Nonselfadjoint semilinear elliptic boundary value problems

Summary In this paper we study the existence of solutions of nonselfadjoint semilinear elliptic boundary value problems with a bounded nonlinear term. We emphasize that this nonlinear term may depend on the derivatives of the function in a nontrivial way. In the proof of our main result we use the Leray-Schauder degree theory.Supported in part by the C.A.I.C.Y.T., Ministry of Education (Spain), under Grant no. 3258/83.     

Multiple solutions for semilinear elliptic boundary value problems with double resonance

In this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of −Δ with zero Dirichlet boundary data. The methods we use here are Morse theory, minimax methods and bifurcation theory.     

Multiple solutions of semilinear degenerate elliptic boundary value problems

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super‐sub‐solution method in the degenerate case, and is distinguished by the extensive use of an L^p Schauder theory elaborated for second‐order, elliptic differential operators with discontinuous zero‐th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub‐ and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications

Assume that I is the functional defined on the Hilbert space concerning the problem: in and u on , where f is sublinear at and superlinear at 0, that is, , and is the first eigenvalue of in . Under very general conditions, I has at least two local minimizers u ₁ and u ₂ and one mountain pass point u ₃ , and . Assuming that u ₁ , u ₂ and u ₃ are the only three nontrivial critical points of I, we prove that the level set I _b is contractible for all . Using this conclusion, we extend one of Hofer's result concerning existence of four nontrivial solutions of the above problem to the case where I is not and the trivial critical point 0 may be degenerate. Since I is not , the local topological degree and the critical groups of u ₃ can not be clearly computed. The lack of topological information about 0 and u ₃ makes it impossible to use topological degree theory or Morse theory in obtaining the fourth nontrivial solution. To overcome these difficulties, we explore a new technique in this paper.     

Inverse boundary value problems for a class of semilinear elliptic equations

We show that in two dimensions the scalar coefficient a(x,p) of the semilinear elliptic equation Δu+u(x,u)=0 is uniquely determined by the Dirichlet to Neumann map of the equation on a bounded domain with smooth boundary.     

kth Order convergence for semilinear elliptic boundary value problems

The method of quasilinearization coupled with the method of upper and lower solutions, is applied to the semilinear elliptic boundary value problems. At the same time, the result of kth convergence for the semilinear boundary value problems is obtained via the idea of Taylors approximation.     

Four versus two solutions of semilinear elliptic boundary value problems

This paper concerns the existence of four (or six) solutions of semilinear elliptic boundary value problems provided that two disorderly solutions are known. The results are obtained under very generic conditions. Received: 26 August 2000 / Accepted: 23 February 2001 / Published online: 23 July 2001     

Generalized boundary value problems for semilinear elliptic equations in weighted function spaces

In a weighted L ₁-space, we prove the solvability of a boundary value problem for a semilinear elliptic equation of order 2m in a bounded domain for the case in which generalized functions with strong power-law singularities at isolated points and with finite-order singularities on the entire boundary are given on the boundary.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.     

Adaptive multilevel finite element approximations of semilinear elliptic boundary value problems

Summary. In this paper we consider two aspects of the problem of designing efficient numerical methods for the approximation of semilinear boundary value problems. First we consider the use of two and multilevel algorithms for approximating the discrete solution. Secondly we consider adaptive mesh refinement based on feedback information from coarse level approximations. The algorithms are based on an a posteriori error estimate, where the error is estimated in terms of computable quantities only. The a posteriori error estimate is used for choosing appropriate spaces in the multilevel algorithms, mesh refinements, as a stopping criterion and finally it gives an estimate of the total error. Received April 8, 1997 / Revised version received July 27, 1998 / Published online September 24, 1999     

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.