Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

Isolated singularities for weighted quasilinear elliptic equations

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|^p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b₀(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.     

Some remarks on Liouville type results for quasilinear elliptic equations

For a wide class of nonlinearities satisfying

0$\space in $(0,a)$\space and $f(u)<0$\space in $(a,\infty)$ ,}\end{displaymath}">

we show that any nonnegative solution of the quasilinear equation over the entire must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.

     

Morrey global bounds and quasilinear Riccati type equations below the natural exponent

We obtain global bounds in Lorentz–Morrey spaces for gradients of solutions to a class of quasilinear elliptic equations with low integrability data. The results are then applied to obtain sharp existence results in the framework of Morrey spaces for Riccati type equations with a gradient source term having growths below the natural exponent of the operator involved. A special feature of our results is that they hold under a very general assumption on the nonlinear structure, and under a mild natural restriction on the boundary of the ground domain.     

Multiplicity and asymptotic behavior of solutions for quasilinear elliptic equations with small perturbations

This paper considers the following general form of quasilinear elliptic equation with a small perturbation:$\{\begin{array}{c}?{\sum }_{i,j=1}^N{D}_j(a_{ij}(x,u)D_{i}u)+\frac{1}{2}{\sum }_{i,j=1}^N{D}_t{a}_{ij}(x,u)D_{i}u{D}_{j}u\hfill \\ =f(x,u)+\epsilon g(x,u),x\in \mathrm{\Omega },u\in H_0^1(\mathrm{\Omega }),\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 3)$ is a bounded domain with smooth boundary and $|\epsilon |$ small enough. We assume the main term in the equation to have a mountain pass structure but do not suppose any conditions for the perturbation term $\epsilon g(x,u)$. Then we prove the equation possesses a positive solution, a negative solution and a sign-changing solution. Moreover, we are able to obtain the asymptotic behavior of these solutions as $\epsilon \to 0$.     

A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

Two types of quasilinear elliptic equations with degenerate coerciveness and slightly superlinear growth

The existence of positive solutions for two types of quasilinear elliptic equations with degenerate coerciveness and slightly superlinear growth is established. Especially, we solve an open problem proposed in the literature (Mercaldo and Peral, 2008).     

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.     

Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions

We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem.Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.