This paper deals with the qualitative analysis of solutions
to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser-Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational methods, we prove the existence of nonnegative solutions.
We consider a fractional Schrödinger-Poisson system with a general nonlinearity in subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.
We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degre…
We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $λ|u|^αu$, where $0<α<2$, and $λ$ is a complex constant satisfying $\text{Im} λ>\frac{α|\text{Re} λ|}{2\sqrt{ α+1}}$. For arbitrary large initial data, we present the uniform time decay estimates when $4/3<α<2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<α<2$. The proof is based on the vector fields method and a semiclassical analysis method.
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
Let $T_ε$ be the lifespan for the solution to the Schrödinger equation on $\mathbb{R}^d$ with a power nonlinearity $λ|u|^{2θ/d}u$ ($λ\in \mathbb{C}$, $0<θ<1$) and the initial data in the form $εφ(x)$. We provide a sharp lower bound estimate for $T_ε$ as $ε\to +0$ which can be written explicitly by $λ$, $d$, $θ$, $φ$ and $ε$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].
In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-Δ)^αu=g(u)+σν\quad{\rm in}\quad Ω,$$ subject to the condition (E2) $$u=\varrhoμ\quad {\rm on}\quad \partialΩ \ {\rm if}\ α=1\qquad {\rm or\ \ in}\ \ Ω^c \ \ {\rm if}\ α\in(0,1),$$ where $σ,\varrho\ge0$, $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ or Laplacian operator if $α=1$, $ν,μ$ are suitable Radon measures and $g:\mathbb{R}_+\mapsto\mathbb{R}_+$ is a continuous function. We introduce an approach to obtain weak solutions for probl…
In this paper, we are concerned with the following elliptic equation $$\left\{\begin{array}{rrl}-Δu&=& |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon\hbox{ in } Ω,\\ u&=&0 \hbox{ on }\partial Ω, \end{array} \right.$$ where $Ω$ is a smooth bounded open domain in $\mathbb{R}^n, \ n\geq 3$ and $\varepsilon>0$. Clapp et al. in Journal of Diff. Eq. (Vol 275) proved that there exists a single-peak positive solution for small $\varepsilon$ if $n \geq 4$. Here we construct positive as well as changing sign solutions concentrated at several points at the same time.
We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by establishing the refined analysis of the spatially decaying nonlinear term.
Initial-boundary value problem for the generalized Zakharov-Kuznetsov equation posed on a bounded rectangle is considered. Critical and subcritical powers in nonlinearity are studied.
We look for ground state solutions to the Schrödinger-type system \[ \begin{cases} -Δu_j + λ_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (λ_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in ]0,\infty[^M$ is prescribed and $(λ,u) = (λ_1,\dots,λ_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, w…
This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case is even new in the classical $p$-Laplacian case.
In this paper, we prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with respect to $t$ of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by Staffilani to c…
Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
Research Institute for Interdisciplinary Science, Okayama University（冈山大学跨学科科学研究所）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …
We investigate a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. The problem is set on the whole space and compactness issues have to be tackled.
In this article we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not require the usual Ambrosetti-Rabinowitz condition. The proof is based on variational methods and perturbation arguments.
We consider short-range mass-subcritical nonlinear Schrödinger equations and we show that the corresponding solutions with initial data in $Σ$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and Tsutsumifrom $L^2$ to $H^1$.We also provide some partial results concerning the scattering of the first order moments, as well as a short proof via lens transform of a classical result due to Tsutsumi and Cazenave-Weissler on the scattering in $Σ$.
In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the $p$-laplacian combined either with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a quasilinear boundary value problem $ -Δ_p u= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and …
In this paper, we study the existence of solutions for the new fractinal Robin equations with variable exponents. Moreover, we deal with the logarithm-type nonlinearity. In particular, we consider two cases: critical and subcritical cases.
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.
Let us consider a semilinear boundary value problem $ - Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $ Ω\subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions guarantying that semi-stable weak positive solutions to subcritical semilinear elliptic equations are smooth in any dimension, and as a consequence, classical solutions. By a subcritical nonlinearity we mean $f(x,s)/s^\frac{N+2}{N-2} \to 0$ as $s\to\infty$, including non-power nonlinearities, and enlarging the class of subcritical nonlinearities, which is usually reserved for power …
This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.
In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method.
In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schrödinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $|u|^{α-1}u$.
We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity $$ i\partial_t u+Δu=\frac{u}{|x|^b}(e^{α|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $α=2π(2-b)$. We establish local and global well-posedness in the subcritical and critical regimes.
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1 \leq s \leq 2$, the local well-posedness follows from a Strichartz estimate. In higher dimensional $H^1$ scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional $H^1$ scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the correspond…
We investigate the initial value problem for a defocusing nonlinear Schrödingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.
Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent
In this paper we discuss global well - posedness and scattering for some initial value problems that are $L^{2}$ supercritical and $\dot{H}^{1}$ subcritical, with radial data. We prove global well - posedness and scattering for radial data in $H^{s}$, $s > s_{c}$, where the problem is $\dot{H}^{s_{c}}$ - critical. We make use of the long time Strichartz estimates of \cite{D2} to do this.
We study the scattering problem for the nonlinear Schrödinger equation $i\partial_t u + Δu = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $Σ$ holds and the wave operator is well-defined on $Σ$. We show that there exists $0<β<p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+β}$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in t…
In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -Δ_{p}u-\baΔ_{q}u & = \la u^{-\de}+ u^{r-1}, \ u>0, \ \text{ in } \Om \\ u&=0 \quad \text{ on } \pa\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1< q< p<r \leq p^{*}$, where $p^{*}=\ds \frac{np}{n-p}$, $0<\de< 1$, $n> p$ and $\la,\, \ba>0$ are parameters. We prove existence, multiplicity and regularity of weak solutions of $(P_\la)$ for suitabl…
In this work, we study a quasilinear elliptic problem involving the 1-laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - β)$. Our approach is based on an analysis of the associated p-laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $β\to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.
In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global …
This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrödinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previou…
This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These space…
In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the non-degenerate as well as the degenerate Kirchhoff case in our setting.
The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. In this paper we choose $15/16<s<1$ and assume that the nonlinearity equals $-|u|^p u$ with $p = 2/{3/2 -s}$, so that this problem is $\dot{H}^s \times \dot{H}^{s-1}$ critical. We will prove the global well-posedness and scattering of the solution under the additional assumption that the $\dot{H}^s \times \dot{H}^{s-1}$ norm of the sol…
We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.
In this paper, a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity under initial-boundary condition is investigated. Based on potential well method and Hardy-Sobolev inequality, the global existence of solutions is derived when the initial energy $J(u_0)$ is subcritical($J(u_0)<d$), critical($J(u_0)=d$) with $d$ being the mountain-pass level. Finite time blow-up results are obtained as well when the initial energy $J(u_0)$ satisfies specific conditions. Moreover, the upper and lower bounds of the blow-up time are given.
We consider a class of nonlinear Schrödinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time depe…
In this paper we study a slightly subcritical Choquard problem on a bounded domain D. We prove that the number of positive solutions depends on the topology of the domain. In particular when the exponent of the nonlinearity approaches the critical one, we show the existence of cat(D)+1 solutions. Here cat(D) denotes the Lusternik-Schnirelmann category.
Let $Ω$ be a bounded domain in $\mathbb{R}^N$. In this paper, we consider the following nonlinear elliptic equation of $N$-Laplacian type: $-Δ_{N}u=f(x,u)$ where $u\in W_{0}^{1,2}\{0}$ when $f$ is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to the above equation without the Ambrosetti-Rabinowitz $(AR)$ condition. Earlier works in the literature on the existence of nontrivial solutions to $N-$Laplacian in $\mathbb{R}^{N}$ when the nonlinear term $f$ has …
We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation \[iu_t+Δu=u(e^{4π|u|^2}-1)\] in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao--Visan--Zhang. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson and the work of Ibrahim--Majdoub--Masmoudi--Nakanishi on a related problem to prove corresponding spacetime estimates which imply sc…
