Quasilinear Elliptic Systems Research Articles

Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis

Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.

Local boundedness under nonstandard growth conditions

This paper deals with local boundedness under nonstandard growth conditions by using De Giorgi's iteration method. We first consider quasilinear elliptic systems, and assume that the coefficients satisfying some boundedness and coercivity conditions, then we derive local boundedness of solutions. Secondly, we consider elliptic equations with a drift term, under some nonstandard growth conditions, we derive that solutions are locally bounded. Finally, we consider variational integrals with the integrand satisfy nonstandard growth conditions, local boundedness of minimizers is derived.

Quasilinear elliptic and parabolic systems with nondiagonal principal matrices and strong nonlinearities in the gradient. Solvability and regularity problems

We consider nondiagonal elliptic and parabolic systems of equations with strongly nonlinear terms in the gradient. We review and comment known solvability and regularity results and describe the last author’s results in this field.

Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System

This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, S μ 1 , μ 2 q 1 , q 2 − div q 1 x ∇ u p − 2 ∇ u = μ 1 u p − 2 u + α + 1 u α − 1 u v β + 1 in Ω − div q 2 x ∇ v p − 2 ∇ v = μ 2 v p − 2 v + β + 1 u α + 1 v β − 1 v in Ω u > 0 , v > 0 in Ω u = v = 0 on ∂ Ω , where Ω ⊂ ℜ N N ≥ 3 , 2 ≤ p < N , q 1 , q 2 ∈ W 1 , p Ω ∩ C Ω ¯ , α , β ≥ 0 , μ 1 , μ 2 , ≥ 0 and α , β > 0 satisfy α + β = p ∗ − 2 with p ∗ = p N / N − p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q 1 , q 2 near their minima and the dimension N . Moreover, we use the well known Pohozaev identity for prove the nonexistence result.

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

Multiple solutions for quasilinear elliptic systems involving variable exponents

We establish the existence of multiple solutions for a nonvariational elliptic systems involving p(x)-Laplacian operator. The approach combines the methods of sub–supersolution and Leray–Schauder topological degree.

Existence of solutions to some quasilinear degenerate elliptic systems with right hand side in a Marcinkiewicz space

<abstract><p>We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum is in a Marcinkiewicz space. The main assumption asks the off-diagonal coefficients to have support in the union of a geometric progression of squares.</p></abstract>

Multiple nonsemitrivial solutions for quasilinear elliptic systems via associated eigenvalue problems

In this paper, the existence, nonexistence and multiplicity of nonnegative solutions for quasilinear elliptic systems of resonant type involving $ (p,q) $-Laplacian are investigated. By using the generalized variant of the Nehari manifold and fibering method, we prove a suitable manifold with associated map for the systems and exploit the relation between them to study the existence and multiplicity of nonnegative solutions and obtain bifurcation results. Furthermore, we also apply the approach to treat a class of $ (p,q) $-Laplacian elliptic systems with concave-convex nonlinearities.

Existence results to Steklov system involving the (p, q)-Laplacian

In this paper, a quasilinear elliptic system involving a pair of (p,q)-Laplacian operators with Steklov boundary value conditions is studied. Using the Mountain Pass Geometry, we prove the existence of at least one weak solution. For the infinitely many weak solutions, we based on Bratsch’s Fountain Theorem.

Existence of weak solutions for some quasilinear degenerated elliptic systems in weighted Sobolev spaces

We consider, for a bounded open domain Ω in Rn; (n ≥ 1) and a function u : Ω → ℝm; (m ≥ 1) the quasilinear elliptic system: (QESw)(f,g) (0.1) Which is a Dirichlet problem. Here, v belongs to the dual space , f and g satisfy some stan- dard continuity and growth conditions. we will show the existence of a weak solution of this problem in the four following cases: σ is mono- tonic, σ is strictly monotonic, σ is quasi montone and σ derives from a convex potential.

A density property for tensor products of gradients of harmonic functions and applications

We show that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C(Ω‾), for any bounded domain Ω in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported k-tensor that is L2-orthogonal to all such products must be zero. This is done by using a Gaussian quasi-mode based construction of harmonic functions in the orthogonality relation. We then demonstrate the usefulness of this result by using it to prove uniqueness in the inverse boundary value problem for a coupled quasilinear elliptic system. The paper ends with a discussion of the corresponding property for products of two gradients of harmonic functions, and the connection of this property with the linearized anisotropic Calderón problem.

Regularity for quasilinear vectorial elliptic systems via an iterative scheme with numerical applications

We consider an iterative procedure to solve quasilinear elliptic systems with p-growth. The scheme was first considered by Koshelev in the quadratic case p = 2. We present numerical applications as well as applications to higher regularity properties.

Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: $$ \begin{cases} \Delta_p u= v^m| \nabla u |^\alpha & \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\ (N\geq 1)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1 < p < \infty$, $m,q > 0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $\delta:=(p-1-\alpha)(p-1-\beta)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that, we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\mathbb{R}^3$.

ON STRONGLY QUASILINEAR DEGENERATE ELLIPTIC SYSTEMS WITH WEAK MONOTONICITY AND NONLINEAR PHYSICAL DATA

This work is devoted to studying the quasilinear elliptic system $$\begin{aligned} -div ~ a(x,u,Du) + \vert u\vert ^{p-2} u +b(x,u,Du)=v(x)+f(x,u)+div ~g(x,u) \end{aligned}$$on a bounded open domain of \(\mathbb {R}^n\) with homogeneous Dirichlet boundary conditions. We show that there is a weak solution to this system under regularity, growth, and coercivity conditions for a, but only with very moderate monotonicity assumptions. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.

Interpolation inequalities for partial regularity

Abstract We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p &gt; 1 {p&gt;1} . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.

Nontrivial solutions for a class of gradient-type quasilinear elliptic systems

The aim of this paper is investigating the existence of weak bounded solutions of the gradient-type quasilinear elliptic system $$(P)\qquad \left\{ \begin{array}{ll} - {\rm div} ( a_i(x, u_i, \nabla u_i) ) + A_{i, t} (x, u_i, \nabla u_i) = G_i(x, \mathbf{u}) &\hbox{ in $\Omega$}\\ \quad\qquad\qquad\qquad\qquad \mbox{ for }\; i\in\{1,\dots,m\},\\ \mathbf{u} = 0 &\hbox{ on $\partial\Omega$,} \end{array} \right.$$ with $m\geq 2$ and $\mathbf{u}=(u_1,\dots, u_{m})$, where $\Omega\subset\mathbb{R}^N$ is an open bounded domain and some functions $A_i:\Omega\times\mathbb{R} \times \mathbb{R}^N\rightarrow\mathbb{R}$, $i\in\{1,\dots,m\}$, and $G:\Omega\times\mathbb{R}^m\rightarrow\mathbb{R}$ exist such that $a_i(x,t,\xi) = \nabla_{\xi} A_i(x,t,\xi)$, $A_{i, t} (x,t,\xi) = \frac{\partial A_i}{\partial t} (x,t,\xi)$ and $G_{i}(x,\mathbf{u}) = \frac{\partial G}{\partial u_i}(x,\mathbf{u})$. We prove that, under suitable hypotheses, the functional $\mathcal{J}$ related to problem $(P)$ is $\mathcal{C}^1$ on a "good" Banach space $X$ and satisfies the weak Cerami-Palais-Smale condition. Then, generalized versions of the Mountain Pass Theorems allow us to prove the existence of at least one critical point and, if $\mathcal{J}$ is even, of infinitely many ones, too.

Existence and Asymptotic Behaviors of Sign-Changing Solutions for Quasilinear Elliptic Systems

Existence and Asymptotic Behaviors of Sign-Changing Solutions for Quasilinear Elliptic Systems

Existence of solutions to some quasilinear degenerate elliptic systems when the datum has an intermediate degree of integrability

We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum has an intermediate degree of summability. The main assumption asks the off-diagonal coefficients to have a crossed ‘party-flags’ support.

Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper, we consider the following coupled gradient-type quasilinear elliptic system -div(a(x,u,∇u))+At(x,u,∇u)=Gu(x,u,v)inΩ,-div(b(x,v,∇v))+Bt(x,v,∇v)=Gvx,u,vinΩ,u=v=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{*{20}l} - {\ ext{div}} ( a(x, u, \ abla u) ) + A_t (x, u, \ abla u) = G_u(x, u, v) &{}{\\hbox { in }}\\Omega ,\\\\ - {\ ext{div}} ( b(x, v, \ abla v) ) + B_t(x, v, \ abla v) = G_v\\left( x, u, v\\right) &{}{\\hbox { in }}\\Omega ,\\\\ u = v = 0 &{}{\\hbox { on }}\\partial \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega is an open bounded domain in {mathbb {R}}^N, Nge 2. We suppose that some mathcal {C}^{1}–Carathéodory functions A, B:Omega times {mathbb {R}}times {mathbb {R}}^Nrightarrow {mathbb {R}} exist such that a(x,t,xi ) = nabla _{xi } A(x,t,xi ), A_t(x,t,xi ) = frac{partial A}{partial t} (x,t,xi ), b(x,t,xi ) = nabla _{xi } B(x,t,xi ), B_t(x,t,xi ) =frac{partial B}{partial t}(x,t,xi ), and that G_u(x, u, v), G_v(x, u, v) are the partial derivatives of a mathcal {C}^{1}–Carathéodory nonlinearity G:Omega times {mathbb {R}}times {mathbb {R}}rightarrow {mathbb {R}}. Roughly speaking, we assume that A(x,t,xi ) grows at least as (1+|t|^{s_1p_1})|xi |^{p_1}, p_1 > 1, s_1 ge 0, while B(x,t,xi ) grows as (1+|t|^{s_2p_2})|xi |^{p_2}, p_2 > 1, s_2 ge 0, and that G(x, u, v) can also have a supercritical growth related to s_1 and s_2. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.

Ground state solution for quasi-linear elliptic systems with general source terms on unbounded domain

Ground state solution for quasi-linear elliptic systems with general source terms on unbounded domain