Behavior of positive radial solutions for quasilinear elliptic equations

We establish a necessary and sufficient condition so that positive radial solutions to 0, \end{equation*}"> having an isolated singularity at , behave like a corresponding fundamental solution. Here, and are continuous functions satisfying some mild growth restrictions.

     

Multibump solutions for quasilinear elliptic equations

The current paper is concerned with constructing multibump type solutions for a class of quasilinear Schrödinger type equations including the Modified Nonlinear Schrödinger Equations. Our results extend the existence results on multibump type solutions in Coti Zelati and Rabinowitz (1992) [17] to the quasilinear case. Our work provides a theoretic framework for dealing with quasilinear problems, which lack both smoothness and compactness, by using more refined variational techniques such as gluing techniques, Morse theory, Lyapunov–Schmidt reduction, etc.     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equation

$\mathrm{\Delta }u+f(u)=0$

under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\mathrm{\Delta }u+u^p=0$ and $\mathrm{\Delta }u+e^u=0$.     

Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations

The existence and uniqueness of positive radial solutions of the equations of the type [IML0001] in B_R, p>1 with Dirichlet condition are proved for λ large enough and f satisfying a condition[IML0002] is non-decreasing on [IML0003] It is also proved that all the positive solutions in C₁ ⁰(B^R) of the above equations are radially symmetric solutions for f satisfying [IML0004] and λ large enough.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

On global bifurcation for quasilinear elliptic systems on bounded domains

General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of [Jacobo Pejsachowicz, Patrick J. Rabier, Degree theory for C¹ Fredholm mappings of index 0, J. Anal. Math. 76 (1998) 289-319] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.     

Mountain pass type solutions for quasilinear elliptic equations

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem where is a bounded domain in , , and the function is an increasing homeomorphism from onto . Under appropriate conditions on , , and the Orlicz-Sobolev conjugate of (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type. Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth

Asymptotic behaviour of small solutions of a class of singularly perturbed quasilinear elliptic Dirichlet problems with critical and supercritical growth is studied as ε→0.     

Existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case

In this paper, we consider a class of semilinear elliptic Dirichlet problems in a bounded regular domain with cylindrical symmetry involving concave-convex nonlinearities with supercritical growth. Using a new Sobolev embedding theorem and variational method, we show the existence of two positive solutions of the problem. Additionally, we study the quasilinear elliptic equation and obtain a similar result.     

R~N上一类拟线性椭圆型方程弱解的多重性

在R~N上研究一类拟线性椭圆型方程,借助不光滑泛函的临界点理论和山路引理.得到该问题具有无穷多个弱解.