Quasilinear Elliptic Operators Research Articles

Strong barriers for weighted quasilinear equations

In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.

A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator

A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator

Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term

Abstract We prove Gibbons’ conjecture for the quasilinear elliptic equation - Δ p ⁢ u + a ⁢ ( u ) ⁢ | ∇ ⁡ u | q = f ⁢ ( u ) in ⁢ ℝ N , -\Delta_{p}u+a(u)|\nabla u|^{q}=f(u)\quad\text{in }\mathbb{R}^{N}, where N ≥ 2 {N\geq 2} , 2 ⁢ N + 2 N + 2 < p < 2 {\frac{2N+2}{N+2}<p<2} , q ≥ 1 {q\geq 1} and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ ( ℝ N ) {u\in C^{1}(\mathbb{R}^{N})} of the equation with | u | ≤ 1 {|u|\leq 1} and lim x N → ± ∞ ⁡ u ⁢ ( x ′ , x N ) = ± 1 {\lim_{x_{N}\to\pm\infty}u(x^{\prime},x_{N})=\pm 1} , uniformly in x ′ ∈ ℝ N - 1 {x^{\prime}\in\mathbb{R}^{N-1}} , must depend only on x N {x_{N}} and ∂ ⁡ u ∂ ⁡ x N > 0 {\frac{\partial u}{\partial x_{N}}>0} in ℝ N {\mathbb{R}^{N}} . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] {[-1,-1+\delta]} and on [ 1 - δ , 1 ] {[1-\delta,1]} and f ⁢ ( u ) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ ( 1 - u 2 ) {f(u)=|u|^{r}u|1-u^{2}|^{s}(1-u^{2})} , where r , s , δ ≥ 0 {r,s,\delta\geq 0} . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator.

Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to L^{infty }, which implies C^{1,alpha } smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

Noncoercive quasilinear elliptic operators with singular lower order terms

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

Mixed boundary value problem for a class of quasi-linear elliptic operators containing p-Laplacian

Mixed boundary value problem for a class of quasi-linear elliptic operators containing p-Laplacian

A limiting problem for local/non-local p-Laplacians with concave–convex nonlinearities

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by $$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega ,\\ u(x)&{} =&{} 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is a bounded and smooth domain, $$s\in (0,1)$$ , $$2 \le p < \infty $$ , $$0<q(p)<p-1<r(p)<\infty $$ and $$0<\lambda _p< \infty $$ , being $$\Delta _p$$ and $$(-\Delta )_p^s$$ the p-Laplace and fractional p-Laplace operators, respectively. We study existence and global uniform and explicit boundedness results to weak solutions. Then, we perform an asymptotic analysis for the limit of a family of weak solutions $$\{u_p\}_{p\ge 2}$$ as $$p \rightarrow \infty $$ , which converges, up to a subsequence (under suitable assumptions on the problem data), to a non-trivial profile with uniform and explicit bounds, enjoying of a universal Lipschitz modulus of continuity, and verifying a nonlinear limiting PDE in the viscosity sense, which exhibits both local/non-local character.

Full and partial regularity for a class of nonlinear free boundary problems

In this paper we classify the nonnegative global minimizers of the functionalJF(u)=∫ΩF(|∇u|2)+λ2χ{u>0}, where F satisfies some structural conditions and χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{u>0} can be seen as a rectifiable n−1 varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if n=3 and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

Abstract We characterize the existence of solutions to the quasilinear Riccati-type equation { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) = | ∇ ⁡ u | q + σ in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &amp;\displaystyle=|\nabla u|^{q}+\sigma&amp;&amp;\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with a distributional or measure datum σ. Here div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ( p &gt; 1 {p&gt;1} ), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that p &gt; 1 {p&gt;1} and q &gt; p {q&gt;p} . For measure data, we assume that they are compactly supported in Ω, p &gt; 3 ⁢ n - 2 2 ⁢ n - 1 {p&gt;\frac{3n-2}{2n-1}} , and q is in the sub-linear range p - 1 &lt; q &lt; 1 {p-1&lt;q&lt;1} . We also assume more regularity conditions on 𝒜 {\mathcal{A}} and on ∂ ⁡ Ω ⁢ Ω {\partial\Omega\Omega} in this case.

On a weighted Trudinger-Moser inequality in [formula omitted

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type Lu:=−r−θ(rα|u′(r)|βu′(r))′, where θ,β≥0 and α>0, are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p-Laplacian and k-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Pólya-Szegö principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.

On minimal decay at infinity of Hardy-weights

We study the behavior of Hardy-weights for a class of variational quasilinear elliptic operators of [Formula: see text]-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.

Kato's inequalities for admissible functions to quasilinear elliptic operators &lt;i&gt;A &lt;/i&gt;

Kato's inequalities for admissible functions to quasilinear elliptic operators &lt;i&gt;A &lt;/i&gt;

Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization

We discuss the existence and uniqueness of the weak solution of the following nonlinear parabolic problem: [Formula: see text] which involves a quasilinear elliptic operator of Leray–Lions type with variable exponents. Next, we discuss the global behavior of solutions and in particular the convergence to a stationary solution as [Formula: see text].

A limit case in non-isotropic two-phase minimization problems driven by $p$-Laplacians

In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p- Laplacian type. The problem in its variational form is as follows: \displaystyle \text{min} \left\{ \int\limits_{\Omega \cap \{v&gt;0\}} \left(\frac{1}{p}|\nabla v|^p + \lambda_{+}^p+ f_{+}v \right)dx + \int\limits_{\Omega \cap \{v\leq 0\}} \left(\frac{1}{q}|\nabla v|^q + \lambda_{-}^q+ f_{-}v\right)dx \right\}. Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v=g on \partial \Omega . First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the \infty -Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.

English

We shall prove a weak comparison principle for quasilinear elliptic operators −div(a(x,∇u)) that includes the negative p-Laplace operator, where a: × ℝN → ℝN satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

Lyapunov-type inequalities for partial differential equations

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in N -dimensional domains Ω. We also consider singular and degenerate elliptic problems with A p coefficients involving the p -Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the p -Laplacian, and compare them with the usual ones in the literature.

Quasilinear Riccati type equations with distributional data in Morrey space framework

In the framework of Morrey or Lorentz–Morrey spaces, we characterize the existence of solutions to the quasilinear Riccati type equation{−divA(x,∇u)=|∇u|q+σinΩ,u=0on∂Ω, with a distributional datum σ. Here divA(x,∇u) is a quasilinear elliptic operator modelled after the p-Laplacian, p>1, but with a very general nonlinear structure, and Ω is a sufficiently flat domain in the sense of Reifenberg. The existence results are obtained in the natural or super-natural range of the gradient growth, i.e., q≥p.Motivated by the analysis of quasilinear Riccati type equation, a substantial part of the paper is also devoted to the Calderón–Zygmund type gradient regularity for the boundary value problem{divA(x,∇u)=div|f|p−2finΩ,u=0on∂Ω. We obtain regularity estimates in some weighted and unweighted function spaces as well as natural Lorentz–Morrey spaces associated to the Riccati type equation above.

Barrier solutions of elliptic variational inequalities

We consider elliptic variational inequalities in a bounded domain Ω⊂RN of the formu∈K:〈Au+λ|u|p−2u+F(u),v−u〉≥〈h,v−u〉,∀v∈K, where A is a second order quasilinear elliptic operator of divergence type, F is the operator generated by lower order terms, and K is a closed convex subset of the Sobolev space W1,p(Ω), 1<p<∞, and λ≥0. The main goal of this paper is to answer the following question: Does the variational inequality possess barrier solutions? Here, solutions u∗ and u∗ of the variational inequality are called barrier solutions if u∗≤u∗, and any solution u of the variational inequality satisfies u∗≤u≤u∗. In other words, we are going to provide sufficient conditions which ensure that the solution set S of the variational inequality is nonempty, and S possesses a greatest and a smallest element with respect to the natural partial ordering of functions. An answer to the raised question is by no means trivial as will be seen by specific examples of the variational inequality under consideration. The obtained results are finally used to derive a priori order bounds as well as existence of barrier solutions for multi-valued variational inequalities including variational–hemivariational inequalities as special case.

Elliptic inequalities with multi-valued operators: Existence, comparison and related variational–hemivariational type inequalities

We study multi-valued elliptic variational inclusions in a bounded domain Ω⊂RN of the form u∈K:0∈Au+∂IK(u)+F(u)+FΓ(u), where A is a second order quasilinear elliptic operator of Leray–Lions type, K is a closed convex subset of some Sobolev space, IK is the indicator function related to K, and ∂IK denoting its subdifferential. The lower order multi-valued operators F and FΓ are generated by multi-valued, upper semicontinuous functions f:Ω×R→2R∖{0̸} and fΓ:Γ×R→2R∖{0̸}, respectively, with Γ⊂∂Ω. Our main goals are as follows: First we provide an existence theory for the above multi-valued variational inequalities. Second, we establish an enclosure and comparison principle based on appropriately defined sub–supersolutions, and prove the existence of extremal solutions. Third, by means of the sub–supersolution method provided here, we are going to show that rather general classes of variational–hemivariational type inequalities turn out to be only subclasses of the above general multi-valued elliptic variational inequalities, which in a way fills a gap in the current literature where these kind of problems are studied independently. Finally, the existence of extremal solutions will allow us to deal with classes of multi-valued function f and fΓ that are neither lower nor upper semicontinuous, which in turn will provide a tool to obtain existence results for variational–hemivariational type inequalities whose Clarke’s generalized directional derivative may, in addition, discontinuously depend on the function we are looking for. This paper, though of surveying nature, provides an analytical framework that allows to present in a unifying way and to extend a number of recent results due to the authors.

Convergence rate for some quasilinear eigenvalues homogenization problems

In this work we study the homogenization problem for nonlinear eigenvalues of some quasilinear elliptic operators. We obtain an explicit order of convergence in k and in ε for the (variational) eigenvalues.