Singular Trudinger Moser Inequality Research Articles

Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Abstract The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ , $$\underset{u\in {H}^{1}({\mathbb{R}}^{2}),\underset{{\mathbb{R}}^{2}}{\int }(\vert \nabla u{\vert }^{2}+V(x){u}^{2})\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\tfrac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x< +\infty ,$$ where 0 < β < 2 $0< \beta < 2$ , V ( x ) ≥ 0 $V(x)\ge 0$ and may vanish on an open set in R 2 ${\mathbb{R}}^{2}$ . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 ${\mathbb{R}}^{2}$ : (0.2) − Δ u + γ u = f ( u ) | x | β , $${-}{\Delta }u+\gamma u=\frac{f(u)}{\vert x{\vert }^{\beta }},$$ where the nonlinearity f $f$ has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${\mathbb{R}}^{n}$ and n $n$ -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 ${\mathbb{R}}^{2}$ : (0.3) − Δ u + V ( x ) u = f ( u ) | x | β . $${-}{\Delta }u+V(x)u=\frac{f(u)}{\vert x{\vert }^{\beta }}.$$

Sharp anisotropic singular Trudinger–Moser inequalities in the entire space

Sharp anisotropic singular Trudinger–Moser inequalities in the entire space

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

Solutions for nonhomogeneous singular fractional p-Laplacian equations via fixed point theorem

In this paper, we study the following fractional p-Laplacian equations with exponential growth and Hardy term: where the nonlinear term satisfies exponential growth, , , λ is a positive parameter, is the fractional p-Laplacian, is a scalar potential, the weight function either falls into some Lebesgue space or is a convolution function, and g is a perturbation term. With the help of a fixed point result and singular Trudinger–Moser inequality in , we obtain the existence of a nontrivial weak solution for the above equations.

Singular Trudinger–Moser inequality with Lp norm and the existence of extremals

In this paper, we establish the singular Trudinger–Moser inequality with norm in a bounded domain and the existence of extremal functions by blow-up analysis. As our first main result, we prove that for , , and , the singular Trudinger–Moser inequality holds, where is the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality with norm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.

A singular Trudinger–Moser inequality on compact Riemann surface with smooth boundary

Let be a compact Riemann surface with smooth boundary. In this paper, using blow-up analysis, we prove the existence of an extremal function for a singular Trudinger–Moser inequality on .

Concentration-compactness principle of singular Trudinger–Moser inequality involving N-Finsler–Laplacian operator

In this paper, suppose [Formula: see text] be a convex function of class [Formula: see text] which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger–Moser Inequalities involving [Formula: see text]-Finsler–Laplacian operator. Let [Formula: see text] be a smooth bounded domain. [Formula: see text] be a sequence such that anisotropic Dirichlet norm[Formula: see text], [Formula: see text] weakly in [Formula: see text]. Denote [Formula: see text] Then we have [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the volume of a unit Wulff ball. This conclusion fails if [Formula: see text]. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space [Formula: see text].

Singular Trudinger–Moser inequality and fractional [formula omitted]-Laplace equations in [formula omitted

In this paper, we establish a singular Trudinger–Moser inequality in RN. Applying that result, we study the existence of solution to N∕s-fractional Laplacian equation LN∕ssu(x)+V(x)|u|Ns−2u=f(x,u)|x|γ,where γ∈[0,N), 0<s<1,LN∕ss is a nonlocal fractional operator with singular kernel K:RN∖{0}→R+,f is a continuous function on RN×R which does not satisfy the Ambrosetti–Rabinowitz condition. By using Mountain Pass Theorem, we obtain the existence of solution to above equation. Furthermore, when f satisfies the Ambrosetti–Rabinowitz condition, we obtain the existence to solution of that equation with negative energy. In our best knowledge, it is the first time the existence to solution of above equation is studied.

Sharp Singular Trudinger–Moser Inequalities Under Different Norms

Abstract The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝ N {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces D N , q ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})} , q ≥ 1 {q\geq 1} , the completion of C 0 ∞ ⁢ ( ℝ N ) {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥ ∇ ⁡ u ∥ N + ∥ u ∥ q {\|\nabla u\|_{N}+\|u\|_{q}} . The case q = N {q=N} (i.e., D N , q ⁢ ( ℝ N ) = W 1 , N ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})} ) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type ∥ ∇ ⁡ u ∥ N ≤ 1 {\|\nabla u\|_{N}\leq 1} and full-norm type ∥ ∇ ⁡ u ∥ N a + ∥ u ∥ q b ≤ 1 {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1} , a &gt; 0 {a&gt;0} , b &gt; 0 {b&gt;0} . We will show that the Trudinger–Moser-type inequalities hold if and only if b ≤ N {b\leq N} . Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when b &gt; N {b&gt;N} .

On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem

In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

AbstractThe main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in . More precisely, let and denote urn:x-wiley:0025584X:media:mana201600528:mana201600528-math-0006then we will prove in this article that there exists such that can be achieved for all , .

An improved singular Trudinger–Moser inequality in [formula omitted] and its extremal functions

Let N≥2, W1,N(RN) be the standard Sobolev space. For any real numbers τ>0 and p≥N, we denoteλN,p,τ=infu∈W1,N(RN),u≢0⁡∫RN(|∇u|N+τ|u|N)dx(∫RN|u|pdx)N/p and define a norm in W1,N(RN) by‖u‖N,p,α,τ=(∫RN(|∇u|N+τ|u|N)dx−α(∫RN|u|pdx)N/p)1/N for any fixed α, 0≤α<λN,p,τ. If τ>0, 0<β<1, 0<α<λN,p,τ, p≥N≥2, and 0≤γ≤αN, then using blow-up analysis, we prove that the supremumsupu∈W1,N(RN),‖u‖N,p,α,τ≤1⁡∫RN1|x|Nβ(eγ(1−β)|u|NN−1−∑k=0N−2γk(1−β)k|u|kNN−1k!)dx can be attained by some function u∈W1,N(RN) with ‖u‖N,p,α,τ≤1, where αN=NωN−11/(N−1) and ωN−1 is the area of the unit sphere in RN. This improves an inequality of Adimurthi and Yang and the result I obtained recently with Yang.

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space RN(N≥2). Precisely, if 0≤β<1 and 0<γ≤1−β, then there holds for any τ>0,supu∈W1,N(RN),∫RN(|∇u|N+τ|u|N)dx≤1⁡∫RN1|x|Nβ(eαNγ|u|NN−1−∑k=0N−2αNkγk|u|kNN−1k!)dx<∞, where αN=NωN−11/(N−1) and ωN−1 is the area of the unit sphere in RN. 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.

Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two

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.

An improved singular Trudinger–Moser inequality in unit ball

Let B⊂Rn(n≥2) be the unit ball centered at the origin with radius 1. Let β, 0≤β<n, be fixed. Defineλβ(B)=infu∈W01,n(B),u≢0⁡∫B|∇u|ndx∫B|x|−β|u|ndx. Suppose that γ satisfies γαn+βn=1, where αn=nωn−11/(n−1), ωn−1 is the area of the unit sphere in Rn. Using rearrangement argument, we prove that for any α, 0≤α<λβ(B), there holdssupu∈W01,n(B),∫B|∇u|ndx≤1⁡∫B|x|−βeγ|u|nn−1(1+α∫B|x|−β|u|ndx)1n−1dx<+∞. Moreover, we prove that the above supremum is infinity for α≥λβ(B). This improves earlier results of Yang [15] and Adimurthi and Sandeep [2] in the unit ball.

On a singular and nonhomogeneous N-Laplacian equation involving critical growth

In this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form − Δ N u = f ( x , u ) | x | a + h ( x ) in Ω , where u ∈ W 0 1 , N ( Ω ) , Δ N u = div ( | ∇ u | N − 2 ∇ u ) is the N-Laplacian, a ∈ [ 0 , N ) , Ω is a smooth bounded domain in R N ( N ⩾ 2 ) containing the origin and h ∈ ( W 0 1 , N ( Ω ) ) ⁎ = W − 1 , N ′ is a small perturbation, h ≢ 0 . Motivated by a singular Trudinger–Moser inequality, we study the case when f ( x , s ) has the maximal growth on s which allows to treat this problem variationally in the Sobolev space W 0 1 , N ( Ω ) .