Supercritical Exponents Research Articles

Variable Supercritical Schrödinger-Poisson system with singular term

In this paper, we consider a class of Schrödinger-Poisson systems on bounded domains that depend on a parameter η. These systems have variable supercritical exponents and a singular term as part of their nonlinearity. For η=1, we proved the existence and uniqueness of a solution using variational methods. In the case η=−1, the structure of the problem changes significantly, and we proved the existence of a solution using non-variational methods based on an approximating scheme. In both cases, we faced difficulties handling the loss of compactness because the variable exponents involve supercritical growth. The supercritical variable growth not only causes the system to lose its homogeneity but also its compactness properties.

The Lane-Emden System on Cartan-Hadamard Manifolds: Asymptotics and Rigidity of Radial Solutions

Abstract We investigate existence and qualitative properties of globally defined and positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard model manifold $ {{\mathbb{M}}}^{n} $. We prove that, for critical or supercritical exponents, there exists at least a one-parameter family of such solutions. Depending on the stochastic completeness or incompleteness of $ {{\mathbb{M}}}^{n} $, we show that the existence region stays one dimensional in the former case, whereas it becomes two dimensional in the latter. Then, we study the asymptotics at infinity of solutions, which again exhibit a dichotomous behavior between the stochastically complete (where both components are forced to vanish) and incomplete cases. Finally, we prove a rigidity result for finite-energy solutions, showing that they exist if and only if is isometric to $ {\mathbb{R}}^{n} $.

Multiple solutions for the logarithmic fractional Kirchhoff equation with critical or supercritical nonlinearity

In this paper, we study the following logarithmic fractional Kirchhoff equation: ( a + b ∫ R 3 | ( − Δ ) s / 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | p − 2 u log ⁡ u 2 + λ | u | q − 2 u , in R 3 , where a, b>0, ( − Δ ) s is the fractional Laplace operator with q ≥ 2 s ∗ = 6 3 − 2 s and V satisfies suitable conditions. Since the presence of supercritical exponents leads to a lack of compactness, we introduce a truncation function to reduce the exponents, allowing compactness to be recovered. We show that the above equation has a ground state solution and a sign-changing solution via the constraint variation method, the quantitative deformation lemma and Moser's iterative method.

Fractional magnetic Schrödinger equations with potential vanishing at infinity and supercritical exponents

This paper focuses on the following class of fractional magnetic Schrödinger equations: where is the fractional magnetic Laplacian, is the magnetic potential, , N>2s, is a parameter, is a potential function that may decay to zero at infinity and is a continuous function with subcritical growth. We deal with supercritical case . Our approach is based on variational methods combined with penalization technique and -estimates.

Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres

This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents (S_{pm varepsilon}): Delta ^{2}u-c_{n}Delta u+d_{n}u = Ku^{ frac{n+4}{n-4}pm varepsilon}, u>0 on S^{n}, where ngeq 5, ε is a small positive parameter and K is a smooth positive function on S^{n}. We construct some solutions of (S_{-varepsilon}) that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation (S_{+varepsilon}).

Fractional Hardy equations with critical and supercritical exponents

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P)(-Δ)su-θu|x|2s=up-uqinRNu>0inRNu∈H˙s(RN)∩Lq+1(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ({\\mathcal {P}}) \\quad \\quad \\left\\{ \\begin{aligned} (-\\Delta )^s u -\ heta \\frac{u}{|x|^{2s}}&=u^p - u^q \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&> 0 \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&\\in {\\dot{H}}^s({\\mathbb {R}}^N)\\cap L^{q+1}({\\mathbb {R}}^N), \\end{aligned} \\right. \\end{aligned}$$\\end{document}where sin (0,1), N>2s, q>pge {(N+2s)}/{(N-2s)}, theta in (0, Lambda _{N,s}) and Lambda _{N,s} is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole mathbb {R}^N, and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

A spectral analysis of the nonlinear Schrödinger equation in the co-exploding frame

The nonlinear Schrödinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon identifying the self-similar solutions in the so-called “co-exploding frame”, a dynamical systems analysis of their stability is natural, yet is complicated by the mixed Hamiltonian-dissipative character of the relevant frame. In the present work, we study the spectral picture of the relevant linearized problem. We examine the point spectrum of 3 eigenvalue pairs associated with translation, U(1) and conformal invariances, as well as the continuous spectrum. We find that two eigenvalues become positive, yet are attributed to symmetries and are thus not associated with instabilities. In addition to a vanishing eigenvalue, 3 more are found to be negative and real, while the continuous spectrum is nearly vertical and on the left-half (spectral) plane. The eigenfunctions and eigenvalues are approximated both asymptotically and numerically, with good agreement between the two approaches. The non-Hamiltonian nature of the co-exploding system results in the 3 eigenvalue pairs failing to be equal-and-opposite by an exponentially small amount. A projection method is used to evaluate this small correction, and at the same time explains the subtle effects of finite boundaries and their role in the observed weak eigenvalue oscillations.

The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<s<1, to the case s≥1. We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, Γ-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Furthermore, we show the same asymptotic behavior as the core-radius vanishes and s→s̄≥1 simultaneously. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics.

Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

We study low regularity behavior of the nonlinear wave equation in $\mathbb {R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb {R}^2)\times H^{s-1}(\mathbb {R}^2)$ is ill-posed whenever $0 < s < s_{cr}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb {R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.

Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents

This work studies the existence of positive solutions for fractional Schrodinger equations of the form $$\begin{aligned} (-\Delta )^s u + V(x) u = g(u)+\lambda |u|^{q-2}u\quad \text{ in }\quad {\mathbb {R}}^N, \end{aligned}$$ where $$s\in (0,1)$$ , $$N>2s$$ , $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ is a potential function which can vanish at infinity, $$g:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ is superlinear and has subcritical growth, the exponent $$q\ge 2^*_s:=2N/(N-2s)$$ and $$\lambda $$ is a nonnegative parameter. Our approach is based on a truncation argument in combination with variational techniques and the Moser iteration method.

STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION

This paper is concerned with the static Choquard equation $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$ where $N,p&gt;2$ and $\max \{0,N-4\}&lt;\unicode[STIX]{x1D6FC}&lt;N$. We prove that if $u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then $u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and $\unicode[STIX]{x1D6FC}\neq 2$.

Existence and shape of the least energy solution of a fractional Laplacian

We discuss the asymptotic behavior of the least energy solution of a half-Laplacian operator with supercritical exponents on an interval. This extends the results obtained by Ren–Wei (Trans Am Math Soc 343(2):749–763, 1994) to the fractional Laplacian case.

Erratum to: Quasilinear Riccati type equations with super-critical exponents

ABSTRACTThis is an erratum to the paper: N. C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations 35 (2010), 1958–1981.

On singular equations with critical and supercritical exponents

We study the problem(Iε){−Δu−μu|x|2=up−εuqin Ω,u>0in Ω,u∈H01(Ω)∩Lq+1(Ω), where q>p≥2⁎−1, ε>0, Ω⊆RN is a bounded domain with smooth boundary, 0∈Ω, N≥3 and 0<μ<μ¯:=(N−22)2. We completely classify the singularity of solution at 0 in the supercritical case. Using the transformation v=|x|νu, we reduce the problem (Iε) to (Jε)(Jε){−div(|x|−2ν∇v)=|x|−(p+1)νvp−ε|x|−(q+1)νvqin Ω,v>0in Ω,v∈H01(Ω,|x|−2ν)∩Lq+1(Ω,|x|−(q+1)ν), and then formulating a variational problem for (Jε), we establish the existence of a variational solution vε and characterize the asymptotic behavior of vε as ε→0 by variational arguments when p=2⁎−1.

Semilinear nonlocal elliptic equations with critical and supercritical exponents

We study the problem $\left\{ \begin{align} &amp;{{(-\Delta lta )}^{s}}u={{u}^{p}}-{{u}^{q}}\ \text{in}\ \text{ }{{\mathbb{R}}^{N}}, \\ &amp;u\in {{{\dot{H}}}^{s}}({{\mathbb{R}}^{N}})\cap {{L}^{q+1}}({{\mathbb{R}}^{N}}), \\ &amp;u&gt;0\ \ \text{in}\ \ {{\mathbb{R}}^{N}}, \\ \end{align} \right.$ where $s∈(0,1)$ is a fixed parameter, $(-Δ)^s$ is the fractional Laplacian in $\mathbb{R}^N$, $q&gt;p≥q \frac{N+2s}{N-2s}$ and $N&gt;2s$. For every $s∈(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all $s∈(0,1)$. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we show that the above problem does not have any solution when $p=\frac{N+2s}{N-2s}$. We also discuss radial symmetry and decreasing property of the solution and prove that when $p&gt;\frac{N+2s}{N-2s}$, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every $p≥q \frac{N+2s}{N-2s}$ and every solution is a classical solution.

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Let BR be a ball of radius R in RN. We analyze the positive solutions to the problem {−Δu+u=|u|p−2u,in BR,∂νu=0,on ∂BR, that branch out from the constant solution u=1 as p grows from 2 to +∞. The nonzero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p→∞ (in particular, for supercritical exponents) or as R→∞ for any fixed value of p>2, partially answering a conjecture in Bonheure et al. (2012). We give explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

Slowly decaying radial solutions of an elliptic equation with subcritical and supercritical exponents

We study radial solutions of the problem $$\Delta u + {u^p} + {u^q} = 0,\;u > 0\;in\;{\mathbb{R}^N}$$ , where N ≥ 3 and $$\frac{N}{{N - 2}} < p < \frac{{N + 2}}{{N - 2}} < q$$ . We show that if p is close to N/(N-2), q is close to (N +2)/(N-2), and a certain relation holds between them, then the problem has slowly decaying solutions.

A class of quasilinear Schrödinger equations with critical or supercritical exponents

We study the existence of nontrivial solutions for the following quasilinear Schrödinger equations −Δu+V(x)u+κ2[Δ∣u∣2]u=l(u),x∈RN, where V:RN→R and l(u) involves critical or supercritical exponents.

Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign

In this study, we establish the existence of solutions for quasilinear Schrödinger equations involving supercritical growth with nonlinearities that are indefinite in sign. By changing the variables, the quasilinear equations are reduced to semilinear and variational methods, which are then used to obtain existence results.

The singular set of stationary solution for semilinear elliptic equations with supercritical growth

In this paper, we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations. For almost all supercritical exponents, we improve the upper bounds of the Hausdorff dimension of the singular set, which are derived by Pacard, to the optimal ones.