Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

1991 ◽  
Vol 118 (3-4) ◽  
pp. 237-270 ◽  
Author(s):  
Hans-Peter Heinz
Keyword(s):  
Nonlinear Boundary ◽  
Eigenvalue Problems ◽  
Real Hilbert Space ◽  
Point Theory ◽  
Elliptic Partial Differential Equations ◽  
Existence Results ◽  
Nonlinear Perturbations ◽  
Nonlinear Boundary Value Problems ◽  
Operator Equations ◽  
Nonlinear Eigenvalue Problems

SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

scholarly journals Existence results for nonlinear boundary value problems

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 609-618 ◽  
Author(s):  
Abdeljabbar Ghanmi ◽  
Samah Horrigue
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Nonlinear Boundary Value Problems ◽  
Krasnoselskii Fixed Point Theorem

In the present paper, we are concerned to prove under some hypothesis the existence of fixed points of the operator L defined on C(I) by Lu(t) = ?w0 G(t,s)h(s) f(u(s))ds, t ? I, ? ? {1,?}, where the functions f ? C([0,?); [0,?)), h ? C(I; [0,?)), G ? C(I x I) and (I = [0,1]; if ? = 1, I = [0,?), if ? = 1. By using Guo Krasnoselskii fixed point theorem, we establish the existence of at least one fixed point of the operator L.

scholarly journals Existence and bifurcation of solutions for Fredholm operators with nonlinear perturbations

1982 ◽  
Vol 86 ◽  
pp. 249-271 ◽  
Author(s):  
Yasuo Niikura
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Nonlinear Operator ◽  
Real Banach Space ◽  
Eigenvalue Problems ◽  
Nonlinear Perturbations ◽  
Fredholm Operators ◽  
Nonlinear Eigenvalue Problems ◽  
Bifurcation Of Solutions ◽  
Nonlinear Eigenvalue

In this paper we shall discuss nonlinear eigenvalue problems for the equations of the formwhere L is a linear operator on a real Banach space X with non-zero kernel, K(-) is a linear or nonlinear operator on X and M(·, ·) is an operator from X X R into X. Equations of the form (1) arise in various fields of physics and engineering.

scholarly journals Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian

2021 ◽  
Vol 11 (1) ◽  
pp. 304-320
Author(s):  
Said El Manouni ◽  
Greta Marino ◽  
Patrick Winkert
Keyword(s):  
Nonlinear Boundary ◽  
Eigenvalue Problems ◽  
Nonlinear Boundary Condition ◽  
Existence Results ◽  
Double Phase ◽  
Steklov Eigenvalue ◽  
Variational Tools ◽  
Steklov Eigenvalues ◽  
Truncation Techniques ◽  
Steklov Eigenvalue Problems

Abstract In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

Solutions for a Kirchhoff Equation with Weight and Nonlinearity with Subcritical and Critical Caffarelli–Kohn–Nirenberg Growth

2016 ◽  
Vol 59 (4) ◽  
pp. 925-944 ◽  
Author(s):  
Giovany M. Figueiredo ◽  
Mateus Balbino Guimarães ◽  
Rodrigo da Silva Rodrigues
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Kirchhoff Equation ◽  
Nonlinear Boundary Value Problems ◽  
Kirchhoff Type ◽  
Nonlinear Boundary Value

AbstractIn this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.

Some New Existence Results for a Class of Four Point Nonlinear Boundary Value Problems

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Nazia Urus ◽  
Amit K. Verma ◽  
Mandeep Singh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Iterative Scheme ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Point Boundary ◽  
Nonlinear Boundary Value Problems ◽  
Monotone Iterative

In this paper we consider the following class of four point boundary value problems—y"(x) = f (x, y), 0 less than x lessthan 1, y'(0) = 0, y(1) = 1y(1) + 2)7(2)’where 1, 2  0 lesstahn 1, 2 less than 1, and f (x, y), is continuous in one sided Lipschitz in y. We propose a monotone iterative scheme and show that under some sufficient conditions this scheme generates sequences which converges uniformly to solution of the nonlinear multipint boundary value problem.

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