Multiplicity of Solutions for a Nonlinear Degenerate Problem in Anisotropic Variable Exponent Spaces

January 2013
Asian Academy of Management Journal of Accounting & Finance;2013, Vol. 9 Issue 1, p117
Academic Journal
We study a nonlinear elliptic problem with Dirichlet boundary condition involving an anisotropic operator with variable exponents on a smooth bounded domain Ω ⊂ RN. For that equation we prove the existence of at least two nonnegative and nontrivial weak solutions. Our main result is proved using as main tools the Mountain Pass Theorem and a direct method in Calculus of Variation.


Related Articles

  • CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. Boutreanu, Maria-Magdalena // Studia Universitatis Babes-Bolyai, Mathematica;2010, Issue 4, p27 

    We work on the anisotropic variable exponent Sobolev spaces and we consider the problem: -ΣN i=11 @xiai (x, @xiu) + b(x)→u→P + + -2u = f(x, u) in , u → 0 in and u = 0 on @, where → RN (N → 3) is a bounded domain with smooth boundary and PN i=1 @xiai (x,...

  • Dirichlet and Neumann problems to critical Emden–Fowler type equations. Alexander Nazarov // Journal of Global Optimization;Mar2008, Vol. 40 Issue 1-3, p289 

    Abstract  We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to...

  • BOUNDARY VALUE PROBLEMS IN PLAN SECTOR WITH CORNERS FOR A CLASS OF SOBOLEV SPACES OF DOUBLE WEIGHT. Benseridi, H.; Dilmi, M. // Journal of Applied Functional Analysis;Jan2008, Vol. 3 Issue 1, p233 

    In this note, we study some boundary value problems of elasticity in plan domain with corners for a class of Sobolev spaces of double weight. We give a generalization of some results of existence and unicity of the solution obteined by P. Grisvard [6] in classical spaces of nul weight, by D....

  • AN EXISTENCE RESULT FOR HEMIVARIATIONAL INEQUALITIES. Dályay, Zsuzsánna; Varga, Csaba // Electronic Journal of Differential Equations;2004, Vol. 2004, p1 

    We present a general method for obtaining solutions for an abstract class of hemivariational inequalities. This result extends many results to the nonsmooth case. Our proof is based on a nonsmooth version of the Mountain Pass Theorem with Palais-Smale or with Cerami compactness condition. We...

  • A remark on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg–Landau equation. Karachalios, Nikos. I. // Journal of Mathematical Physics;Aug2009, Vol. 50 Issue 8, p082701 

    Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proven by Melas [Proc. Am. Math. Soc. 131, 631 (2003)], we remark on a modified estimate of the dimension of the global attractor associated with the complex Ginzburg–Landau (CGL) equation...

  • Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent. Changshou Lin; Liping Wang; Juncheng Wei // Calculus of Variations & Partial Differential Equations;Oct2007, Vol. 30 Issue 2, p153 

    We consider the following critical elliptic Neumann problem $${- \Delta u+\mu u=u^{\frac{N+2}{N-2}}, u > 0 in \Omega; \frac{\partial u}{\partial n}=0}$$ on $${\partial\Omega;}$$ , O; being a smooth bounded domain in $${\mathbb{R}^{N}, N\geq 7, \mu > 0}$$ is a large number. We show that at a...

  • Elliptic Operators in a Refined Scale of Functional Spaces. Mikhailets, V.; Murach, A. // Ukrainian Mathematical Journal;May2005, Vol. 57 Issue 5, p817 

    We study the theory of elliptic boundary-value problems in a refined two-sided scale of the Hormander spaces H s , ?, where s ? R and ? is a functional parameter slowly varying at +8. For the Sobolev spaces H s , the function ?(|?|) = 1. We establish that the considered operators possess the...

  • A priori estimates of the solution for the Dirichlet problem. VILLACAMPA, Y.; BALAGUER, A. // IMA Journal of Applied Mathematics;Aug2002, Vol. 67 Issue 4, p371 

    We study the Dirichlet problem defined by ?u = f in O and u = g in dO when O is the half-space or the unitary rectangle, obtaining an a priori estimate of the solution. Furthermore, in both cases a concrete numerical estimation is arrived at. First we get the a priori estimate in the case of the...

  • Boundary value problems for quasielliptic systems. Bondar, L.; Demidenko, G. // Siberian Mathematical Journal;Mar2008, Vol. 49 Issue 2, p202 

    We consider the boundary value problems in the half-space for a class of quasielliptic systems with variable coefficients. Assuming that the boundary value problems satisfy the LopatinskiÄ­ condition, we establish sufficient conditions for unique solvability in Sobolev spaces.


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics