Class Of Semilinear Elliptic Equations Research Articles

Structure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity

We consider a semilinear elliptic equation with asymptotic linear nonlinearity applying bifurcation theory and spectral analysis. We obtain the exact multiplicity of the positive solutions and a very precise structure of the solution set, which improves the previous knowledge of the problem.

Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy–Sobolev critical exponents

Existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with Hardy term, Hardy–Sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques.

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence, uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations. A necessary and sufficient condition for the existence of positive solutions to problems is given. We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations, then they are still valid when one perturbs the differential operator a little bit. As consequences, some uniqueness results of positive solutions under the domain perturbation are also obtained.

Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

We establish that for n ⩾ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p = 0 in R n possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K ⩾ 0 in R n ∖ { 0 } , satisfies K ( x ) = O ( | x | σ ) at x = 0 for some σ > − 2 , and | x | 2 K ( x ) = c + O ( [ log | x | ] − θ ) near ∞ for some constants c > 0 and θ > 1 . The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in R n , J. Differential Equations 185 (2002) 225–250]. The crucial observation is that in the radial case of K ( r ) = K ( | x | ) , two fundamental weights, ( log r ) p p − 1 and r n − 2 ( log r ) − p p − 1 , appear in analyzing the asymptotic behavior of solutions.

On the existence of entire solutions to a class of semilinear elliptic equations on noncompact Riemann manifolds

On the existence of entire solutions to a class of semilinear elliptic equations on noncompact Riemann manifolds

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form $$\label{eq:abs} -\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\mathbb{R}^{2}$$ where \({a:\mathbb{R}\to\mathbb{R}}\) is a periodic, positive function and \({W:\mathbb{R}\to\mathbb{R}}\) is modeled on the classical two well Ginzburg-Landau potential \({W(s)=(s^{2}-1)^{2}}\) . We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W'(q(x))=0,\quad x\in\mathbb{R},\quad q(\pm\infty)=\pm 1,$$ has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions \({u(x,y)\to\pm 1}\) as \({x\to\pm\infty}\) uniformly with respect to \({y\in\mathbb{R}}\) .

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 that contains m disjoint large enough balls has m 2 2-nodal solutions and m positive solutions.

Exact multiplicity of solutions and S-shaped bifurcation curve for a class of semilinear elliptic equations

The set of steady state solutions to a reaction–diffusion equation modeling an autocatalytic chemical reaction is completely determined, when the reactor has spherical geometry, and the spatial dimension is n = 1 or 2 for any reaction order, or n ⩾ 3 for subcritical reaction order. Bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states.

Liouville-type results for semilinear elliptic equations in unbounded domains

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.

Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations

Abstract We consider a class of semilinear elliptic equations of the form −Δu(x,y) + a(εx)Wʹ(u(x,y)) = 0, (x,y) ∊ ℝ2 (0.1) where ε > 0, a : ℝ → ℝ is an almost periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W(s) = (s2 - 1)2. We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∊ ℝ.

Remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations

The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively.

Existence and uniqueness results for a class of elliptic equations with exponential nonlinearity

We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

Existence and uniqueness results for a class of elliptic equations with exponential nonlinearity

We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

Entire solutions in ${\mathbb{R}}^{2}$ for a class of Allen-Cahn equations

We consider a class of semilinear elliptic equations of the form 15.7cm - where , is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions as uniformly with respect to . We show via variational methods that if e is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

We study the existence of nontrivial solutions for the problem Δu = u, in a bounded smooth domain Ω ⊂ ℝℕ, with a semilinear boundary condition given by ∂u/∂ν = λu − W(x)g(u), on the boundary of the domain, where W is a potential changing sign, g has a superlinear growth condition, and the parameter λ ∈ ]0, λ1]; λ1 is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min‐max methods.

Semilinear elliptic equations and fixed points

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain Q ⊂ R N , N ≥ 3, with C 1,1 boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS

This paper deals with the Neumann problem for a class of semilinear elliptic equations −Δu+u=|u|2*−2u+μ|u|q−2u in Ω,∂u∂γ=|u|s*−2u on ∂ΩΣ, where 2*=2NN−2,S*=2(N−1)N−2,1<q<2,N≥3,μ>0, γ denotes the unit outward normal to boundary ∂ΩΣ. By variational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.

Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques

A class of semilinear elliptic equations with dependence on the gradient is considered. The existence of a positive and a negative solution is stated through an iterative method based on mountain-pass techniques.

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 Δ u+ u( x,∇ u)=0 is uniquely determined by the Dirichlet to Neumann map of the equation on a bounded domain with smooth boundary.

A note on the positive energy solutions for elliptic equations involving critical sobolev exponents

In this paper, we study the Neumann problem for a class of semilinear elliptic equations −Δu = |u| 2 ∗ −2u + μ|u| q−2u in Ω, ∂u ∂v = 0 on ∂Ω where Ω is the unit ball in R N centered at the origin, N 〉- 3, 2 ∗ = 2N/(N −2), 1 < q < 2, μ > 0 . By cutting the unit ball into angular sectors and using the variational method to deal with a mixed boundary value problem on such domains, we prove the existence of infinitely many nonradial solutions with positive energy for small μ > 0.