Singular Semilinear Elliptic Problems Research Articles

Singular semilinear elliptic problems on unbounded domains in [formula omitted

Singular semilinear elliptic problems on unbounded domains in [formula omitted

Strong solutions for singular Dirichlet elliptic problems

We prove an existence result for strong solutionsu∈W2,q(Ω)of singular semilinear elliptic problems of the form−Δu=g(⋅,u)inΩ,u=τon∂Ω,where1<q<∞,Ωis a bounded domain inRnwithC2boundary,0≤τ∈W2−1q,q(∂Ω),and withg:Ω×(0,∞)→[0,∞)belonging to a class of nonnegative Carathéodory functions, which may be singular ats=0and also atx∈Sfor some suitable subsetsS⊂Ω¯.In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.

Positive solution to singular semilinear elliptic problems

In this paper, we study the existence and regularity of a non-trivial weak solution to the following semilinear elliptic problem with singular weights and singular nonlinearity: under assumptions and .

Singular semilinear elliptic problems with asymptotically linear reaction terms

We consider the problem { − Δ u = λ K ( x ) f ( u ) in B 1 c , u = 0 on ∂ B 1 , u ( x ) → 0 as | x | → ∞ , where B 1 c = { x ∈ R n | | x | > 1 } , n > 2 , λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter λ . Existence results to similar problems in R n are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions.

Identification of anti-aging natural products from southwestern Indian Ocean marine sponges and their microbial symbionts

In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem where F(x,s) is a Carathéodory function which can take the value +∞ when s=0. Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set {x∈Ω:u(x)=0}, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F(x,0)=0. Third, we consider the case where a zeroth order term μu, with μ a nonnegative bounded Radon measure which also belongs to H−1(Ω), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μu of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.

On the definition of the solution to a semilinear elliptic problem with a strong singularity at [formula omitted

In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem u≥0inΩ,−divA(x)Du=F(x,u)inΩ,u=0on∂Ω,where F(x,s) is a Carathéodory function which can take the value +∞ when s=0. Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set {x∈Ω:u(x)=0}, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F(x,0)=0. Third, we consider the case where a zeroth order term μu, with μ a nonnegative bounded Radon measure which also belongs to H−1(Ω), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μu of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.

Positive solutions of singular semilinear elliptic problems in NTA-cones

We study the existence, the uniqueness and the asymptotic behavior of positive solutions of the nonlinear equation \Delta v+ f(.,v)=0, in an NTA-cone in \mathbb R^n( n\geq 3 ), when a positive Borel measurable function f(.,.) is continuous and non-increasing with respect to the second variable and satisfies a certain condition related to a Kato class.

Some elliptic problems with singular nonlinearity and advection for Riemannian manifolds

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary condition. We prove uniform Lebesgue estimates and we determine the critical dimensions for these problems with nonlinearities of the type Gelfand, MEMS and power case. As an application, we show that extremal solutions are classical whenever the dimension of the manifold is below the critical dimension of the associated problem. Moreover, we analyze the branch of minimal solutions and we prove multiplicity results when the parameter is close to critical threshold and we obtain uniqueness on it. Furthermore, for the case of Riemannian models we study properties of radial symmetry and monotonicity for semi-stable solutions.

The moving plane method for singular semilinear elliptic problems

Abstract We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of the solutions, under general assumptions on the nonlinearity.

Nonnegative solutions to some singular semilinear elliptic problems

Nonnegative solutions to some singular semilinear elliptic problems

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

Abstract We prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equation - Δ ⁢ u = g ⁢ ( x ) u γ + h ⁢ ( x ) ⁢ f ⁢ ( u ) $-\Delta u=\frac{g(x)}{u^{\gamma}}+h(x)f(u)$ in bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

On a singular semilinear elliptic problem with an asymptotically linear nonlinearity

In this paper we deal with a singular elliptic problem involving an asymptotically linear nonlinearity and depending on two positive parameters. We investigate the existence, uniqueness and non-existence of the minima of the functional associated with the problem and, by employing a natural and very general definition of a weak solution, we also obtain a bifurcation-type result.

Positive Solutions to Singular Semilinear Elliptic Problems

We obtain a characterization of all locally bounded functions p ≥ 0 for which the equation (E) ∆u + p(x)ψ(u) = 0 has a positive solution in Ω vanishing on the boundary, where Ω is a domain of R N and ψ> 0 is a nonin- creasing continuous function on )0, ∞(. In particular, for Ω = R N with N ≥ 3, it is shown that (E) has a (unique) positive solution in R N which decays to zero at infinity if and only if the set { p> 0} has positive Lebesgue measure and lim |x|→∞ � RN p(y)|x − y| 2−N dy =0 .

Weak solution of a singular semilinear elliptic problem

We study the singular semilinear elliptic equation $\Delta u + f(.,u)$ $= 0$ in ${\cal D}'({\mathbb R}^N)$, $N \geq 3$. $f: {\mathbb R}^N \times (0,\infty) \to [0,\infty)$ is such that $f(.,u) \in L^1({\mathbb R}^N)$ for $u > 0$ and $u \to f(x,u)$ is continuous and nonincreasing for a.e. $x$ in ${\mathbb R}^N$. We assume that there exists a subset $\Omega \subset {\mathbb R}^N$ with positive measure such that $f(x,u) > 0$ for $x \in \Omega$ and $u > 0$ and that $\int_{\rl^N}f(x,c|x|^{2-N})dx < \infty$ for some $c > 0$. Then we show that there exists a unique solution $u$ in the Marcinkiewicz space $M^{N/(N-2)}({\mathbb R}^N)$ such that $\Delta u \in L^1({\mathbb R}^N)$,$u > 0$ a.e. in ${\mathbb R}^N$.

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.

Existence of solution for a singular critical elliptic equation

In this paper, a singular semilinear elliptic problem involving the critical Sobolev exponent is studied by variational method, the existence of a solution is proved under certain conditions. The Hardy inequality is used and plays an important role in the discussion.

Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem

The following singular elliptic boundary value problem is studied: Δ u+λu −γ +u p =0 in Ω, u>0 in Ω, u=0 on ∂Ω, where Ω⊂R N (N⩾3) is a bounded domain with smooth boundary ∂Ω , 0<γ<1<p⩽ N+2 N−2 , and λ >0 is a real parameter. The existence, multiplicity and asymptotic behavior (as p →1) of solutions of this equation are discussed by combining variational and sub-supersolution methods.

A Remark on the Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

It is proved that the singular semilinear elliptic equation −Δ u = p ( x ) g ( u ), 0 ≤ p ( x ), x ∈ R n , 2 ≤ n , lim s → 0 + g ( s ) = +∞, and g ∈ C 1 ((0, ∞), (0, ∞)) which is strictly decreasing in (0, ∞), has a unique positive C 2 + α loc ( R n ) solution that decays to zero near ∞ provided ∫ ∞ 0 t ϕ( t ) dt < ∞, where ϕ( t ) = @ imax @ i | x | = t g ( x ).

Classical and Weak Solutions of a Singular Semilinear Elliptic Problem

The singular semilinear elliptic equation Δ u + p ( x ) f ( u ) = 0 is shown to have a unique positive classical solution in R n that decays to zero at ∞ if p ( x ) is simply a nontrivial nonnegative continuous function satisfying ∫ ∞ 0 t max | x | = t p ( x ) dt < ∞, provided f is a non-increasing continuously differentiable function on (0, ∞). It is also shown that the equation has a unique weak H 1 0 -solution on a bounded domain provided ∫ ε 0 f ( s ) ds < ∞ and p ( x ) ∈ L 2 .

Entire Solution of a Singular Semilinear Elliptic Problem

The authors prove that the singular semilinear elliptic equation Δ u + p ( x ) u −γ =0, γ>0, p ( x )≥0, x ∈ R n , n ≥3, has a unique positive C loc 2+α ( R n ) solution that decays to zero near ∞ provided ∫ 0 ∞ t φ( t ) dt <∞, where φ( t )=max | x |= t p ( x ). Furthermore, they show that this condition on p is nearly optimal.