Class Of Nonlinear Elliptic Problems Research Articles

On a class of elliptic equations with critical perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞, if N = 2, λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

A Constant Sign Theorem for a Class of Semi-linear Elliptic Equations

Many problems arising from physics and analytic and geometric properties of solutions to partial differential equations have strong connections with convexity. In the present manuscript, we use the level curve curvature formula to derive a differential inequality and prove a invariant sign property for a class of nonlinear elliptic problems.

Existence of solutions for a class of nonlinear elliptic problems with measure data in the setting of Musielak–Orlicz –Sobolev spaces

Existence of solutions for a class of nonlinear elliptic problems with measure data in the setting of Musielak–Orlicz –Sobolev spaces

Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.

Quasi-Newton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces

Abstract Quasi-Newton-type iterative solvers are developed for a wide class of nonlinear elliptic problems. The presented generalization of earlier efficient methods covers various nonuniformly elliptic problems arising, e.g., in non-Newtonian flows or for certain glaciology models. The robust estimates are reinforced by several examples.

Bifurcation for nonlinear elliptic problems with singular potentials

We study the bifurcation phenomena for a class of nonlinear elliptic problems with Hardy term and subcritical nonlinearity. Sufficient conditions for bifurcation are established. The existence of a positive solution is also obtained for an asymptotically linear problem.

On a class of nonlinear elliptic problems with obstacle

Abstract This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type - div ⁡ ( a ⁢ ( x , u , ∇ ⁡ u ) ) + H ⁢ ( x , u , ∇ ⁡ u ) = f {-\operatorname{div}(a(x,u,\nabla u))+H(x,u,\nabla u)=f} .

Nonlinear problems with unbounded coefficients and $$L^1$$ data

We consider a class of nonlinear elliptic problems whose prototype involves a coefficients matrix blowing up for a finite value m of the unknown u. Since datum is in $$L^1$$ , a suitable notion of renormalized solutions is introduced taking into account that u can reach the value m and the existence of such solutions is proved. We study the corresponding evolution problem as well.

Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation

Abstract We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations

In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to one or two. This topological instability is shown to result from a local deformation of the global bifurcation diagram associated with the corresponding elliptic problems. Such a deformation is shown to systematically occur via the creation of either a multiple-point or a new fold-point on this diagram when an appropriate small perturbation is applied to the nonlinear term. More precisely, it is shown that for a broad class of nonlinear elliptic problems, one can always find an arbitrary small perturbation of the nonlinear term, that generates a local S on the bifurcation diagram whereas the latter is e.g. monotone when no perturbation is applied; substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem. The rigorous proof of the latter instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for the applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily $C^1$ nonlinear terms, that prevent in particular the use of linearization techniques, and for which the family of semigroups may exhibit non-dissipative properties.

Combined perturbation effects for a class of nonlinear fourth-order Navier problems

We study a class of nonlinear elliptic problems with Navier boundary condition and involving the Laplace and the biharmonic operators. The main result of this paper establishes a sufficient condition for the existence of nontrivial weak solutions, in relationship with the values of a certain real parameter with respect to the principal eigenvalue of the Laplace operator.

Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.

A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems (Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937-955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461-1483.) is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

Existence of a renormalized solution of nonlinear degenerate elliptic problems

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta (u)-{\rm div}(a(x,Du)+F(u))\ni f $ in $\varOmega $ where $f\in L^{\infty }(\varOmega )$. The vector field $a(\cdot ,\cdot )$ is a Carathéodory functi

Existence and multiplicity of positive solutions for a class of nonlinear elliptic problems

We study the existence and multiplicity of nonnegative solutions for the nonlinear elliptic problem, -\Delta u+v(x)u=a(x)u^p+\lambda f(x,u) for x\in\Omega and u=0 on \partial\Omega, where \Omega is a bounded region in R^N, N>2, 1

Three Solutions for Inequalities Dirichlet Problem Driven byp(x)-Laplacian-Like

A class of nonlinear elliptic problems driven byp(x)-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

Multiplicity results for a class of elliptic problems with nonlinear boundary condition

This paper provides multiplicity results for a class of nonlinear elliptic problems under a nonhomogeneous Neumann boundary condition. We prove the existence of three nontrivial solutions to these problems which depend on the Fucik spectrum of the negative $p$-Laplacian with a Robin boundary condition. Using variational and topological arguments combined with an equivalent norm on the Sobolev space $W^{1,p}$ it is obtained a smallest positive solution, a greatest negative solution, and a sign-changing solution.

Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

We study a general class of nonlinear elliptic problems associated with the differential inclusion β(x,u)−div(a(x,∇u)+F(u))∋f, where f∈L1(Ω). The vector field a(⋅,⋅) is monotone in the second variable and satisfies a non-standard growth condition described by an x-dependent convex function that generalizes both Lp(x) and classical Orlicz settings. Using truncation techniques and a generalized Minty method in the functional setting of non-reflexive spaces we prove existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established. Sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem.

Infinitely many solutions for a class of nonlinear elliptic problems on fractals

We study the nonlinear problem Δu+a(x)u=λg(x)f(u) in V∖V0, u=0 on V0, where V is the Sierpiński gasket, V0 is its intrinsic boundary, Δ denotes the weak Laplace operator, λ is a positive parameter, and f has an oscillatory behaviour either near the origin or at infinity. In both cases, we establish the existence of infinitely many solutions, which either converge to zero or have larger and larger energies.

Three solutions to inequalities of Dirichlet problem driven by p(x)-Laplacian

A class of nonlinear elliptic problems driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential is considered. By applying the version of the nonsmooth three-critical-point theorem, the existence of three solutions to the problems is proved.