Quasilinear Elliptic Systems Research Articles

Existence of Multiple Solutions for a Quasilinear Elliptic System Involving Sign-Changing Weight Functions and Variable Exponent

Existence of Multiple Solutions for a Quasilinear Elliptic System Involving Sign-Changing Weight Functions and Variable Exponent

On quasilinear elliptic systems with growth conditions in Orlicz–Sobolev spaces

On quasilinear elliptic systems with growth conditions in Orlicz–Sobolev spaces

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.

Existence of solutions for a p-Laplacian system with a nonresonance condition between the first and the second eigenvalues

In this article, we study the existence of positive solutions for the quasilinear elliptic system −∆_p u(x) = f_1(x, v(x)) + h_1(x) in Ω,−∆_p v(x) = f_2(x, u(x)) + h_2(x) in Ω,u = v = 0 on ∂Ω,where f_i(x, s), (i = 1, 2) locates between the first and the second eigenvalues of the p-Laplacian. To prove the existence of solutions, we use a topological method the Leray-Schauder degree.

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.

Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

<p style='text-indent:20px;'>This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.</p>

Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips

<p style='text-indent:20px;'>By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b20">20</xref>] to the system, in which substantial differences with the single cases are presented.</p>

Existence of minimizers for a quasilinear elliptic system of gradient type

<p style='text-indent:20px;'>The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb R^N $\end{document}</tex-math></inline-formula> is an open bounded domain, <inline-formula><tex-math id="M2">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ A(x,t,\xi) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ B(x,t, {\xi}) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}^1 $\end{document}</tex-math></inline-formula>–Carathéodory functions on <inline-formula><tex-math id="M6">\begin{document}$ \Omega \times \mathbb R \times { \mathbb R}^{N} $\end{document}</tex-math></inline-formula> with partial derivatives <inline-formula><tex-math id="M7">\begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ a = {\nabla}_{\xi}A $\end{document}</tex-math></inline-formula>, respectively <inline-formula><tex-math id="M9">\begin{document}$ B_t = \frac{\partial B}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ b = {\nabla}_{{\xi}}B $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M11">\begin{document}$ g_1(x,t,s) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ g_2(x,t,s) $\end{document}</tex-math></inline-formula> are given Carathéodory maps defined on <inline-formula><tex-math id="M13">\begin{document}$ \Omega \times \mathbb R\times \mathbb R $\end{document}</tex-math></inline-formula> which are partial derivatives with respect to <inline-formula><tex-math id="M14">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ s $\end{document}</tex-math></inline-formula> of a function <inline-formula><tex-math id="M16">\begin{document}$ G(x,t,s) $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We prove that, even if the general form of the terms <inline-formula><tex-math id="M17">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ B $\end{document}</tex-math></inline-formula> makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space <inline-formula><tex-math id="M19">\begin{document}$ X $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.</p>

Quasilinear elliptic systems with nonlinear physical data

Using the theory of Young measures, we prove the existence of weak solutions to the following quasilinear elliptic system: A(u)=f(x)+div σ0(x,u), where A(u)=−div σ(x,u,Du) and f∈W−1LM¯(Ω;ℝm). This problem corresponds to a diffusion phenomenon with a source f in a moving and dissolving substance, where the motion is described by σ0.

Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

Abstract We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

Existence of solutions for a quasilinear elliptic system with variable exponent

We consider the following quasilinear elliptic system in a Sobolev space with variable exponent: [-text{div}(a(|Du|)Du)=f,] where $a$ is a $C^1$-function and $fin W^{-1,p'(x)}(Omega;R^m)$. We use the theory of Young measures and weak monotonicity conditions to obtain the existence of solutions.

Existence of solutions for a quasilinear elliptic system with local nonlinearity on ℝN

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system. By developing the Moser iteration technique, we obtain that system has a nontrivial solution (uλ, vλ) with ‖(uλ, vλ)‖∞ ≤ 2 for every λ large enough when the nonlinear term F satisfies some growth conditions only in a circle with center 0 and radius 4, and the families of solutions {(uλ, vλ)} satisfy that ‖(uλ, vλ) ‖ → 0 as λ → ∞. Moreover, because the interaction of u and v in elliptic system causes that the estimate of ‖u‖∞ cannot vary with ‖u‖, the conclusion for the elliptic system is weaker than the corresponding result for the quasilinear elliptic equation, which is given in the end as a comparison.

On a quasilinear Schrödinger-Poisson system

In this paper we study a quasilinear elliptic system coupled by a Schrödinger equation with p-Laplacian operator and a Poisson equation. Some scaling transformation and ingenious methods are applied to produce the bounded Palais-Smale sequences and the existence of nontrivial solutions for the system is obtained by the mountain pass theorem.

On a class of quasilinear elliptic systems

On a class of quasilinear elliptic systems

Local Regularity of Weak Solutions to Quasilinear Elliptic Systems with One-Sided Condition on Quadratic Nonlinearity in the Gradient

We consider a quasilinear elliptic system of equations with nondiagonal principal matrix and additional terms admitting quadratic nonlinearity in the gradient of the solution. Under a one-sided condition on the strongly nonlinear term, we study the local smoothness of the (possibly, unbounded) solution.

Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.

Existence and Multiplicity Results for a Class of Coupled Quasilinear Elliptic Systems of Gradient Type

Abstract The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \textup{(P)} { - div ⁡ ( A ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 - 2 ⁢ ∇ ⁡ u ) + 1 p 1 ⁢ A u ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 = G u ⁢ ( x , u , v ) in Ω , - div ⁡ ( B ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 - 2 ⁢ ∇ ⁡ v ) + 1 p 2 ⁢ B v ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 = G v ⁢ ( x , u , v ) in Ω , u = v = 0 on ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1% }-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(% x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{% p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle% \phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is an open bounded domain, p 1 {p_{1}} , p 2 > 1 {p_{2}>1} and A ⁢ ( x , u ) {A(x,u)} , B ⁢ ( x , v ) {B(x,v)} are 𝒞 1 {\mathcal{C}^{1}} -Carathéodory functions on Ω × ℝ {\Omega\times\mathbb{R}} with partial derivatives A u ⁢ ( x , u ) {A_{u}(x,u)} , respectively B v ⁢ ( x , v ) {B_{v}(x,v)} , while G u ⁢ ( x , u , v ) {G_{u}(x,u,v)} , G v ⁢ ( x , u , v ) {G_{v}(x,u,v)} are given Carathéodory maps defined on Ω × ℝ × ℝ {\Omega\times\mathbb{R}\times\mathbb{R}} which are partial derivatives of a function G ⁢ ( x , u , v ) {G(x,u,v)} . We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional 𝒥 {{\mathcal{J}}} , related to problem (P), admits at least one critical point in the “right” Banach space X. Moreover, if 𝒥 {{\mathcal{J}}} is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition, a “good” decomposition of the Banach space X and suitable generalizations of the Ambrosetti–Rabinowitz Mountain Pass Theorems.

Generalizations of Stampacchia Lemma and applications to quasilinear elliptic systems

We present two generalizations of the classical Stampacchia Lemma. As applications, we consider Dirichlet problem for elliptic systems of the form −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,β(x,u(x))∂uβ(x)∂xj=fα,in Ω,u(x)=0,on ∂Ω.Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive some global regularity results.

Multiple solutions for a class of nonlocal quasilinear elliptic systems in Orlicz\u2013Sobolev spaces

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.