Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- \mbox{div} \big( a(x,u,\nabla u)\big)+ g(x,u,\nabla u) + |u|^{p_{0}(x)-2}u = f-\mbox{div} \phi(u),\quad \mbox{ in } \Omega,$$ where $-\mbox{div}\big(a(x,u,\nabla u)\big)$ is a Leray-Lions operator, $\phi \in C^{0}(I\!\!R,I\!\!R^{N})$. The function $g(x,u,\nabla u)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(\Omega)$.     

Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type:-div(a(x, u, u) + φ(u)) + g(x, u, u) = μ, where the right-hand side belongs to L1(Ω) + W-1,p(x)(Ω),-div(a(x, u, u)) is a Leray–Lions oper- ator defined from W-1,p(x)(Ω) into its dual and φ∈ C0(R, RN). The function g(x, u, u) is a non linear lower order term with natural growth with respect to |u| satisfying the sign condition, that is,g(x, u, u)u ≥ 0.     

Entropy solutions for nonlinear elliptic anisotropic problems with homogeneous Neumann boundary condition

This study is about a nonlinear anisotropic problem with homogeneous Neumann boundary condition. We first prove, by using the technic of monotone operators in Banach spaces, the existence of weak solution, and by approximation methods, we achieve a result of existence and uniqueness of entropy solution     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

On solutions of anisotropic elliptic equations with variable exponent and measure data

ABSTRACT

The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established. It is proved that the obtained entropy solution is a renormalized solution of the considered problem.     

On a class of nonlinear obstacle problems with measure data

We study the notion of solution to an obstacle problem for a strongly monotone and Lipschitz operator A, when the forcing term is a bounded Radon measure. We obtain existence and uniqueness results. We study also some properties of the obstacle reactions associated with the solutions of the obstacle problems, obtaining the Lcwy­Stampacchia inequality. Moreover we investigate the interaction between obstacle and data and the complementarity conditions     

Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      11. Regularity for anisotropic solutions to some nonlinear elliptic system      Hongya GAO  Shuang LIANG  Yi CUI 《Frontiers of Mathematics in China》2016,11(1):77-87 This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system −Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N, We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.      12. Entropy solutions for nonlinear elliptic equations in L1      《Nonlinear Analysis: Theory, Methods & Applications》1998,32(3):325-334       13. Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(3):305-337 We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.      14. Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data      Olivier Guibé    Anna Mercaldo 《Transactions of the American Mathematical Society》2008,360(2):643-669 In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is where is a bounded open subset of , , is the so-called Laplace operator, , is a Radon measure with bounded variation on , , , and and belong to the Lorentz spaces , , and , respectively. In particular we prove the existence under the assumptions that , belongs to the Lorentz space , , and is small enough.       15. On some nonlinear elliptic problems with an unknown measure data      J.F. Padial 《PAMM》2007,7(1):2040037-2040038 We prove the existence of a solution of some nonlinear elliptic problems with a Radon measure data. In contrast with the usual elliptic problem, this measure will be an unknown of the problem depending on the solution. We shall use a Minimax Ambrosetti–Rabinowitz argument. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)      16. Existence of solutions for an anisotropic elliptic problem with variable exponent and singularity          下载免费PDF全文 Mei Yu  Li Wang  Sufang Tang 《Mathematical Methods in the Applied Sciences》2016,39(10):2761-2767 In this paper, we study the following anisotropic problem with singularity: where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.      17. High-energy solutions for a nonlinear elliptic problem with slightly supercritical exponent      《Nonlinear Analysis: Theory, Methods & Applications》1999,38(4):527-546       18. Renormalized solutions to elliptic equations with measure data in unbounded domains      Annalisa Malusa  Maria Michaela Porzio 《Nonlinear Analysis: Theory, Methods & Applications》2007       19. Positive solutions for a nonlinear nonlocal elliptic transmission problem      《Applied Mathematics Letters》2003,16(2):243-248 We show the existence and nonexistence of positive solutions for a transmission problem given by a system of two nonlinear elliptic equations of Kirchhoff type.      20. Renormalized solutions of nonlinear parabolic equations with diffuse measure data      Dominique Blanchard  Francesco Petitta  Hicham Redwane 《manuscripta mathematica》2013,141(3-4):601-635 Given a parabolic cylinder Ω × (0, T), where Ω is a bounded domain of ${\mathbb{R}^N}$ , we consider IBV problems involving equations of the type $$b(u)_{t} - \Delta_{p} u = \mu$$ where b is a increasing C 1-function and μ is a diffuse measure. We prove the existence and uniqueness of a renormalized solution for this class of nonlinear parabolic equations.

Regularity for anisotropic solutions to some nonlinear elliptic system

This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system −Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N, We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.     

Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.     

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

where is a bounded open subset of , , is the so-called Laplace operator, , is a Radon measure with bounded variation on , , , and and belong to the Lorentz spaces , , and , respectively. In particular we prove the existence under the assumptions that , belongs to the Lorentz space , , and is small enough.

     

On some nonlinear elliptic problems with an unknown measure data

We prove the existence of a solution of some nonlinear elliptic problems with a Radon measure data. In contrast with the usual elliptic problem, this measure will be an unknown of the problem depending on the solution. We shall use a Minimax Ambrosetti–Rabinowitz argument. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions for an anisotropic elliptic problem with variable exponent and singularity

In this paper, we study the following anisotropic problem with singularity: where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive solutions for a nonlinear nonlocal elliptic transmission problem

We show the existence and nonexistence of positive solutions for a transmission problem given by a system of two nonlinear elliptic equations of Kirchhoff type.     

Renormalized solutions of nonlinear parabolic equations with diffuse measure data

Given a parabolic cylinder Ω × (0, T), where Ω is a bounded domain of ${\mathbb{R}^N}$ , we consider IBV problems involving equations of the type $$b(u)_{t} - \Delta_{p} u = \mu$$ where b is a increasing C ¹-function and μ is a diffuse measure. We prove the existence and uniqueness of a renormalized solution for this class of nonlinear parabolic equations.