Mousomi Bhakta, Souptik Chakraborty, Patrizia Pucci
This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.
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This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {}= λ\frac{f(u)}{u^γ}, \ u>0 && \quad\mbox{ in } \, Ω, u & {}= 0 && \quad\mbox{ on } \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $N \geq 1$, $λ>0$ is a real parameter, $$\mathcal{L}_{p,q} u := div(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} …
Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems
本文研究了具非齐次非线性项的$χ^{(2)}$型抛物型系统，通过精细积分估计和修改尺度变换与爆破论证，推导出刘维尔型定理、周期解存在性及通用界与衰减估计，是该领域的首个结果。
NLS with mass-subcritical combined nonlinearities: small mass $L^2$-scattering
本文研究了具有聚焦主导项与低阶散焦扰动项的双重非线性NLS方程，利用伪共形变换与变分论证证明了仅需初始数据质量足够小即可实现质量次临界区域中的散射。
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
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 …
In this paper we study the existence of at least two positive weak solutions for an inhomogeneous fourth order equation with Navier boundary data involving nonlinearities of critical growth with a bifurcation parameter $λ$ in $\mathbb{R}^{2m}$. We establish here the lower and upper bound for $λ$ which determine multiplicity and non-existence respectively.
Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
本文研究带有非齐次微分算子的拟线性椭圆方程在凹-凸非线性项下解的多重性，利用Nehari流形和纤维化映射分析，证明方程至少存在一个正能量解和一个作为基态解的负能量解。
We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.
We construct multibump nodal solutions of the elliptic equation $$ -Δu=a^+[λu+ f(\, \cdot\,, u)]-μa^- g(\, \cdot\,, u) $$ in $H^1_0(Ω)$, when $μ$ is large, under appropriate assumptions, for $f$ superlinear and subcritical and such that the eigenvalues of the associated linearized operator on $H^1_0(\{x\inΩ:\: a(x)>0\})$ at zero, $u\longmapsto u-λ(-Δ)^{-1}(a^+ u)$, are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in $H^1_0(Ω)$.
In this note, we deal with a problem of the type $$\cases {-h\left ( \int_Ω|\nabla u(x)|^2dx\right ) Δu=f(u) & in $Ω$\cr & \cr u_{|\partialΩ}=0\ .\cr}$$ As an application of a new general multiplicity result, we establish the existence of at least three solutions, two of which are global minima of the related energy functional. The only condition assumed on $f$ is that it be subcritical and superlinear: no condition on the behaviour of $f$ at $0$ is requested.
In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -Δu+λ_1u=μ_1 |u|^{p-2}u+βr_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -Δv+λ_2v=μ_2 |v|^{q-2}v+βr_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $β$, we obtain the existence of the positive n…
Two nontrivial solutions for a nonhomogeneous quasilinear elliptic system with sign-changing weight functions
该文研究带变号权函数和凹凸非线性项的非齐次拟线性椭圆系统，利用山路引理和Ekeland变分原理，证明系统存在正、负能量各一个非平凡解。
In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional $(p,q)$-problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish new Hopf type maximum principle and strong comparison principle in both situations.
Mass-subcritical Half-Wave Equation with mixed nonlinearities: existence and non-existence of ground states
本文研究带散焦质量次临界扰动的聚焦质量次临界半波方程，证明存在临界质量决定约束极小元的存在性，并在一维情形下证明基态轨道稳定。
Least energy solutions for affine $p$-Laplace equations involving subcritical and critical nonlinearities
本文研究含仿射$p$-拉普拉斯非局部算子的两类椭圆问题，探讨其最小能量型正$C^1$解的存在性与不存在性，分析了由能量泛函非凸及不等式严格性带来的困难。
We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $λ>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $λ>0$ varies. In the sem…
In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find …
Critical quasilinear elliptic systems with convex subcritical perturbations
本文研究由p-拉普拉斯算子驱动的临界拟线性椭圆系统，通过变分方法和非线性特征值理论，证明了在适当维数下对任意参数存在非平凡解，并利用抽象扰动定理处理非齐次情形得到两个解。
In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev ve…
Local well-posedness for a class of semilinear Moore-Gibson-Thompson equations with subcritical nonlinearities
本文研究一类由Moore-Gibson-Thompson模型启发的高阶半线性发展方程，利用扇形算子与分数幂空间，在次临界非线性下证明了半线性Cauchy问题的局部适定性，并阐明了外推空间与分数域的作用。
Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data
本文研究带测度数据的分数阶 $p$-Laplace 型非局部方程 $Lu\pm g(u)=\mu$，给出其可解性的充分条件，并在幂非线性情形用 Bessel 容量刻画。
We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.
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 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.
In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -Δu+u=(I_α\ast|u|^{p})|u|^{p-2}u+λ|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq 3,\ α\in(0,N)$, $λ>0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+α}{N}$ or $\frac{N+α}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_α$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasi…
We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of the region where the nonlinearity is localized that imply, respectively, existence or nonexistence of ground states. In order to obtain these results we adapt to the context of metric graphs some classical techniques from the Calculus of Variations.
Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities
本文研究双曲空间上带临界和次临界非线性的椭圆方程，建立了当λ<(N-1)^2/4且p=2^*-1时非径向变号解的存在性与多重性，并证明了当λ>(N-1)^2/4时一大类对称解的部分不存在性。
The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by Hirose and Ohta in 2007, the problem has remained open for years in the supercritical case. In Hadj Selem et al., the authors obtained interesting numerical computations suggesting that for some bifurcating parameter $λ$, the equation has many positive solutions that vanish at infinity. In this paper, we provide a proof to this claim by constructing an accountabl…
Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities
研究具有凹-凸非线性的次临界哈密顿椭圆系统，证明存在严格递减曲线划分参数区域，使得系统在曲线下方有两个解、曲线上有一个解、上方无解，并推广到对称情形。
This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on …
Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach
该研究采用全局分岔方法，探究有界区域中具一般非线性项的薛定谔方程指定质量正解的存在性等性质，推广了已有自治齐次非线性情形的结果，并得到相关轨道稳定性结论。
We consider the following Schrödinger equation $$ - \hslash ^2 Δu + V(x)u = Γ(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $Γ$ attains global maximum at the same point or $V$ attains global minimum and $Γ$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash>0$.
In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…
Bifurcation for indefinite weighted $p$-laplacian problems with slightly subcritical nonlinearity
本文研究带变号权重的超线性 $p$-laplacian 椭圆边值问题，证明两个主特征值处正解从平凡解分岔，并借助 Orlicz 空间处理紧性，将 Drabek 结果推广至轻微次临界非线性。
We consider the problem -Δu - g(u) = λu, u \in H^1(\R^N), \int_{\R^N} u^2 = 1, λ\in\R, in dimension $N\ge2$. Here $g$ is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the $L^2$-unit sphere, and we show the existence of infinitely many solutions.
We consider a class of nonlinear Schrodinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a small initial data. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that "shadows" the non…
In this paper, we look for solutions to the following coupled Schrödinger system \begin{equation*} \begin{cases} -Δu+λ_{1}u=α_{1}|u|^{p-2}u+μ_{1}u^{3}+ρv^{2}u & \text{in} \ \ \mathbb{R}^{N}, -Δv+λ_{2}v=α_{2}|v|^{p-2}v+μ_{2}v^{3}+ρu^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the additional conditions $\int_{\mathbb{R}^{N}}u^{2}dx=b^{2}_{1}$ and $\int_{\mathbb{R}^{N}}v^{2}dx=b^{2}_{2}.$ Here $b_1, b_2>0$ are prescribed, $N\leq3$, $μ_{1}, μ_{2}, α_{1},α_{2},ρ>0$, $p\in (2,4)$ and the frequencies $λ_{1},λ_{2}$ are unknown and will appear as Lagrange multipliers. In the on…
Uniform estimates for elliptic equations with Carathéodory nonlinearities at the interior and on the boundary
本文针对内部和边界同时具有Carathéodory非线性的略次临界椭圆问题，建立了弱解的显式一致先验估计，该估计以$H^1$范数的幂次表示，通过结合De Giorgi-Nash-Moser迭代、椭圆正则性和Gagliardo-Nirenberg插值不等式证明。
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…
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (ϕ1, ϕ2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allo…
Multiplicity and Nonrelativistic limit of Bound States of Nonlinear Dirac Equations on Noncompact Metric Graphs with Localized Nonlinearities
研究非紧度量图上具有局部非线性项的狄拉克方程在L^2约束下归一化解的多重性，并证明在光速趋于无穷时解收敛到非线性薛定谔方程的解，涵盖质量次临界、临界和超临界情形。
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case
针对径向数据，研究组合非线性薛定谔方程在阈值以下的全局适定性与散射，利用轮廓分解和集中紧性方法，在$d\leq4$时建立结果。
Uniform a priori estimates for slightly subcritical fractional problems
研究有界凸区域中分数阶半线性狄利克雷问题正弱解的一致\(L^\infty\)先验界，针对轻微次临界非线性项\(f(t) = t^q L(t)\)，通过特定结构条件防止质量集中，建立全局一致界，扩展了此类估计成立的非线性项类别。
Nondegeneracy and asymptotic behavior of type I blow-up for semilinear heat equations with general nonlinearities
该研究针对具有一般非线性项的半线性热方程I型爆破解，通过拟相似变换建立局部一致收敛性，在弱假设下证明内部爆破点归一化解的正下界，覆盖多种非线性项情形。
Positive solutions of elliptic systems with superlinear nonlinearities on the boundary
本文研究边界条件为超线性次临界的椭圆方程组，结合重缩放方法、度理论与分岔理论，证明了正弱解分支从无穷远处及零点分岔的存在性，并分析了分支行为及解的数量随参数的变化。
This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enoug…
In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schrödinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the correspondng solution of which blows up. This result partially gives a positive answer to the open problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we obtain the lower blow-up rate and concentration rate for the blow-up solutions in the s…
We study small data scattering of solutions to Nonlinear Klein-Gordon equations with suitable pure power nonlinearities, posed on $\mathbb{R}^d\times \mathcal{M}^k$ with $k\leq2$ and $d\geq1$ and $\mathcal{M}^k$ a compact Riemannian manifold. As a special case we cover the $H^1-$critical NLKG on $\mathbb{R} \times M^2$.
Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
研究直线上两族聚焦非线性Schrödinger方程在固定质量下基态的存在性与唯一性，针对幂次为次临界、临界及双重临界等情形给出了相应条件与结论。
Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities
本文针对含梯度非线性源项的非齐次 $(p,q)$-Laplacian 椭圆方程，利用 Ishii-Lions 技术与改进的 Bernstein 方法证明了有界解的常数性与梯度先验估计，从而建立 Liouville 型定理，推广了经典结果。
A Note on Sharpened Singular Adams-Type Inequalities
建立带奇异权重的尖锐Adams型不等式，证明改进的奇异集中紧致原理，并应用于含奇异指数增长的非齐次拟线性椭圆方程非平凡解的存在性。
Fractional p-Kirchhoff equation with Sobolev and Choquard singular nonlinearities
本文研究全空间上含双重非线性项与临界奇异权重的分数阶p-Kirchhoff方程，利用Cerami条件克服慢增长非线性项带来的紧性证明困难，得到非平凡弱解的存在性结果。
We look for solutions to a fractional Schrödinger equation of the following form $$ (-Δ)^{α/ 2} u + \left( V(x) - \fracμ{|x|^α} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$ is bounded and close-to-periodic potential and $- \fracμ{|x|^α}$ is a Hardy-type potential. We assume that $V$ is positive and $f$ has the subcritical growth but not higher than $|u|^{q-2}u$. If $μ$ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If $μ$ is negative we show that there is no grou…
In this note two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case, when the damping and the mass terms make both equations in some sense "wave-like". In the proof of the subcritical case an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case we employ a test function type method, that has been developed …
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable co…
High-Velocity Inverse Scattering for Nonlinear Schrödinger Equations with Spatially Dependent Nonlinearities
针对d≥3维具有空间依赖非线性的非线性薛定谔方程，通过引入适应高加速初始数据的移动框架，构造了伽利略加速生成的大入射态的散射算子，并利用高速渐近性推导了系数的X射线变换重建公式，证明了散射算子唯一确定非线性指数和空间系数。
Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator
本文研究了涉及对数Schrödinger算子的临界半线性非局部椭圆方程，证明了最小能量解的存在性和收敛性，采用变分方法和渐进行为分析。
The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous crit…
Local well-posedness for the nonlinear heat and Schrödinger equations with nonlinear Wentzell boundary conditions on time-dependent compact Riemannian manifolds
该研究针对时间依赖紧黎曼流形上带Wentzell型非线性动力边界条件的次临界半线性热方程与薛定谔方程，证明了其局部适定性，丰富了相关偏微分方程边界问题的适定性理论。
We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subc…
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