Multiple Solutions for Perturbed Elliptic Equations in Unbounded Domains

We give a short survey of some existence results concerning positive solutions of semilinear elliptic equations in unbounded domains. We consider only solutions which vanish at infinity. We also investigate the existence of “minimum-type” solutions as well as solutions given by “higher” critical points.

Monotone methods for semilinear elliptic equations in unbounded domains

We study the existence of positive solutions of the nonlinear equation Δu+f(⋅,u)=0, in D with u=0 on ∂D, where D is an unbounded domain in R2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.

The main focus in this chapter is on multi-valued quasilinear elliptic variational inequalities of the form in all of \(\mathbb {R}^N\) as well as in the exterior domain \(\Omega =\mathbb {R}^N\setminus \overline {B(0,1)}\), where only for simplifying our presentation \(\mathcal {A}: X\to X^*\) is given by the p-Laplacian, that is, \(\mathcal {A}=-\Delta _p\), see Section 6.3. The study of elliptic equations in unbounded domains and even more the study of multi-valued quasilinear elliptic variational inequalities in unbounded domains causes a number of additional difficulties, and therefore cannot be considered as just a straightforward extension of the bounded domain problems. That is why the multi-valued lower order term \(\mathcal {F}_a\) is of the form \(\mathcal {F}_a=aF(u)\) with \(a: \mathbb {R}^N\to \mathbb {R}\) (resp. \(a: \Omega \to \mathbb {R}\)) satisfying a certain decay property at infinity.

Soit Ω un domaine exterieur dans R N , N≥2, a frontiere lisse Γ=∂Ω et soeint #7B-D et B un operateur elliptique et un operateur frontiere. On etudie l'existence de solutions positives aux problemes aux valeurs limites suivants: −#7B-Du+c(x)u=λm(x)u dans Ω, Bu=0 sur Γ et −#7B-Du+c(x)u=λm(x)uγ dans Ω, Bu=0 sur Γ, ou c(x) et m(x) sont des fonctions donnees, λ est un parametre reel et γ est une constante non nulle avec γ¬=1

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: $$ -a_{ij}(x)\partial_{ij}u(x) -- q_i(x)\partial_iu(x)=f(x,u(x)),\quad x\in{\mathbb R}^N. $$ The aim is to prove uniqueness of positive- bounded solutions—Liouville-type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation.

It is known that in order for a solution of a boundary value problems for elliptic equations in unbounded domains to be unique, it is necessary to impose additional constraints on the behavior, of solutions at infinity. Such constrains may be qualitatively different.In [1] it was noticed that the assumption that the solutions should be positive for sufficiently large arguments may also serve as a condition that guarantees the uniqueness of solutions. The present report explains an investigation of the conditions on boundary value problems whose fulfillment guarantees the validity of such uniqueness theorems for the biharmonic equation. For harmonic functions that satisfy the Dirichlet or Neumann boundary conditions, there are no such uniqueness theorems. But they can exist for simplest nonlocal boundary value problems for harmonic functions.

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solutions. The global-in-time existence and uniqueness of a small-data strong solution is proved. For the proof of this result, we propose a new approach. Our approach is based on the operator treatment of the problem with the consequent application of a theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations.

We investigate the well-posedness of an initial boundary value problem for the Kelvin–Voigt–Brinkman–Forchheimer equations with memory and variable viscosity under a non-homogeneous Dirichlet boundary condition. A theorem about the global-in-time existence and uniqueness of a strong solution of this problem is proved under some smallness requirements on the size of the model data. For obtaining this result, we used a new technique, which is based on the operator treatment of the initial boundary value problem with the consequent application of an abstract theorem about the local unique solvability of an operator equation containing an isomorphism between Banach spaces with two kind perturbations: bounded linear and differentiable nonlinear having a zero Fréchet derivative at a zero element. Our work extends the existing frameworks of mathematical analysis and understanding of the dynamics of non-Newtonian fluids in porous media.

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem, 2013) and study numerically the minimization problem at low and mid-high frequencies. Numerical examples in some relevant cases are also shown.

We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

In this paper we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the auxiliary function that leads to a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains $(\Omega,\omega)$ that satisfy these sufficient conditions. Finally, when $\Omega\setminus \overline \omega$ is bounded, we prove the null controllability of the semilinear heat equation when the nonlinearity $f(y,\nabla y)$ grows more slowly than $|y|\log^{3/2}(1+|y|+|\nabla y|)+ |\nabla y|\log^{1/2}(1+|y|+|\nabla y|)$ at infinity (generally in this case in the absence of control, blow-up occurs). In this aim we also prove the linear null controllability problem with $L^\infty$-controls.

Asymptotic behaviour of solutions to the 2D stochastic Navier-Stokes equations in unbounded domains —new developments

The goal of this paper is to consider the asymptotic behavior of solutions of nonautonomous classical reaction-diffusion equations in unbounded domains with nonlinearity having a polynomial growth of arbitrary order. The existence and structure of a uniform attractor are obtained in the spaces L2(Rn) and Lp(Rn), respectively.