Multiplicity of nodal solutions for elliptic problems involving non-odd nonlinearities

In this paper, we study the effect of domain shape on the number of 2-nodal solutions for the semilinear elliptic equation involving non-odd nonlinearities. We prove that a semilinear elliptic equation in an m$m$-bump domain (possibly unbounded) has m²$m^2$ 2-nodal solutions and we can find a least energy nodal solution in those solutions. Furthermore, we can describe the bump location of these solutions.     

The effect of domain shape on the number of positive and nodal solutions for semilinear elliptic equations

In this paper, we study the effect of domain shape on the number of positive and nodal (sign-changing) solutions for a class of semilinear elliptic equations. We prove a semilinear elliptic equation in a domain Ω$\Omega$ that contains m$m$ disjoint large enough balls has m²$m^2$ 2-nodal solutions and m$m$ positive solutions.     

Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H⁴ spaces using the fixed‐point technique where the elliptic equation contains fourth‐order differential operators with and without Fredholm property, generalizing the previous results.     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δ _p u = a(x)u ^m −b(x)f(u) with p >  1 and 0 <  m <  p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n , II

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.     

Four 2-nodal solutions for a semilinear elliptic equation in a finite strip with a hole

In this paper, we study the decomposition of the filtration of the Nehari manifold via the variation of domain shape. We use this result to prove that the semilinear elliptic equation in a finite strip with hole has at least four 2-nodal solutions (solutions with precisely two nodal domains). Furthermore, we can describe the bump location of these solutions.     

Finding multiple solutions to elliptic systems with polynomial nonlinearity

Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L² norm and H¹ norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D₄ symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.     

NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

On the existence of a solution to a class of variational inequalities

In this note we consider a class of semilinear elliptic variational inequalities on H ¹(Ω) space. With the aid of the mountain-pass principle and the Ekeland variational principle we prove the existence of solutions.     

Quasilinearization Method for Nonlinear Elliptic Boundary-Value Problems

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, L^p-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.     

On Uniqueness of Boundary Blow-Up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations Lu = u ₊ ^p with p > 1, or Lu = e ^au with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.     

A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function

In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (Eλ,μ) involving nonlinear boundary condition and sign-changing weight function. With the help of the Nehari manifold, we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ) belongs to a certain subset of R².     

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.     

Critical point theorems and applications to a semilinear elliptic problem

The objective of this article is to establish the existence of critical points for functionals of classC ²defined on real Hilbert spaces. The argument is based on the infinite dimensional Morse theory introduced by Gromoll-Meyer [13]. The abstract results are applied to study the existence of nonzero solutions for a class of semilinear elliptic problems where the nonlinearity possesses a superlinear growth on a direction of the real line.This research was partially supported by CNPq/Brazil     

Finite element methods for semilinear elliptic problems with smooth interfaces

The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface, optimal order error estimates in L ² and H ¹-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then an extension to the semilinear problem is also considered and optimal error estimate in H ¹ norm is achieved.     

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.     

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W ₂ ^m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L ₂(G), and defined for functions from the space W ₂ ^m (G) that satisfy homogeneous nonlocal conditions, is established.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 665–682.Original Russian Text Copyright ©2005 by P. L. Gurevich.     

A note on the radial symmetry of solutions with spherical nodal sets

We will investigate the radial symmetry of solutions with spherical nodal sets of semilinear elliptic equations.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.