On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems

We study a quasilinear elliptic problem

Nonlinear eigenvalue problems for quasilinear equations in unbounded domains

We prove the existence of a solution of the nonlinear equation in IR^N and in exterior domains, respectively. We concentrate to the case when p ≥ N and the nonlinearity f(x, · ) is “superlinear” and “subcritical”.     

Some uniqueness results for a class of quasilinear elliptic eigenvalue problems

Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in aN-ball or an annulus without monotone assumptions on the nonlinear termf. It is also proved that there is no non-radial positive solution. Supported by the Youth Foundations of National Education Commuttee and the Committee on Science and Technology of Henan Province     

Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.     

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

We study the existence versus absence of nontrivial weak solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing lower order nonlinearities with potentials decaying to zero at infinity.     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

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.     

Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems

In this paper, a two-dimensional quasilinear elliptic problem of the form -divF(x,▽u)=g(x)$-\mathit{divF}(\mathbf{x}\text{,}▽u)=g(\mathbf{x})$ is considered. This problem is ill-conditioned and we therefore propose a modified iterative algorithm based on coupling of the Sobolev space gradient method and WEB-spline finite element method. Applying the preconditioned iterative method, which has been already provided by Farago and Karatson (2001) [1] reduces the our considered problem to a sequence of linear Poisson’s problems. Then the WEB-spline finite element method is applied to the approximate solution of these Poisson’s problems. In this sense, a convergence theorem is proved and the advantages of this technique than the gradient finite element method (GFEM) is also described. Finally, the presented method is tested on some examples and compared with GFEM. It is shown that the gradient WEB-spline finite element method gives better test results.     

Multiplicity result for a double eigenvalue quasilinear problem on unbounded domain

In this paper we prove a multiplicity result for a double eigenvalue quasilinear problem on unbounded domain with nonlinear boundary conditions. We use a recent Ricceri-type result of Bonanno [G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. TMA 54 (2003) 651–665]. This result completes some recent results obtained in this direction.     

Principal frequency and existence of solutions for quasilinear elliptic obstacle problems

In this paper, we are interested in the first eigenvalue of p-Laplacian and the relation between the first eigenvalue and the existence (or nonexistence) of nontrivial (positive) solution for quasilinear elliptic obstacle problems. Utilizing the fact that obstacle problem have consanguineous relations with corresponding equation, we get a simple approach to study the properties of solutions of obstacle problems, such as existence and nonexistence, regularity and stability, etc. In this paper we are mainly concerned with the existence and nonexistence.     

A unified approach to monotone iterative technique for quasilinear elliptic problems

A unified approach Tor the miinatone Itera!ive technique is discussed relative to quasilinear elliptic boundary value problems when the nonlinear term involved admits a splitting of the difference of two monotone functions. This setting includes several results in one framework and is applicable to a variety of nonlinear problems.     

W2,2 interior convergence for some class of elliptic anisotropic singular perturbations problems

In this paper, we deal with anisotropic singular perturbations of some class of elliptic problems. We study the asymptotic behavior of the solution in a certain second-order pseudo Sobolev space.     

Adaptive control of local errors for elliptic problems using weighted Sobolev norms

We develop an a posteriori error estimator which focuses on the local H¹ error on a region of interest. The estimator bounds a weighted Sobolev norm of the error and is efficient up to oscillation terms. The new idea is very simple and applies to a large class of problems. An adaptive method guided by this estimator is implemented and compared to other local estimators, showing an excellent performance. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1266–1282, 2017     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).