Nonlinearities In RN Research Articles

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.

Classification of solutions for mixed order conformally system with Hartree-type nonlinearity in ℝn

In this paper, we consider the following mixed order conformally invariant system with Hartree-type nonlinearity : [Formula: see text] where [Formula: see text], [Formula: see text] is a integer, [Formula: see text], [Formula: see text], [Formula: see text]. We first prove the equivalence of the PDEs system and the IEs system. Then we give the classification of the nonnegative solutions to the system ( 0.1 ) by using the method of moving spheres. Finally, we prove Liouville-type theorems results for system ( 0.1 ) in the critical and supercritical-order cases (i.e. [Formula: see text]), respectively.

The Ground State Solutions for Kirchhoff-Schrödinger Type Equations with Singular Exponential Nonlinearities in ℝN

The Ground State Solutions for Kirchhoff-Schrödinger Type Equations with Singular Exponential Nonlinearities in ℝN

Solutions for nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities

AbstractIn this article, we aimed to study a class of nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities inRN{{\mathbb{R}}}^{N}. By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p,q)-Laplacian systems in the case ofN=sp=lqN=sp=lq. It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p,q)-Laplacian systems is the main novelty of this article.

Multiple solutions for a quasilinear Choquard equation with critical nonlinearity

AbstractIn the present work, we are concerned with the multiple solutions for quasilinear Choquard equation with critical nonlinearity inRN{{\mathbb{R}}}^{N}. We show multiplicity results for this problem, which are characterized, respectively, by the new version of symmetric mountain-pass theorem and the mountain-pass theorem for even functionals. The novelty of our work is the appearance of the convolution terms as well as critical nonlinearities.

Existence of Solutions of Anisotropic Elliptic Equations with Variable Indices of Nonlinearity in ℝn

In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.

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 < s < 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.

Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Supercritical problems with concave and convex nonlinearities in ℝN

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole [Formula: see text] with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev embeddings.

Existence results for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

In this paper, we study a class of Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity in RN. More precisely, we consider −a+b∫RN|∇u|2dxΔu−a[Δ(u2)]u=λIμ∗|u|p|u|p−2u+Iμ∗|u|22μ∗|u|22μ∗−2u, where a>0, b≥0, N≥3, 0<μ<4N3N+4, 2(N+μ)N≤p<2μ∗≔2(N−μ)N−2, λ>0 and Iμ is a Riesz potential. We prove the existence and multiplicity of solutions for the equation by variational methods together with concentration–compactness principle.

Solutions for a class of quasilinear Choquard equations with Hardy–Littlewood–Sobolev critical nonlinearity

In this article, we consider the quasilinear Choquard equation with critical nonlinearity in RN: −ε2Δu+V(x)u−ε2uΔu2=∫RN|u|22μ∗|x−y|μdy|u|22μ∗−2u+h(x,u)u,where 0<μ<N, N≥3, 2μ∗=(2N−μ)∕(N−2) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, ε is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on V and h, we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle.

Multiplicity of solutions to the generalized extensible beam equations with critical growth

In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in RN: Δ2u−M∫RN|∇u|2dxΔu+V(x)u=k(x)|u|q−2u+λ|u|2∗∗−2u,x∈RN, where Δ2u=Δ(Δu) is the biharmonic operator, M:R+→R+ is a continuous function, λ>0 is a parameter, V:RN→R+ is a potential function, and k(x) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions’ second Concentration-compactness principle and Concentration-compactness principle at infinity.

Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

Erratum: “Multiplicity results for stationary Kirchhoff problems involving fractional elliptic operator and critical nonlinearity in RN” [J. Math. Phys. 59, 071512 (2018)

Erratum: “Multiplicity results for stationary Kirchhoff problems involving fractional elliptic operator and critical nonlinearity in RN” [J. Math. Phys. 59, 071512 (2018)

Multiplicity results for stationary Kirchhoff problems involving fractional elliptic operator and critical nonlinearity in RN

In this paper, we study a class of stationary Kirchhoff problems involving a fractional elliptic operator and critical nonlinearity in RN: g[u]s2(−Δ)su=αk(x)|u|q−2u+β|u|2s*−2u. By using a fractional version of Lions’ second concentration compactness principle and concentration compactness principle at infinity to prove that the (PS)c condition holds locally and by minimax methods and Krasnoselskii genus theory, we establish the multiplicity of solutions for suitable positive parameters α, β.

Multiplicity of positive solutions for the Kirchhoff-type equations with critical exponent in [formula omitted

In this work we study the existence and multiplicity of solutions to the following Kirchhoff-type problem with critical nonlinearity in RN−a+b∫RN∇updxΔpu=μup∗−1+λf(x,u);x∈RN,u∈D1,p(RN),where N≥2p, μ,λ,a,b>0 and the nonlinearity f(x,u) satisfies certain subcritical growth conditions. By using topological and variational methods, infinitely many positive solutions are obtained.

Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities

In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system:[see formula in PDF]where (–Δ)spis the fractionalp–Laplace operator with 0 &lt;s&lt; 1 &lt;p&lt;N/s,α,β&gt; 1 withα+β=p*s,M: ℝ+0 → ℝ+0is a continuous function,V: ℝN→ ℝ+is a continuous function, λ &gt; 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrödinger–Kirchhoff systems involving critical nonlinearities in ℝN.

Multiple symmetric results for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in [formula omitted

This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

Solutions of stationary Kirchhoff equations involving nonlocal operators with critical nonlinearity in RN

In this paper, we consider the existence and multiplicity of solutions for fractional Schrödinger equations with critical nonlinearity in RN. We use the fractional version of Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PSc) condition holds locally. Under suitable assumptions, we prove that it has at least one solution and, for any m ∈ N, it has at least m pairs of solutions. Moreover, these solutions can converge to zero in some Sobolev space as ε → 0.

Infinitely many solutions forp-Kirchhoff equation with concave-convex nonlinearities in RN

In this paper, we study the existence of infinitely many solutions to p-Kirchhoff-type equation (0.1) where f(x,u) = λh1(x)|u|m − 2u + h2(x)|u|q − 2u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h1(x),h2(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0 such that problem (0.1) admits infinitely many nonnegative high-energy solutions provided that λ∈[0,λ0) and . Also, we prove that problem (0.1) has at least a nontrivial solution under the assumption f(x,u) = h2|u|q − 2u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h1|u|m − 2u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.