Trudinger Moser Nonlinearities Research Articles

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

On a Schrödinger equation involving fractional [formula omitted]-Laplacian with critical growth and Trudinger–Moser nonlinearity

A nonlinear Schrödinger equation of fractional (N/s1,q)-Laplacian is considered with the Rabinowitz potential, critical Sobolev growth and Trudinger–Moser nonlinearity in RN−ΔN/s1s1u+−Δqs2u+V(ɛx)(uNs1−2u+uq−2u)=λfu+uqs2∗−2u.We establish the global existence of nonnegative ground-state solution for suitable parameter values primarily through variational analysis, fractional Trudinger–Moser inequality and mountain pass approach. It is a crucial ingredient to handle three aspects concerning the limiting setting s1p=N, the critical Sobolev growth and Trudinger–Moser nonlinearity.

Blow Up for Klein–Gordon Equation with Nonlocal Operator and Trudinger–Moser Nonlinearity

Blow Up for Klein–Gordon Equation with Nonlocal Operator and Trudinger–Moser Nonlinearity

Concentration of solutions for ( N , q ) -Laplacian equation with Trudinger–Moser nonlinearity

In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity: { − Δ N u − Δ q u + V ( ε x ) ( | u | N − 2 u + | u | q − 2 u ) = f ( u ) , x ∈ R N , u ∈ W 1 , N ( R N ) ∩ W 1 , q ( R N ) , x ∈ R N , where V is a positive continuous function and has a local minimum, ε > 0 is a small parameter, 2 ≤ N < q < + ∞ , f is C 1 with subcritical growth. When V and f satisfy some appropriate assumptions, we construct the solution u ε that concentrates around any given isolated local minimum of V by applying the penalization method for the above equation.

On critical N‐Kirchhoff type equations involving Trudinger–Moser nonlinearity

In the present paper, we deal with the existence of nontrivial solutions and Nehari‐type ground state solutions to the Kirchhoff type elliptic equations of the form: where is a smooth bounded domain, is a Kirchhoff function, and f(x, s) has critical exponential growth on s which behaviors as with α0 > 0. Based on a deep analysis and some detailed estimate, we can determine a fine upper bound for the minimax level under weaker assumption on . Our results generalize and improve the ones in Chen et al. (2021) (Lemma 3.1) and Figueiredo and Severo (2016) (Theorems 1.3 and 1.4) for and Goyal et al. (2016) (Theorem 1.3) for N ≥ 2.

Existence of ground state solutions for fractional Kirchhoff Choquard problems with critical Trudinger–Moser nonlinearity

Existence of ground state solutions for fractional Kirchhoff Choquard problems with critical Trudinger–Moser nonlinearity

Schrödinger–Kirchhof-type problems involving the fractional p-Laplacian with exponential growth

In this paper, we study the existence of a solution to Schrödinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with the Trudinger–Moser nonlinearity. As a particular case, we consider the following fractional problem: where is a continuous function with some appropriate assumptions, is the fractional p-Laplacian, with sp = N, K, V are positive continuous functions satisfying some additional conditions, f is a continuous function on with exponential growth. By using the mountain pass theorem, we obtain the existence of solutions to the above problem in suitable Sobolev space. A novel feature of our paper is that the above problem may be degenerate, that is, the Kirchhoff function .

Multiplicity and concentration of solutions to a fractional (p,p1)-Laplace problem with exponential growth

In this paper, we study the Schrödinger equation involving the fractional (p,p1)-Laplacian and Trudinger-Moser nonlinearity as follows(−Δ)N/ssu+(−Δ)p1su+V(εx)(|u|Ns−2u+|u|p1−2u)=f(u)inRN, where ε is a positive parameter, N=ps,s∈(0,1),2≤Ns=p<p1. The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the fractional (p,p1)-Laplacian problem involving exponential growth is studied.

Existence of solutions to Kirchhoff type equations involving the nonlocal p 1& … &p m fractional Laplacian with critical Sobolev-Hardy exponent

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal fractional Laplacian with critical Sobolev-Hardy exponent where are nonnegative constants and is called the critical Sobolev-Hardy exponent, . Here with is the fractional r-Laplace operator. Ω is an open bounded subset of with smooth boundary and . are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when . We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, we consider the case and study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.

On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] is a bounded open interval, [Formula: see text] is a continuous potential, the nonlinear term [Formula: see text] has exponential growth of Trudinger–Moser type, [Formula: see text] and [Formula: see text] denotes the standard Gagliardo seminorm of the fractional Sobolev space [Formula: see text]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [Formula: see text] as a positive parameter, we study the behavior of the nodal solutions as [Formula: see text].

Positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities

Positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities

Existence of solution to singular Schrödinger systems involving the fractional p‐Laplacian with Trudinger–Moser nonlinearity in ℝN

In this paper, we study the existence of weak solution for singular fractional Schrödinger system in involving Trudinger–Moser nonlinearity as follows: where N ≥ 1, 0 &lt; s &lt; 1, N = ps, γ ∈ [0, N), and H has exponential growth and does not satisfy the Ambrosetti–Rabinowitz condition. Note that our problem is the lack of compactness. First, we give a version of vanishing lemma due to Lions; using that result and a version of Mountain pass theorem without (PS) condition, we obtain the existence of nontrivial solution to the above system. When H satisfies the Ambrosetti–Rabinowitz condition, motivated by the work of Chen et al., we study the existence of nontrivial solution to singular Schrödinger system involving the fractional (p1, p)‐Laplace and Trudinger–Moser nonlinearity. In our best knowledge, this is the first time the above problems are studied.

On a class of semipositone problems with singular Trudinger-Moser nonlinearities

<p style='text-indent:20px;'>We prove the existence of positive solutions for a class of semipositone problems with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.

On critical elliptic problems with singular Trudinger–Moser nonlinearities

We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.

Multi-bump analysis for Trudinger–Moser nonlinearities. I. Quantification and location of concentration points

In this paper, we carefully investigate the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension 2 containing a nonlinearity with Trudinger–Moser growth. A quantification result was obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers of Del Pino–Musso–Ruf [12],Malchiodi–Martinazzi [30] and Martinazzi [34]. We answer all of them, proving in particular that the blow-up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving results on existence of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what was done for the Liouville equation in the beautiful work of Chen–Lin [8].

Positive solutions of semipositone elliptic problems with critical Trudinger-Moser nonlinearities

We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,\alpha}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.

Correction to: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity

The estimate obtained in [1, Lemma 3.3] is wrong. We correct this error and we point out the corresponding modifications in accordance with this estimate.

Existence and multiplicity of solutions for fractional Schrödinger–Kirchhoff equations with Trudinger–Moser nonlinearity

We study the existence and multiplicity of solutions for a class of fractional Schrödinger–Kirchhoff type equations with the Trudinger–Moser nonlinearity. More precisely, we consider M(‖u‖N∕s)(−Δ)N∕ssu+V(x)|u|Ns−1u=f(x,u)+λh(x)|u|p−2uinRN,‖u‖=∬R2N|u(x)−u(y)|N∕s|x−y|2Ndxdy+∫RNV(x)|u|N∕sdxs∕N,where M:[0,∞]→[0,∞) is a continuous function, s∈(0,1), N≥2, λ>0 is a parameter, 1<p<∞, (−Δ)N∕ss is the fractional N∕s-Laplacian, V:RN→(0,∞) is a continuous function, f:RN×R→R is a continuous function, and h:RN→[0,∞) is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when f satisfies exponential growth conditions and λ is large enough, and we prove that the solution converges to zero in WVs,N∕s(RN) as λ→∞. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when λ is small enough, and we show that the solution converges to zero in WVs,N∕s(RN) as λ→0. Furthermore, using the genus theory, infinitely many solutions are obtained when M is a special function and λ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function M(0)=0.

Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the \begin{document} $N$ \end{document} -Laplacian operator.

N-Laplacian problems with critical Trudinger-Moser nonlinearities

We prove existence and multiplicity results for a $N$-Laplacian problem with a critical exponential nonlinearity that is a natural analog of the Brezis-Nirenberg problem for the borderline case of the Sobolev inequality. This extends results in the literature for the semilinear case $N = 2$ to all $N \ge 2$. When $N > 2$ the nonlinear operator $- \Delta_N$ has no linear eigenspaces and hence this extension requires new abstract critical point theorems that are not based on linear subspaces. We prove new abstract results based on the ${\mathbb Z}_2$-cohomological index and a related pseudo-index that are applicable here.