Singular Quasilinear Elliptic Systems Research Articles

Singular quasilinear elliptic systems with gradient dependence

In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder’s fixed point theorem.

Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity

In the present paper, we investigate the following singular quasilinear elliptic system: where is an open‐bounded domain with smooth boundary, s1, s2 ∈ (0, 1), p1, p2 ∈ (1, + ∞), and α1, α2, β1, β2 are positive constants. We first discuss the nonexistence of positive classical solutions to system . Next, constructing suitable ordered pairs of subsolutions and supersolutions, we apply Schauder's fixed‐point theorem in the associated conical shell and get the existence of a positive weak solutions pair to , turn to be Hölder continuous. Finally, we apply a well‐known Krasnosel'skiı̆'s argument to establish the uniqueness of such positive pair of solutions.

Singular quasilinear convective elliptic systems in ℝ N

Abstract The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

Maximal domains of the $$(\lambda ,\mu )$$-parameters to existence of entire positive solutions for singular quasilinear elliptic systems

In this paper, we establish maximal domains on the real parameters $$\lambda ,\mu >0$$ to existence of $$C^{1}({\mathbb {R}}^{N})$$ -entire positive solutions for the quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u = \eta a(x)f_1(u) + \lambda b(x)g_1(u)h_1(v)~in ~ {\mathbb {R}}^N,\\ -\Delta _p v = \theta c(x)f_2(v) + \mu d(x)g_2(v)h_2(u)~in~ {\mathbb {R}}^N,\\ u, v > 0~in~ {\mathbb {R}}^N,~~ u(x), v(x) {\mathop {\longrightarrow }\limits ^{|x|\rightarrow \infty }} 0, \end{array} \right. \end{aligned}$$ where $$\Delta _p$$ is the $$p-$$ Laplacian operator with $$1< p< N $$ ( $$3\le N$$ ); $$0<a, b, c, d\in C({\mathbb {R}}^{N})$$ ; either $$\eta =\theta =1$$ or $$\eta =\lambda ,\theta =\mu $$ and $$f_i, g_i, h_i~(i=1,2)$$ are positive continuous functions that satisfy some technical conditions, which allow $$f_{i}$$ to behave in a singular way at 0 and $$g_ih_i$$ as a $$(p-1)$$ -superlinear term at 0 and infinity. The main difficulties in approaching our problem come from its non-variational structure, building ordered sub-supersolutions and from the lack of a well-defined spectral theory. Using appropriate truncation, a generalization of the first eigenvalue in $${\mathbb {R}}^{N}$$ and a priori estimates, we are able to prove our principal results.

Singular quasilinear elliptic systems involving gradient terms

In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.

Singular Quasilinear Elliptic Systems with (super-) Homogeneous Condition

In this paper we establish existence, nonexistence and regularity of positive solutions for a class of singular quasilinear elliptic systems subject to (super-) homogeneous condition. The approach is based on sub-supersolution methods for systems of quasilinear singular equations combined with perturbation arguments involving singular terms

Double critical surfaces of singular quasilinear elliptic systems

In this paper, we mainly study the asymptotic behaviors at the origin and infinity of positive weak solutions to quasilinear elliptic systems in RN, which involve strongly–coupled critical nonlinearities and different Hardy–type terms. One critical surface is found, above and below which the asymptotic properties at the origin of solutions are different. Another critical surface is also found, above and below which the asymptotic properties at the infinity of solutions are different. The conclusions are new even in the case of p=2.

Singular quasilinear elliptic systems in $${\mathbb {R}}^{N}$$RN

The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder's fixed point theorem.

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Weak solutions for singular quasilinear elliptic systems

We investigate the quasilinear elliptic system(Section.Display) where is a bounded and smooth domain, , . Under certain conditions imposed on the exponents, we obtain the existence and uniqueness of a weak solution (u, v) with u, . We also investigate the regularity of solution and determine the optimal range of for such regularity.

Singular quasilinear elliptic systems and Hölder regularity

We investigate the following singular quasilinear elliptic system, \begin{equation} \nonumber \tag{P} \left. \begin{alignedat}{2} - \Delta_p u & = \frac{1}{ u^{a_1} v^{b_1} }\, \quad\mbox{ in }\,\Omega ;\quad u\vert_{\partial\Omega} & = 0 ,\quad u > 0\quad\mbox{ in }\,\Omega , \\ - \Delta_q v & = \frac{1}{ v^{a_2} u^{b_2} }\, \quad\mbox{ in }\,\Omega ;\quad v\vert_{\partial\Omega} & = 0 ,\quad v > 0\quad\mbox{ in }\,\Omega , \end{alignedat} \quad \right\} \end{equation} where $\Omega$ is an open bounded domain with smooth boundary, $1 < p, q < \infty$, and the numbers $a_1, a_2, b_1, b_2 > 0$ satisfy certain upper bounds. We employ monotonicity methods in order to prove the existence and uniqueness of a pair of positive solutions to (P). While following a standard fixed point approach with ordered pairs of sub- and super solutions, we need to prove a new regularity result of independent interest for solution pairs to problem (P) in $C^{0,\beta}(\overline{\Omega})$ with some $\beta\in (0,1)$.

Symmetric solutions for singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents in a bounded symmetric domain. Based upon the symmetric criticality principle of Palais and variational methods, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the weighted functions and the parameters.

An existence result for a class of quasilinear singular competitive elliptic systems

In this paper we establish existence and regularity of positive solutions for a singular quasilinear elliptic system with competitive structure. The approach is based on comparison properties, a priori estimates and the Schauder’s fixed point theorem.

A quasilinear singular elliptic system without cooperative structure

In this article, we investigate the existence of positive solutions of a singular quasilinear elliptic system for which the cooperative structure is not required. The approach is based on the Schauder fixed point theorem combined with perturbation arguments that involve the singular terms.

Multiple Solutions for a Singular Quasilinear Elliptic System

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system −div(|x|−ap|∇u|p−2∇u) + f 1(x)|u|p−2 u = (α/(α + β))g(x)|u|α−2 u|v|β + λh 1(x)|u|γ−2 u + l 1(x), −div(|x|−ap|∇v|p−2∇v) + f 2(x)|v|p−2 v = (β/(α + β))g(x)|v|β−2 v|u|α + μh 2(x)|v|γ−2 v + l 2(x), u(x) > 0, v(x) > 0, x ∈ ℝN, where λ, μ > 0, 1 < p < N, 1 < γ < p < α + β < p* = Np/(N − pd), 0 ≤ a < (N − p)/p, a ≤ b < a + 1, d = a + 1 − b > 0. The functions f 1(x), f 2(x), g(x), h 1(x), h 2(x), l 1(x), and l 2(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.

Existence of multiple solutions for singular quasilinear elliptic system with critical sobolev-hardy exponents and concave-convex terms

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.

Singular quasilinear elliptic equations and Hölder regularity

We prove the Hölder regularity (Theorem 2.1) for weak solutions to singular quasilinear elliptic equations whose prototype is(P){−Δpu=K(x)uδ+g(x)in Ω;u|∂Ω=0,u>0in Ω, where Ω is an open bounded domain with smooth boundary, 1<p<∞, δ>0, K∈Lloc∞(Ω) satisfies 0⩽K(x)⩽const⋅dist(x,∂Ω)−ω for a.e. x∈Ω, 0<ω<1+(1−δ)(1−1p), and 0⩽g∈L∞(Ω). Theorem 2.1 together with the Schauder fixed point theorem can be used to obtain the existence of weak solutions to the singular quasilinear elliptic system (PS) described in the Introduction.

On positive solutions for a class of singular quasilinear elliptic systems

We study through the lower and upper-solution method, the existence of positive weak solution to the quasilinear elliptic system with weights { − div ( | x | − a p | ∇ u | p − 2 ∇ u ) = λ | x | − ( a + 1 ) p + c 1 u α v γ in Ω , − div ( | x | − b q | ∇ v | q − 2 ∇ v ) = λ | x | − ( b + 1 ) q + c 2 u δ v β in Ω , u = v = 0 on ∂ Ω , where Ω is a bounded smooth domain of R N , with 0 ∈ Ω , 1 < p , q < N , 0 ⩽ a < N − p p , 0 ⩽ b < N − q q , 0 ⩽ α < p − 1 , 0 ⩽ β < q − 1 , δ , γ , c 1 , c 2 > 0 and θ : = ( p − 1 − α ) ( q − 1 − β ) − γ δ > 0 , for each λ > 0 .