Solutions Of Nonlinear Elliptic Problems Research Articles

Multiplicity of asymmetric solutions for nonlinear elliptic problems

In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem: \[ { − Δ u + ( λ − h ( x ) ) u = ( 1 − f ( x ) ) u p , a m p ; x ∈ R N , u ( x ) > 0 , a m p ; x ∈ R N , u ∈ H 1 ( R N ) , \begin {cases} -\Delta u+(\lambda -h(x))u=(1-f(x))u^p, & x\in \mathbb {R}^N,\\ u(x)>0, &\quad x\in \mathbb {R}^N,\\ u\in H^1(\mathbb {R}^N), \end {cases} \] where λ > 0 \lambda >0 is a parameter, h ( x ) h(x) and f ( x ) f(x) are nonnegative radially symmetric functions in L ∞ ( R N ) L^\infty (\mathbb {R}^N) , h ( x ) h(x) and f ( x ) f(x) have compact support in R N \mathbb {R}^N , f ( x ) ≤ 1 f(x)\leq 1 for all x ∈ R N x\in \mathbb {R}^N , 1 > p > + ∞ 1>p>+\infty for N = 1 , 2 N=1,2 , 1 > p > N + 2 N − 2 1>p>\frac {N+2}{N-2} for N ≥ 3 N\geq 3 . We prove that for any k = 1 , 2 , … k=1,2,\,\ldots \, , if λ \lambda is large enough the above problem has positive solutions u λ u_\lambda concentrating at k k distinct points away from the origin as λ \lambda goes to ∞ \infty .

Degree theoretic methods in the study of positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian differential operator and with a nonsmooth potential (hemivariational inequalities). The hypotheses, in the case $p=2$ (semilinear problems), incorporate in our framework of analysis the so-called asymptotically linear problems. The approach is degree theoretic based on the fixed-point index for nonconvex-valued multifunctions due to Bader [3].

Preconditioning operators and Sobolevgradients for nonlinear elliptic problems

A preconditioning framework is presented for the iterative solution of nonlinear elliptic problems based on the preconditioning operator approach. Various fixed preconditioning operators are used in the iteration, which can be interpreted as a weighted Sobolev gradient method.

Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures

The discrete maximum principles are proved for finite element solutions of some nonlinear elliptic problems with mixed boundary conditions. The effect of quadrature rules, used for the construction of the stiffness matrices, is taken into account.

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.

On the Morse indices of sign changing solutions of nonlinear elliptic problems

On the Morse indices of sign changing solutions of nonlinear elliptic problems

On weighted estimates of solutions of nonlinear elliptic problems

The paper is devoted to the estimate \vert u(x,k)\vert\leq K\vert k\vert\left\{\mathop cap\nolimits_{p,w}(F)\frac{\rho^p}{w(B(x,\rho))}\right\} ^{\frac1{p-1}}, $2\<p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain ${B(x_0,1)\setminus F}$, $F\subset B(x_0,d)=\{x\in\Bbb R^n |x_0-x|<d\}$, $d<\frac12$, under the boundary-value conditions $u(x,k)=k$ for $x\in\partial F$, $ u(x,k)=0$ for $x\in\partial B(x_0,1)$ and where $0<\rho\leq\mathop dist(x,F)$, $w(x)$ is a weighted function from some Muckenhoupt class, and $\mathop cap_{p,w}(F)$, $w(B(x,\rho))$ are weighted capacity and measure of the corresponding sets.

An Approach to The Numerical Verification of Solutions for Nonlinear Elliptic Problems With Local Uniqueness

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kaiitorovich type. Numerical examples are presented.

On the effect of domain topology in a singular perturbation problem

where Ω ⊂ R (N ≥ 2) is a smooth bounded domain, 1 0 is a positive small parameter. Our interest in (1.1) arises from two aspects. First, (1.1) is a typical singular perturbation problem. Singular perturbation problems have received much attention lately due to their significances in applications such as chemotaxis (see [18] and [19]), population dynamics (see [1], [16]) and chemical reaction theory (see [1]), etc. Secondly, we are interested in the effect of the properties of the domain, such as geometry, topology on the solutions of nonlinear elliptic problems. Problem (1.1) can be a prototype. Recently, the geometry of the domain on the solutions of (1.1) has been a subject of study. Beginning in [20], Ni and Wei studied the “least-energy solutions” of (1.1) and showed that for e sufficiently small, the least-energy solution has only one local maximum point Pe and Pe must lie in the most centered part of Ω, namely, d(Pe, ∂Ω)→ maxP∈Ω d(P, ∂Ω), where d(P, ∂Ω) is the distance from

Numerical Verification of Solutions for Nonlinear Elliptic Problems Using anL∞Residual Method

We consider a numerical enclosure method with guaranteedL∞error bounds for the solution of nonlinear elliptic problems of second order. By using an a posteriori error estimate for the approximate solution of the problem with a higher orderC0-finite element, it is shown that we can obtain the guaranteedL∞error bounds with high accuracy. A particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation, but it follows as the result of computation itself. A numerical example that confirms the effectiveness of the method is presented.

Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN

We consider the following problem:wherefor x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.

SOLUTIONS OF A NONLINEAR DIRICHLET PROBLEM IN WHICH THE NONLINEAR PART IS BOUNDED FROM ABOVE AND BELOW BY POLYNOMIALS

In this paper we study the existence and multiplicity solutions of nonlinear elliptic problem of the form Here Ω is a smooth and bounded domain in RN , N ≥ 2, λ ∈ R and f : R → R is a continuous, even function satisfying the following condition for some c 1, c 2, c 3, p, α ∈ R, c 1, c 2, c 3, α &gt; 0 and p &gt; 1+ α. We shall show that, for λ ∈ R, g ∈ Lr (Ω) if N = 2, r &gt; 1, p &gt; 1 + α or the above problem has solutions. Assuming additionally that, λ ≤ λ1 and f is decreasing for t ≤ 0, we shall show that, this problem have exctly one solution. We take advantage of the fact, that a continuous, proper and odd (injective) map of the form I + C (where C is compact) is suriective (a homeomorphism).

Multiplicity of positive and nodal solutions for nonlinear elliptic problems in ℝN

We are concerned with the multiplicity of positive and nodal solutions of{−Δu+μu=Q(x)|u|p−2uinℝNu∈H1(ℝN) where 2<p<2NN−2, N ≥ 3, μ > 0, Q ∈ C(ℝN) and Q(x) ≥ 0 for x ∈ ℝN. We show how the “shape” of the graph of Q(x) affects the number of both positive and nodal solutions.

Positive solutions of nonlinear elliptic problems approximating degenerate equations

where Ω is a smooth domain in R , N ≥ 3, and p ≥ 2N/(N − 2) (the critical Sobolev exponent for the embedding H 0 (Ω) ↪→ L(Ω)). Many papers have been devoted to such problems (see [2], [5], [6], [10]–[12], [14], [15], [18]–[23], [25], [27], [28], and the references therein). Here the lack of compactness, due to the presence of the critical exponent, is just associated with concentration phenomena and, when it is possible to overcome the difficulties due to the lack of compactness, one can often obtain multiplicity results for positive solutions.

On averaging of nonlinear elliptic problems with inhomogeneous boundary conditions

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.

The symmetry of positive solutions of periodic-parabolic problems

In this paper, we generalized a result of Gidas, Ni and Nirenberg (1979) on the forced symmetry of positive solutions of nonlinear elliptic problems to the periodic (in time) problem for nonlinear parabolic equations with Dirichlet boundary conditions.

Multiplicity Results for Semilinear Elliptic Problems at Resonance and with Jumping Nonlinearities

In the present paper, we develop a method to deal with multiple existence of solutions of nonlinear elliptic problems and give some multiple existence results for elliptic boundary value problems at resonance and problems with jumping nonlinearities.

Multiple solutions for a neumann problem in an exterior domain

In this paper, we study the existence of multiple solutions of nonlinear elliptic problems in an exterior domain with Neumann boundary conditions.

Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with “rich” topology

Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with “rich” topology