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.     

Multiple Sign-Changing Solutions for Quasilinear Equations of Bounded Quasilinearity

The existence of an infinite sequence of sign-changing solutions are proved for a class of quasilinear elliptic equations under suitable conditions on the quasilinear coefficients and the nonlinearity■ where ? ? R~N is a bounded domain with smooth boundary, and we use■ The main interest of this paper is for the case of bounded quasilinearity bij. The result is proved by an elliptic regularization method involving truncations of both u and the gradient of u.     

关于算子方程AXB－X＝C的解─—纪念C．Apostol

本文给出了算子方程ＡＸＢ－Ｘ＝Ｃ可解的若干充要条件，其中（Ａ，Ｂ）为下列情形之一：Ａ或Ｂ有闭值域；Ａ（Ｂ＊）有闭值域并且是单射或者相似于一个协亚正规算子并且Ｂ（Ａ＊）是单侧移位；Ａ＋（Ｂ（＊＋））幂有界，其值域Ｒ（Ａ＋）　Ｒ（Ａ）（Ｒ（Ｂ）　Ｒ（Ｂ＋）并且Ｂ（Ａ＊）是单侧移位；Ａ＝Ｕ＊且Ｂ＝Ｕ是Ｈａｒｄｙ空间上重数为１的单侧移位．而且，给出了解的表达式．     

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 $\epsilon \to 0$.     

Maximum Principles for a Class of Nonlinear Elliptic Boundary Value Problems

The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.     

Quasilinear elliptic equations with subquadratic growth

In this paper we consider nonlinear boundary value problems whose simplest model is the following:

(0.1)     

Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.     

Generalized G-convergence for quasilinear elliptic differential operators

Generalized G-convergence for a quasilinear elliptic differential equation is defined and studied. The equation describes heat conduction in the cores of large electric transformers. The coefficients of the equation depend on temperature and the corresponding differential operator is neither potential nor monotone. A theory which generalizes the classical G-convergence is proposed. The theory is applied to the homogenization of the quasilinear elliptic differential equation with periodic coefficients.     

Solvability for nonlinear elliptic equation with boundary perturbation

The solvability of nonlinear elliptic equation with boundary perturbation is consid- ered.The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.     

Nonexistence results for some nonlinear nonlocal elliptic inequalities with variable exponents

We provide sufficient conditions for the nonexistence of nontrivial nonnegative solutions for some nonlinear elliptic inequalities involving the fractional Laplace operator and variable exponents. The used techniques are based on the test function method. Copyright © 2016 John Wiley & Sons, Ltd.