Global L∞L^\infty-estimate for general quasilinear elliptic equations in arbitrary domains of RN\mathbb{R}^N

In this paper our main goal is to present a new global $L^\infty$-estimate for a general class of quasilinear elliptic equations of the form $$-div \mathcal{A}(x,u,\nabla u)=\mathcal{B}(x,u,\nabla u)$$ under minimal structure conditions on the functions $\mathcal{A}$ and $\mathcal{B}$, and in arbitrary domains of $\mathbb{R}^N$. The main focus and the novelty of the paper is to prove $L^\infty$-estimate of the form $$|u|_{\infty, \Omega}\le C \Phi(|u|_{\beta,\Omega})$$ where $\Phi: \mathbb{R}^+\to \mathbb{R}^+$ is a data independent function with $\lim_{s\to 0^+}\Phi(s)=0$

Multi-valued parabolic variational inequalities and related variational-hemivariational inequalities

In this paper we study multi-valued parabolic variational inequalities involving quasilinearparabolic operators, and multi-valued integral terms over the underlying parabolic cylinderas well as over parts of the lateral parabolic boundary, where the multi-valued functionsinvolved are assumed to be upper semicontinuous only. Note, since lower semicontinuousmulti-valued functions allow always for a Carath ́eodory selection, this case can be consid-ered as the trivial case, and therefore will be omitted. Our main goal is threefold: First,we provide an analytical frame work and an existence theory for the problems under con-sideration. Unlike in recent publications on multi-valued parabolic variational inequalities,the closed convex setKrepresenting the constraints is not required to possess a nonemptyinterior. Second, we prove enclosure and comparison results based on a recently developedsub-supersolution method due to the authors. Third, we consider classes of relevant gen-eralized parabolic variational-hemivariational inequalities that will be shown to be specialcases of the multi-valued parabolic variational inequalities under consideration

Elliptic pp-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } \Omega,\newline -\Delta_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in } \Omega,\newline |\nabla u_1|^{p_1-2}\nabla u_1 \cdot \nu &=g_1(x,u_1,u_2) && \text{on } \partial\Omega,\newline |\nabla u_2|^{p_2-2}\nabla u_2 \cdot \nu &=g_2(x,u_1,u_2) && \text{on } \partial\Omega, \end{aligned} \end{equation*} where $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, $\Delta_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partial\Omega \times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution

EXISTENCE OF EXTREMAL PERIODIC SOLUTIONS FOR QUASILINEAR PARABOLIC EQUATIONS

Abstract. In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction

Quasilinear noncoercive parabolic bilateral variational inequalities in Lp(0, τ ;D1,p(RN))

In this paper, we prove existence results for quasilinear parabolic bilateral variational inequalities of the form: Find u ∈ K ⊂ X with u(・, 0) = 0 satisfying 0 ∈ ut − Δpu + aF(u) + ∂IK(u) in X∗ in the unbounded cylindrical domain Q = RN × (0, τ ), where Δp is the p-Laplacian acting on X = Lp(0, τ ;D1,p(RN)) with its dual space X∗, and with D1,p(RN) denoting the Beppo-Levi space (or homogeneous Sobolev space). The bilateral constraint is represented by the closed convex set K ⊂ X given by K = {v ∈ X : ϕ(x, t) ≤ v(x, t) ≤ ψ(x, t) for a.a. (x, t) ∈ Q} and IK is the indicator function related to K with ∂IK denoting its subdifferential in the sense of convex analysis. The main goal and the novelty of this paper is to prove existence and directedness results without assuming coercivity conditions on the operator −Δp + aF : X → X∗, and without supposing the existence of sub- and supersolutions. Moreover, additional difficulties we are faced with arise due to the lack of compact embedding of D1,p(RN) into Lebesgue spaces Lσ(RN), and the fact that the domain K of ∂IK has empty interior, which prevents us to use recent results on evolutionary variational inequality. Instead our approach is based on an appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as pseudomontone operator theory