Supercritical Problems Research Articles

Normalized solutions for Sobolev critical fractional Schrödinger equation

Abstract In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: ( − Δ ) s u + λ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R N , ( P m ) ∫ R N ∣ u ∣ 2 d x = m 2 , \hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" voffset="-2.9ex">\left({P}_{m})</mml:mpadded>\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , N ≥ 2 N\ge 2 , m > 0 m\gt 0 , 2 s * ≔ 2 N N − 2 s {2}_{s}^{* }:= \frac{2N}{N-2s} , λ \lambda is an unknown parameter that will appear as a Lagrange multiplier, and f f is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f f , with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to ( P m ) \left({P}_{m}) . We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s = 1 s=1 .

Existence and symmetry breaking results for positive solutions of elliptic Hamiltonian systems

Abstract In this paper, we are interested in positive solutions of $$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$ where $\Omega $ is a bounded annular domain (not necessarily an annulus) in ${\mathbb {R}}^N (N \ge 3)$ and $ a(x), b(x)$ are positive continuous functions. We show the existence of a positive solution for a range of supercritical values of p and q when the problem enjoys certain mild symmetry and monotonicity conditions. We shall also address the symmetry breaking phenomena where the system is fully symmetric. Indeed, as a consequence of our results, we shall show that problem (1) has $\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $ (the floor of $\frac {N}{2}$ ) positive non-radial solutions when $ a(x)=b(x)=1$ and $\Omega $ is an annulus with certain assumptions on the radii. In general, for the radial case where the domain is an annulus, we prove the existence of a non-radial solution provided $$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$ where $\lambda _H$ is the best constant for the Hardy inequality on $\Omega .$ We remark that the best constant $\lambda _H$ for the Hardy inequality is just the characteristic of the domain, and is independent of the choices of p and $q.$ For this reason, the aforementioned inequality plays a major role to prove the existence and multiplicity of non-radial solutions when the problem is fully symmetric. Our proofs use a variational formulation on appropriate convex subsets for which the lack of compactness is recovered for the supercritical problem.

Global solutions to the tangential Peskin problem in 2-D

We introduce and study the tangential Peskin problem in 2D, which is a scalar drift-diffusion equation with a nonlocal drift. It is derived with a new Eulerian perspective from a special setting of the 2D Peskin problem where an infinitely long and straight 1D elastic string deforms tangentially in the Stokes flow induced by itself in the plane. For initial datum in the energy class satisfying natural weak assumptions, we prove existence of its global solutions. This is considered as a super-critical problem in the existing analysis of the Peskin problem based on Lagrangian formulations. Regularity and long-time behaviour of the constructed solution is established. Uniqueness of the solution is proved under additional assumptions.

Ground states in spatially discrete non-linear Schrödinger models

In the seminal work (Weinstein 1999 Nonlinearity 12 673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on . More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed power P (i.e. l 2 mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedke et al 1994 Phys. Rev. Lett. 73 1055 and Laedke et al 1996 Phys. Rev. E 54 4299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g. l 2 supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity . In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter ω varies, in contrast with the corresponding continuous NLS model.

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

Asymptotics for a high-energy solution of a supercritical problem

In this paper we deal with the equation −Δpu+|u|p−2u=|u|q−2ufor 1<p<2 and q>p, under Neumann boundary conditions in the unit ball of RN. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for q large is proved in Colasuonno et al. (2022). We detect the limit profile as q→∞ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant 1. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest.

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

In this paper we are interested in positive classical solutions of(1){−Δu=a(x)up−1 in Ω,u>0 in Ω,u=0 on ∂Ω, where Ω is a bounded annular domain (not necessarily an annulus) in RN(N≥3) and a(x) is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of p when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall show that (1) has ⌊N2⌋ (the floor of N2) positive nonradial solutions when a(x)=1 and Ω is an annulus with certain assumptions on the radii. We also obtain the existence of positive solutions in the case of toroidal domains. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space at the expense of a mild invariance property.

A class of supercritical Sobolev type inequalities with logarithm and related elliptic equations

In this paper, we first deduce the following Sobolev inequality with logarithmic term:(0.1)sup⁡{∫B|u|2⁎|ln⁡(τ+|u|)||x|βdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<∞, where β>0, τ≥0 are constants, B is the unit ball in RN, N≥3, and 2⁎=2N/(N−2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0<β<min⁡{N/2,N−2} and 1≤τ<∞. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem:(0.2){−Δu=u2⁎−1(ln⁡(τ+u))|x|β+g(|x|,u),u>0inB,u=0on∂B, where g(r,u)∈C([0,1)×R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity:(0.3){−Δu=u2⁎−1(ln⁡(τ+u))|x|β,u>0inB,u=0on∂B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent |x|β is replaced by any non-negative constant.

An existence result for super-critical problems involving the fractional [formula omitted]-Laplacian in [formula omitted

In this paper we establish an existence result for the following super-critical problems involving the fractional p-Laplacian (−Δ)psu+V(x)|u|p−2u=|u|ς−2u+λ|u|q−2u,inRN,where 0<s<1, 1<q<p<ς and sp<N. Here the nonlinearity |u|ς−2u imposes no any growth conditions. This is a new result for super-critical values of ς.

Liouville-Type Theorems for Fractional and Higher-Order Hénon–Hardy Type Equations via the Method of Scaling Spheres

Abstract In this paper, we aim to develop the (direct) method of scaling spheres, its integral forms, and the method of scaling spheres in a local way. As applications, we investigate Liouville properties of nonnegative solutions to fractional and higher-order Hénon–Hardy type equations $$ \begin{align*}&amp; (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\, \mathbb{R}^{n}_{+} \,\,\, \text{or} \,\,\, B_{R}(0) \end{align*}$$with $n&amp;gt;\alpha $, $0&amp;lt;\alpha &amp;lt;2$ or $\alpha =2m$ with $1\leq m&amp;lt;\frac {n}{2}$. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $a\in (-\alpha ,\infty )$ and $0&amp;lt;p&amp;lt;p_{c}(a):=\frac {n+\alpha +2a}{n-\alpha }$. By using the method of scaling spheres, we prove Liouville theorems for the above Hénon–Hardy equations and equivalent integral equations (IEs). In $\mathbb {R}^{n}$, our results improve the known Liouville theorems for some especially admissible subranges of $a$ and $1&amp;lt;p&amp;lt;\min \big \{\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big \}$ to the full range $a\in (-\alpha ,\infty )$ and $p\in (0,p_{c}(a))$. In particular, when $a&amp;gt;0$, we covered the gap $p\in \big [\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big )$. For bounded domains (i.e., balls), we also apply the method of scaling spheres to derive Liouville theorems for super-critical problems. Extensions to PDEs and IEs with general nonlinearities $f(x,u)$ are also included (Theorem 1.31). In addition to improving most of known Liouville type results to the sharp exponents in a unified way, we believe the method of scaling spheres developed here can be applied conveniently to various fractional or higher order problems with singularities or without translation invariance or in the cases the method of moving planes in conjunction with Kelvin transforms do not work.

On some supercritical problems involving the fractional Laplacian operator

On some supercritical problems involving the fractional Laplacian operator

Qualitative properties of stable solutions to some supercritical problems

&lt;abstract&gt;&lt;p&gt;In this paper, we study symmetry properties of stable solutions to the Lane-Emden equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \Delta u+|u|^{p-1}u = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;with $ n\geq 11 $, $ p $ in a suitable range and the Liouville equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ \Delta u+e^{u} = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;with $ n = 10 $.&lt;/p&gt;&lt;/abstract&gt;

A supercritical variable exponent problem

Suppose B1 is the open unit ball centered at the origin in RN with N≥3 and 1<p<N+2N−2. Let 0<σ1<1 and suppose 0≤γ(x)≤−σ1ln⁡(|x|) with γ=0 near |x|=1 and γ→∞ as |x|→0. We show for sufficiently small ε>0 there is a positive weak solution u∈Cloc2,α(B1‾﹨{0})∩C1,α(B1‾) of{−Δu=up+εγ(x)in B1∖{0},u=0on ∂B1. Note since γ(x) blows up at the origin that this problem can be viewed as a supercritical problem near the origin.

Fast and slow decay solutions for supercritical fractional elliptic problems in exterior domains

We consider the fractional elliptic problem: where B1 is the unit ball in ℝN, N ⩾ 3, s ∈ (0, 1) and p &gt; (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists Ps &gt; (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) &lt; p &lt; Ps, the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

High Speed Rail in Northern England Tactics and Policies for Implementing Mega Plans by Modular Incrementalism

Britain lags behind many other countries in its provision of high-speed rail. This paper looks in depth at the challenges of providing high-speed rail links, east–west, across Northern England, identi fied as a key issue by former Chancellor of the Exchequer George Osborne in his Northern Powerhouse speech in 2014. We ask what can be learned from the politics of Britain's successful motorway programme in the 1960s and 1970s, and from the plan advanced for High Speed North by the late Professor Sir Peter Hall and colleagues, published some weeks before Osborne's speech. Introducing the concept of centripetal urban dynamics, we doubt whether the suppression of public transport demand by the Covid 19 virus will be long lasting. Thus the Hall Plan still has remarkable relevance, especially in its tactics for sequencing investment, which we term modular incrementalism. Some updating is needed, so that the investment strategy focuses on super critical problems for rail investment. We conclude with recommendations for the High Speed North project itself and re flect on wider implications for decision-making processes.

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.

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.

On supercritical problems involving the Laplace operator

AbstractWe discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

Multiple solutions for some symmetric supercritical problems

The aim of this paper is to investigate the existence of one or more critical points of a family of functionals which generalizes the model problem [Formula: see text] in the Banach space [Formula: see text], where [Formula: see text] is an open bounded domain, [Formula: see text] and the real terms [Formula: see text] and [Formula: see text] are [Formula: see text] Carathéodory functions on [Formula: see text]. We prove that, even if the coefficient [Formula: see text] makes the variational approach more difficult, if it satisfies “good” growth assumptions then at least one critical point exists also when the nonlinear term [Formula: see text] has a suitable supercritical growth. Moreover, if the functional is even, it has infinitely many critical levels. The proof, which exploits the interaction between two different norms on [Formula: see text], is based on a weak version of the Cerami–Palais–Smale condition and a suitable intersection lemma which allow us to use a Mountain Pass Theorem.