Advances in Nonlinear Analysis 

About: Advances in Nonlinear Analysis is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Nonlinear system & Geometry and topology. It has an ISSN identifier of 2191-9496. It is also open access. Over the lifetime, 487 publications have been published receiving 6244 citations.

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations

Abstract: Abstract The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent

Abstract: Abstract This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: ‖ u ‖λ(θ−1)p[ λ(−Δ)psu+V(x)| u |p−2u ]=| u |ps⋆−2u+f(x,u) in ℝN,‖ u ‖λ=(λ∫ℝ∫2N| u(x)−u(y) |p| x−y |N+psdxdy+∫ℝNV(x)| u |pdx)1/p $$\\begin{align}& \\left\\| u \\right\\|_{\\lambda }^{\\left( \\theta -1 \\right)p}\\left[ \\lambda \\left( -\\Delta \\right)_{p}^{s}u+V\\left( x \\right){{\\left| u \\right|}^{p-2}}u \\right]={{\\left| u \\right|}^{p_{s}^{\\star }-2}}u+f\\left( x,u \\right)\\,in\\,{{\\mathbb{R}}^{N}}, \\\\ & {{\\left\\| u \\right\\|}_{\\lambda }}={{\\left( \\lambda \\int\\limits_{\\mathbb{R}}{\\int\\limits_{2N}{\\frac{{{\\left| u\\left( x \\right)-u\\left( y \\right) \\right|}^{p}}}{{{\\left| x-y \\right|}^{N+ps}}}}dxdy+\\int\\limits_{{{\\mathbb{R}}^{N}}}{V\\left( x \\right){{\\left| u \\right|}^{p}}dx}} \\right)}^{{1}/{p}\\;}} \\\\ \\end{align}$$ where (−Δ)ps $\\left( -\\Delta \\right)_{p}^{s}$is the fractional p–Laplacian with 0 < s < 1 < p < N/s, ps⋆=Np/(N−ps) $p_{s}^{\\star }={Np}/{\\left( N-ps \\right)}\\;$is the critical fractional Sobolev exponent, λ > 0 is a real parameter, 1<θ≤ps⋆/p, $1<\\theta \\le {p_{s}^{\\star }}/{p}\\;,$and f : ℝN × ℝ → ℝ is a Carathéodory function satisfying superlinear growth conditions. For θ∈(1,ps⋆/p), $\\theta \\in \\left( 1,{p_{s}^{\\star }}/{p}\\; \\right),$by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For θ=ps⋆/p, $\\theta ={p_{s}^{\\star }}/{p}\\;,$by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infx∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.

Regularity for minimizers for functionals of double phase with variable exponents

Abstract: The functionals of double phase type \[ \mathcal{H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ( q > p > 1, a(x)\geq 0) \] are introduced in the epoch-making paper by Colombo-Mingione for constants $p$ and $q$, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of $x$ and partly generalize their regularity results.

Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

Abstract: Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

The Yamabe equation in a non-local setting

Abstract: Aim of this paper is to study the following elliptic equation driven by a gen- eral non-local integrodifferential operator LK such that LKuCu Cjuj 2 2 uD0 in, uD0 in R n n, wheres2.0;1/, is an open bounded set of R n ,n > 2s, with Lipschitz boundary, is a positive real parameter,2 D2n=.n 2s/ is a fractional critical Sobolev exponent, while LK is the non-local integrodifferential operator