Singular Dirichlet Problem Research Articles

Positive solutions for a class of singular (p, q)-equations

Abstract We consider a nonlinear singular Dirichlet problem driven by the ( p , q ) \left(p,q) -Laplacian and a reaction where the singular term u − η {u}^{-\eta } is multiplied by a strictly positive Carathéodory function f ( z , u ) f\left(z,u) . By using a topological approach, based on the Leray-Schauder alternative principle, we show the existence of a smooth positive solution.

Solvability of solution of singular and degenerate fractional nonlinear parabolic Dirichlet problems

Solvability of solution of singular and degenerate fractional nonlinear parabolic Dirichlet problems

Kipriyanov–Beltrami Operator with Negative Dimension of the Bessel Operators and the Singular Dirichlet Problem for the $$B $$-Harmonic Equation

For the Kipriyanov operator $$\Delta _B$$ written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator $$\Delta _B$$ on the sphere and the corresponding spherical functions ( $$B $$ -harmonics) are introduced. The main properties of the operator $$ \Delta _B$$ on the sphere and the differential equation of $$B $$ -harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in $$\mathbb {R}^n $$ is given. The solution is obtained by the Fourier method in the form of Laplace series in $$B$$ -harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in $$\Delta _B $$ belong to the interval $$(-1,2/n-1) $$ , $$n\in \mathbb {N}$$ , $$n\geq 3 $$ .

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝn, g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$, b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

Abstract This paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models - △ ⁢ u ± λ ⁢ | ∇ ⁡ u | 2 u β = b ⁢ ( x ) ⁢ u - α {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}} , u > 0 {u>0} , x ∈ Ω {x\in\Omega} , u | ∂ ⁡ Ω = 0 {u|_{\partial\Omega}=0} , where Ω is a bounded domain with smooth boundary in ℝ N {\mathbb{R}^{N}} , λ > 0 {\lambda>0} , β > 0 {\beta>0} , α > - 1 {\alpha>-1} , and b ∈ C loc ν ⁢ ( Ω ) {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some ν ∈ ( 0 , 1 ) {\nu\in(0,1)} , and b is positive in Ω but may be vanishing or singular on ∂ ⁡ Ω {\partial\Omega} . Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

Positive solutions for a class of singular Dirichlet problems

We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter λ>0 and it need not satisfy the AR-condition. Having as our starting point the work of Diaz-Morel-Oswald (1987) [3], we show that there is a critical parameter value λ⁎ such that for all λ>λ⁎ the problem has two positive solutions, while for λ<λ⁎ there are no positive solutions. What happens in the critical case λ=λ⁎ is an interesting open problem.

Multiplicity of positive radial solutions of singular Minkowski-curvature equations

This paper is about a singular Dirichlet problem involving the mean curvature operator in Minkowski space. We get the existence of triple and arbitrarily many positive radial solutions by using the Leggett-Williams fixed point theorem. Recent results in the literature are significantly improved.

Continuity results for parametric nonlinear singular Dirichlet problems

AbstractIn this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameterλ&gt; 0 that was considered in [32]. Denoting bySλthe set of positive solutions of the problem corresponding to the parameterλ, we establish the following essential properties ofSλ:there exists a smallest element$\begin{array}{} u_\lambda^* \end{array}$inSλ, and the mappingλ↦$\begin{array}{} u_\lambda^* \end{array}$is (strictly) increasing and left continuous;the set-valued mappingλ↦Sλis sequentially continuous.

Radial Convex Solutions of a Singular Dirichlet Problem with the Mean Curvature Operator in Minkowski Space

In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

Abstract This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation det ⁡ D 2 ⁢ u = b ⁢ ( x ) ⁢ g ⁢ ( - u ) , u &lt; 0 , x ∈ Ω , u | ∂ ⁡ Ω = 0 , \operatorname{det}D^{2}u=b(x)g(-u),\quad u&lt;0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0, where Ω is a strictly convex and bounded smooth domain in ℝ N {\mathbb{R}^{N}} , with N ≥ 2 {N\geq 2} , g ∈ C 1 ⁢ ( ( 0 , ∞ ) , ( 0 , ∞ ) ) {g\in C^{1}((0,\infty),(0,\infty))} is decreasing in ( 0 , ∞ ) {(0,\infty)} and satisfies lim s → 0 + ⁡ g ⁢ ( s ) = ∞ {\lim_{s\rightarrow 0^{+}}g(s)=\infty} , and b ∈ C ∞ ⁢ ( Ω ) {b\in C^{\infty}(\Omega)} is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition on g which plays a crucial role in the boundary behavior of such solution.

Corrigendum to Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Corrigendum to Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Existence and boundary behavior of solutions to p-Laplacian elliptic equations

Under appropriate conditions on $b(x)$ and $g(u)$ , we consider the singular Dirichlet problems $-\Delta_{p} u=b(x)g(u)$ , $u>0$ , $x \in \Omega $ , $u\vert_{\partial \Omega }=0$ . These problems are shown to admit weak solutions, and we analyze their exact asymptotic behavior near the boundary. As the main tools, we use Karamata regular variation theory and the method of upper and lower solutions.

Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space

In this article, by using the Leggett–Williams’ fixed point theorem, we prove the existence of at least three positive radial solutions of the singular Dirichlet problem for the prescribed mean curvature equation in Minkowski space {div(∇v1−|∇v|2)+f(|x|,v)=0inΩ;v=0on∂Ω, and the corresponding one-parameter problem. Here Ω is a unit ball in RN.

Second order expansion for the solution to a singular Dirichlet problem

In this paper, we analyze the second order expansion for the unique solution near the boundary to the singular Dirichlet problem −▵u=b(x)g(u),u>0,x∈Ω,u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN,g ∈ C1((0, ∞), (0, ∞)), g is decreasing on (0, ∞) with lims→0+g(s)=∞ and g is normalized regularly varying at zero with index −γ (γ > 1), b∈Clocα(Ω) (0 < α < 1), is positive in Ω, may be vanishing or singular on the boundary and belongs to the Kato class K(Ω). Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

Well-posedness of the second-order linear singular Dirichlet problem

Abstract Conditions guaranteeing well-posedness of the problem u ' ' = p 0 ( t ) u + q 0 ( t ) ${u^{\prime \prime }=p_0(t)u+q_0(t)}$ , u ( a ) = 0 ${u(a)=0}$ , u ( b ) = 0 ${u(b)=0}$ , are established. Here p 0 , q 0 : ] a , b [ → ℝ ${p_0,q_0\colon ]a,b[\rightarrow \mathbb {R}}$ are locally Lebesgue integrable functions and may have singularities at t = a ${t=a}$ and t = b ${t=b}$ .

Lower and upper functions in a singular Dirichlet problem with ø-Laplacian

The paper investigates the Dirichlet problem with o-Laplacian of the form $$(\varphi (u'))' + f(t,u,u') = 0,u(0) = u(T) = 0.$$ An existence principle which can be used for problems where f(t, x, y) may have singularities at t = 0, t = T and also at x = 0, y = 0, is proved here. As an application of this principle, new conditions that guarantee the solvability of the above problem are found.

The second order expansion of solutions to a singular Dirichlet boundary value problem

In this paper, we mainly study the second order expansion of classical solutions in a neighborhood of ∂Ω to the singular Dirichlet problem −Δu=b(x)g(u)+λa(x)f(u), u>0, x∈Ω, u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, λ≥0. The weight functions b,a∈Clocα(Ω) are positive in Ω and both may be vanishing or be singular on the boundary. The function g∈C1((0,∞),(0,∞)) satisfies limt→0+⁡g(t)=∞, and f∈C([0,∞),[0,∞)). We show that the nonlinear term λa(x)f(u) does not affect the second order expansion of solutions in a neighborhood of ∂Ω to the problem for some kinds of functions b and a.

Asymptotic Behavior of Solutions to Singular Quasilinear Dirichlet Problem with a Convection Term

In this paper, we study the boundary behavior of solution to the singular Dirichlet problem 8 : div(jruj m 2 ru) = b(x)g(u) + jru(x)j q(m 1) ; x2 ; u > 0; x2 ; uj@ = 0; where is a bounded domain with smooth boundary in R N , 2 R;m > 1; 0 < q m=(m 1), lims!0+ g(s) = +1, and b 2 C () , which is non-negative on and may be vanishing on the boundary, mainly, we investigate the exact asymptotic behavior of solution to the above problem.

Some consequences of an existence result by Kiguradze and Partsvania for singular Dirichlet problems

A sharp theorem by Kiguradze and Partsvania ensures the existence of extremal solutions between given lower and upper solutions for singular Dirichlet problems. This paper has a twofold purpose: first, we present a new sufficient condition for one of Kiguradze and Partsvania’s assumptions, and we illustrate its applicability in the study of a new family of examples; second, we combine Kiguradze and Partsvania’s theorem with Heikkila’s iterative technique to obtain a new result on the existence of extremal solutions for a more general class of discontinuous and singular functional boundary value problems. In particular, our framework includes classical equations with delay (or advance), singularities with respect to the independent variable, and implicit functional boundary conditions.

The exact boundary behavior of the unique solution to a singular Dirichlet problem with a nonlinear convection term

In this paper we analyze the exact boundary behavior of the unique solution to the singular nonlinear Dirichlet problem −△u=b(x)g(u)+λ|∇u|q,u>0,x∈Ω,u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, q∈(0,2], λ≥0, g∈C1((0,∞),(0,∞)), lims→0+g(s)=∞, g is decreasing on (0,∞), and b∈Clocα(Ω), is positive in Ω, may be vanishing or singular on the boundary. We reveal that the nonlinear convection term λ|∇u|q does not affect the first expansion of the unique solution near the boundary to the problem.