the standard and conical metrics on ℝn. Then the extremal family {uβ}0<β<1 of the following singular Trudinger-Moser functionals ... is compactness as β→0. This extends earlier result of Wang and Yang [33] and complements that of Li and Yang [16]. [ABSTRACT FROM AUTHOR]

Combining Carleson–Chang's result [9] with blow-up analysis, we prove existence of extremal functions for certain Trudinger–Moser inequalities in dimension two. This kind of inequality was originally proposed by Adimurthi and O. Druet [1] , extended by the author to high dimensional case and Riemannian surface case [40,41] , generalized by C. Tintarev to wider cases including singular form [36] and by M. de Souza and J.M. do Ó [14] to the whole Euclidean space R 2 . In addition to the Euclidean case, we also consider the Riemannian surface case. The results in the current paper complement that of L. Carleson and A. Chang [9] , M. Struwe [35] , M. Flucher [16] , K. Lin [19] , and Adimurthi and O. Druet [1] , our previous ones [26,41] , and part of C. Tintarev [36] . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish a blow-up analysis for the Finsler–Liouville equation in two dimensions. Here the operator Q is the so-called Finsler–Laplacian, which is a generalized Laplacian. As a byproduct, we obtain an existence result. [Copyright &y& Elsevier]

Abstract: In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball. [Copyright &y& Elsevier]

Abstract: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball. [Copyright &y& Elsevier]

Abstract: In this paper we study low energy sign changing solutions of the critical exponent problem in Ω, on ∂Ω, where Ω is a smooth bounded domain in and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions. [Copyright &y& Elsevier]

In this paper, we perform a fine blow-up analysis for a boundary value elliptic equation involving the critical trace Sobolev exponent related to the conformal deformation of the metrics on the standard ball, namely the problem of prescribing the boundary mean curvature. From this analysis some a priori estimates in low dimension are obtained. With these estimates, we prove the existence of at least one solution when an index-counting formula associated to the prescribed mean curvature is different from zero. [Copyright &y& Elsevier]

Assume that (X,g+) is an asymptotically hyperbolic manifold, (M,[h¯]) is its conformal infinity, ρ is the geodesic boundary defining function associated to h¯ and g¯=ρ²g+. For any γ∈(0,1), we prove that the solution set of the γ-Yamabe problem on M is compact in C²(M) provided that convergence of the scalar curvature R[g+] of (X,g+) to -n(n+1) is sufficiently fast as ρ tends to 0 and the second fundamental form on M never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant to the geometric assumption imposed on X, our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem. [ABSTRACT FROM AUTHOR]

Let (Σ g) be a compact Riemannian surface without boundary and W1;2(Σ) be the usual Sobolev space. For any real number p > 1 and α ∊ R, we define a functional... has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu (DOI: 10.1007/s11425-017-9086-6). [ABSTRACT FROM AUTHOR]