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)      2. The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations      Alkis S. Tersenov  Aris S. Tersenov 《Journal of Differential Equations》2007,235(2):376-396 We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.      3. 椭圆外区域各向异性问题基于人工边界条件的Schwarz交替算法      冯丹  杜其奎 《高校应用数学学报(A辑)》2011,26(2):158-168 研究一类二维各向异性外问题的重叠型区域分解.基于自然边界归化,对各向异性外问题提出了一种Schwarz交替算法,并给出其离散形式,分析了算法的收敛性.给出数值试验以示算法的可行性与有效性.      4. The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow      Rodica Chirila-Socolescu 《Journal of Mathematical Analysis and Applications》1977,60(2):449-460 We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.      5. Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations      Agnese Di Castro  Eugenio Montefusco 《Nonlinear Analysis: Theory, Methods & Applications》2009 We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.      6. Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain      Caisheng Chen  Qi Zhang 《Mathematical Methods in the Applied Sciences》2012,35(7):782-794 In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition (1) where Δp denotes the p‐Laplacian operator, defined by △ pu = div( | ? u | p ? 2 ? u),1 < p < N, Ω is a smooth exterior domain in . is the outward normal derivative, . The parameters p,q,r are either or . The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ1 for some λ1. Copyright © 2012 John Wiley & Sons, Ltd.      7. Positive solutions of quasilinear elliptic equations in exterior domains      E.S Noussair  C.A Swanson 《Journal of Mathematical Analysis and Applications》1980,75(1):121-133       8. Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions      Caisheng Chen  Shuai Liu  Huaping Yao 《Journal of Mathematical Analysis and Applications》2011,383(1):111-119 In this paper, we consider the following quasilinear elliptic exterior problem       9. The Dirichlet problem for quasilinear uniformly elliptic equations      J.H. Michael 《Nonlinear Analysis: Theory, Methods & Applications》1985,9(5):455-467       10. On a jumping problem for quasilinear elliptic equations      Annamaria Canino 《Mathematische Zeitschrift》1997,226(2):193-210       11. Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent      Mihai Mih?ilescu 《Journal of Mathematical Analysis and Applications》2008,340(1):687-698 In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.      12. The ground state problem for quasilinear elliptic equations      James Serrin 《Milan Journal of Mathematics》1992,62(1):137-137 Conferenza tenuta il 29 settembre 1992      13. The problem with an obstacle for certain classes of quasilinear elliptic equations      A. A. Arkhipova 《Journal of Mathematical Sciences》1978,10(6):805-824       14. Constant boundary value problem for a quasilinear elliptic system      Yaotian Shen  Shusen Yan 《manuscripta mathematica》1993,79(1):99-112 We study the constant boundary value problem for a class of quasilinear elliptic systems. The existence of a nonconstant solution is proved by discussing the minimization problem for the relaxed energy. As a by-product, we show that generally the Pohozaev identity does not hold for H1-weak solutions.      15. A degenerate Neumann problem for quasilinear elliptic integro-differential operators      D.K. Palagachev  P.R. Popivanov  K. Taira 《Mathematische Zeitschrift》1999,230(4):679-694 This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998      16. The Dirichlet problem for singularly perturbed quasilinear elliptic equations      Walter G Kelley 《Journal of Differential Equations》1981,40(1):37-52       17. The nonlinear oblique derivative problem for quasilinear elliptic equations      Gary M. Lieberman 《Nonlinear Analysis: Theory, Methods & Applications》1984,8(1):49-65       18. Existence for bounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains      Jiqin Deng 《Nonlinear Analysis: Theory, Methods & Applications》2008 In this paper, by using the fixed point theory, we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two-dimensional exterior domains.      19. Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions      Leiga Zhao  Anran LiJiabao Su 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2520-2533 In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.      20. On the exterior two-dimensional Dirichlet problem for elliptic equations      Remigio Russo  Christian G. Simader 《Ricerche di matematica》2009,58(2):315-328 A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

椭圆外区域各向异性问题基于人工边界条件的Schwarz交替算法

研究一类二维各向异性外问题的重叠型区域分解.基于自然边界归化,对各向异性外问题提出了一种Schwarz交替算法,并给出其离散形式,分析了算法的收敛性.给出数值试验以示算法的可行性与有效性.     

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.     

Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.     

Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain

In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition (1) where Δ_p denotes the p‐Laplacian operator, defined by △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N, Ω is a smooth exterior domain in . is the outward normal derivative, . The parameters p,q,r are either or . The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ₁ for some λ₁. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions

In this paper, we consider the following quasilinear elliptic exterior problem

     

Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.     

The ground state problem for quasilinear elliptic equations

Conferenza tenuta il 29 settembre 1992     

Constant boundary value problem for a quasilinear elliptic system

We study the constant boundary value problem for a class of quasilinear elliptic systems. The existence of a nonconstant solution is proved by discussing the minimization problem for the relaxed energy. As a by-product, we show that generally the Pohozaev identity does not hold for H¹-weak solutions.     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

Existence for bounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains

In this paper, by using the fixed point theory, we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two-dimensional exterior domains.     

Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.     

On the exterior two-dimensional Dirichlet problem for elliptic equations

A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.