Isabeau Birindelli, Giulio Galise, Fabiana Leoni, Filomena Pacella
We study the asymptotic behaviour of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.
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By means of a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$ recently obtained, we give some necessary conditions for locating the concentration points for a class of quasi-linear elliptic problems in divergence form. More precisely we show that the points where the concentration occurs must be critical, either in a generalized or in the classical sense, for a suitable ground state function.
Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $Ω$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$.
In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation $$ ε^{2s} (-Δ)^{s}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $ε$ is a positive parameter, $f$ has a subcritical growth, $V$ possesses a local minimum, $N > 2s,$ $s \in (0,1),$ and $ (-Δ)^{s}u$ is the fractional laplacian.
In this paper, we study the following fourth order elliptic problem $$ Δ^2 u=(1+εK(x)) u^{2^*-1}, \quad x\in \mathbb{R}^N $$ where $2^*=\frac{2N}{N-4}$,$N\geq5$, $ ε>0$. We prove that the existence of two peaks solutions for the above problem, if $K(x)$ has two critical points satisfying certain conditions, provided $ε$ is small enough.
Concentration analysis for elliptic critical equations with no boundary control: ground-state blow-up
本文研究带边界流形上临界非线性椭圆方程极小化型解的集中现象，在无边界条件下假设 $L^2$-集中，并揭示非齐次非线性促使集中点快速收敛。
We consider a reaction-diffusion-advection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competition system is also discussed.
The formation of codimension-one interfaces for multiwell gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extend related to the absence of known appropriate monotonicity formula. In this paper, we focus on the elliptic case in two dimensions, and introduce some methods which allow to circumvent the lack of monotonicity formula. This methods lead, as expected, to concentration on one-dimensional rectifiable se…
In this paper, we deal with the following singular perturbed fractional elliptic problem $ ε^{} (-Δ)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-Δ)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of $V$ as $ε$ goes to $0.$
In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \begin{align*} \Bigg\{\begin{split} & Δ^2_p u -Δ_p u + λV(x)|u|^{p-2}u = f(x,u)\,,\,x\in \mathbb{R}^N, & u \in W^{2,p}(\mathbb{R}^N), \end{split} \end{align*} where $λ>0$, $p>1, N>2p$ and $V\in C(\mathbb{R}^N, \mathbb{R}^+)$, $f\in C(\mathbb{R}^N \times \mathbb{R},\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $λ\rightarrow \infty$.
We investigate symmetry properties of solutions to equations of the form $$ -Δu = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best cons…
We revisit an argument due to Lesch (Topology 32 (1993), no. 3, 611-623) for proving the cobordism invariance of the index of Dirac operators on even-dimensional closed manifolds and combine this with recent work by the author (New York J. Math. 28 (2022), 705-772) to show vanishing results for the spectral flow for families of selfadjoint Fredholm realizations of elliptic operators in case the family is induced on the boundary by an elliptic operator on a compact space. This work is motivated by studying the behavior of the index of realizations of elliptic operators under cobordisms of stati…
This paper is concerned with the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of the evolution of surface water waves. We prove a mass concentration property for $H^{1}$-solutions, analogous to the one known for the $L^{2}$-critical nonlinear Schrödinger equation. We also prove a mass concentration result for $L^{2}$-solutions.
In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -ε^2Δu +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -ε^2Δv +c(x)v=a(x) |u|^{p-1}u &&\text{in } Ω\newline u>0, \ v>0 \text{ in } Ω, &\text{ and }\quad\frac{\partial u}{\partialν} = 0 = \frac{\partial v}{\partialν} &&\text{on }\partialΩ \end{aligned} \right. \end{equation} where $Ω$ is a smooth bounded domain in $R^n, n\geq 3$. The coefficients $a(x), b(x)$ and $c(x)$ are positive bounded smooth functions. We shall study the existence of point concentrating solutions and discuss …
We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter $1/\varepsilon$ in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a…
We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies obtained for linear systems without zero order term in bounded domains and quasilinear systems of special form in bounded and in a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic ho…
We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -Δv= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and $p_\eps={(m+1)+2\over(m+1)-2}-\eps$, $\eps>0.$ We prove the existence of positive and sign changing solutions of $(P_\eps)$ concentrating and blowing-up, as $\eps\to0$, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem $(P_\eps)$ into a nonhomogeneous problem in an…
For a domain $Ω\subset\dR^N$ we consider the equation $ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $Ω$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\inΩ$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground …
Infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs. II
本文基于第一部分的集中估计，研究圆盘上超临界半线性椭圆方程爆破解在原点附近的无限振荡现象，刻画渐近形状并给出分歧图无限振荡的充分条件，从而证明无穷多解的存在性。
Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance
本文将刘维尔型定理应用于无尺度不变性的椭圆系统以获得普适奇性和衰减估计，提出了基于 Pohozaev 恒等式等方法证明新定理，并综述改进了现有证明方法。
Concentration and oscillation analysis of positive solutions to semilinear elliptic equations with exponential growth in a disc
本文研究圆盘上具指数增长半线性椭圆方程正解的集中与振荡，在超临界情形 p>2 下发现无穷气泡序列并精确刻画其极限形态、能量和位置，进而证明解的无穷振荡及分岔图的无穷振荡现象。
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…
This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results when dealing with the parabolic-elliptic case are: local existence without smallness assumption on the initial density, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon criteria for the fully parabolic case is also given. The analysis is co…
Infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs. I
本文在圆盘上超临界半线性椭圆方程中建立无限集中与振荡估计，将作者先前结果推广到一般超临界情形，并通过伸缩与逐点技巧对爆破解的无限集中行为进行分类，揭示多重指数增长下的新行为。
We proove some inequalities concerning the product, sup * inf for some elliptic operators of order 2 and 4. Using those inequalities and the concentration phenomena we can describe the asymptotic behavior of those PDE solutions.
Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction
该研究针对带任意形状收缩自聚焦核的薛定谔方程，分析其正解与结点解的存在性和集中性，通过尺度变换得到对应非自治半线性椭圆方程的极限轮廓，并刻画了最小能量极限轮廓的对称性与衰减特性。
Quantization for sequences of blow-up solutions to an elliptic equation having nonlocal exponential nonlinearity
本文研究了具有非局部指数非线性的椭圆方程解序列的渐进行为，通过集中-紧致性替代方案和能量量化结果揭示了解的爆破现象。
We investigate the next Trudinger-Moser critical equations, \[ \begin{cases} -Δu=λue^{u^2+α|u|^β}&\text{ in }B,\\ u=0&\text{ on }\partial B, \end{cases} \] where $α>0$, $(λ,β)\in(0,\infty)\times(0,2)$ and $B\subset \mathbb{R}^2$ is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow--up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary cond…
Let $Ω$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partialΩ$. We consider the equation $ Δu + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,Ω$, under zero Dirichlet boundary condition, where $\varepsilon$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partialΩ$, which is non-degenerate, and along which a certain weighted average of sectional curvatures of $\partialΩ$ is negative. Under these assumptions, we prove existence of a sequence $\varepsilon=\varepsilon_j$ and a solution $u_{\varepsil…
This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray-Schauder degree.
We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u) = 0$ in a bounded open set $Ω\subset\mathbf{R}^n$. The vector-valued function $\mathcal{A}$ satisfies the standard ellipticity assumptions with a parameter $1<p<\infty$ and a $p$-admissible weight $w$. We show that arbitrary perturbations on sets of $(p,w)$-capacity zero of continuous (and certain quasicontinuous) boundary data $f$ are resolutive and that the Perron solutions for $f$ and such perturbations coincide. As a consequence, we prove that the Perron solu…
In this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result.
Some superlinear fourth order elliptic equations are considered. Ground states are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of nontrivial solutions are related to a suitable truncated equation.
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga-Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates i…
We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media leading to concentration of measure, we prove an almost sure and local uniform homogenization result with a rate of convergence in probability.
In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then…
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
Compensation phenomena for concentration effects via nonlinear elliptic estimates
本文研究点态与线性微分约束下场的补偿现象，通过非线性椭圆估计证明可积性增益取决于算子与锥，反驳了近期一个猜想，并扩展了补偿可积性理论。
We verify the existence of radial positive solutions for the semi-linear equation $$ -\,Δu=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a radial smooth potential. If $q$ is super-critical, namely $q>p^*$, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided $V(0)<0$. On the other hand, if $N/(N-2)<q<p^*$, we prove that this Problem has a radial solution behaving like a super-position of {\it …
In this paper we study the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Using a reduction method we prove that doubly symmetric positive solutions in a $2m$-dimensional ball must concentrate and blow up on $(m-1)$-spheres as the concentration parameter tends to infinity. We also consider axially symmetric positive solutions in a ball in $\mathbb{R}^N$, $N \geq 3$, and show that concentration and blow up occur on two antipodal points, as the concentration parameter tends to infinity.
We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of ne…
By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi-spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci-Serrin type identity plays an important role.
It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $Φ$-Laplacian operator. One of these solutions is obtained as ground state solution by applying the well known Nehari method. The semilinear term in the quasilinear equation is a concave-convex function which presents a critical behavior at infinity. The concentration compactness principle is used in order to recover the compactness required in variational methods.
We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e., integral) comparison for fractional elliptic equations of the type $(-Δ)^{s}u=f$ $(0<s<1)$ in a bounded domain $Ω$, equipped with homogeneous boundary conditions. The classical pointwise Talenti rearrangement inequality is recovered in the limit $s\rightarrow1$. Finally, explicit counterexamples constructed for all $s\in(0,1)$ highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.
Spike layered solutions for elliptic systems on Riemannian Manifolds
本文研究紧致无边黎曼流形上一类哈密顿椭圆方程组的最小能量临界点，证明当参数趋于零时解在标量曲率最大点处发生点集中。
This article concerns the fractional elliptic equations \begin{equation*}(-Δ)^{s}u+λV(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-Δ)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $λ>0$ is a parameter, $V\in C(\mathbb{R}^N)$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of nontrivial solutions. Moreover, the concentration of solutions is also explored on the set $V^{-1}(0)$ as $λ\to\infty$.
We derive in this note a high-order corrector estimate for the homogenization of a microscopic semi-linear elliptic system posed in perforated domains. The major challenges are the presence of nonlinear volume and surface reaction rates. This type of correctors justifies mathematically the convergence rate of formal asymptotic expansions for the two-scale homogenization settings. As main tool, we follow the standard approach by the energy-like method to investigate the error estimate between the micro and macro concentrations and micro and macro concentration gradients. This work aims at gener…
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in [18, Theorem 31]. Some interesting consequences are Lp regularity results and nonlocal energy estimates for solutions.
We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.
In this paper we consider nonautonomous elliptic operators ${\mathcal A}$ with nontrivial potential term defined in $I\times\mathbb R^d$, where $I$ is a right-halfline (possibly $I=\mathbb R$). We prove that we can associate an evolution operator $(G(t,s))$ with ${\mathcal A}$ in the space of all bounded and continuous functions on $\mathbb R^d$. We also study the compactness properties of the operator $G(t,s)$. Finally, we provide sufficient conditions guaranteeing that each operator $G(t,s)$ preserves the usual $L^p$-spaces and $C_0(\mathbb R^d)$.
On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to scaling, in all directions not parallel to that axis. Elliptic examples include generalized 3-body Hamiltonians at zero energy such as $Δ_x+V_0(x')+V(x)$ where $Δ_x$ is the Laplace operator on $\mathbb{R}^n_x=\mathbb{R}^{n-1}_{x'}\times\mathbb{R}_{x''}$, and $V_0$ and $V$ are potentials with at least inverse quadratic decay: this operator is approximately translati…
By means of a penalization scheme due to del Pino and Felmer, we prove the existence of single-peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness. We don't require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity. Compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the Pucci-Serrin variational identity plays an important role.
In this paper we consider the initial Neumann boundary value problem for a degenerate Keller--Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos--Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the moti…
Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance
本文针对无尺度不变性的超线性抛物问题，利用椭圆刘维尔定理、能量估计和分量比例化约化方法建立全空间与半空间中的刘维尔型定理，并将其应用于普适奇异性与衰减估计。
We study entire bounded solutions to the equation $Δu - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.
Recent developments on elliptic equations from composites
本文综述了过去三十年复合材料中椭圆方程和系统的最新进展，重点在于最优估计和精确渐进行为，同时指出了一些开放性问题。
Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system
本文研究有界光滑区域中弱耦合奇异摄动椭圆系统，证明在竞争机制下完全非平凡解的数量随方程数增加而增加，方法结合变分法、渐近分析、Lusternik-Schnirelman理论及构型空间范畴估计。
We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration ten…
In this paper, we are concerned with the existence and asymptotic behavior of minimizers for a minimization problem related to some quasilinear elliptic equations. Firstly, we proved that there exist minimizers when the exponent $q$ equals to the critical case $q^*=2+\frac{4}{N}$, which is different from that of \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends to the critical case $q^*$ when $a<a^*$ is fixed. Moreover, we studied the concentration behavior of minimizers as the exponent $q$ tends to the critical case $q^*$ for any fixed $a>a^*$.
Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method
本文研究了(p,N)-拉普拉斯方程在不连续非线性和惩罚方法下的解的存在性与集中现象，结合变分方法和Moser型迭代技术，揭示了临界增长下的解行为。
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