Search results for "Unbounded solutions"

showing 3 items of 3 documents

Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term

2001

Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.

Bounded and unbounded solutionsQuasi-linear elliptic problemsDirichlet problemMathematics(all)Pure mathematicsApplied MathematicsGeneral MathematicsWeak solutionMathematical analysisQuadratic functionWeak formulationNonlinear systemElliptic curveQuadratic equationBounded functionQuadratic gradient termMathematicsJournal de Mathématiques Pures et Appliquées
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Nonlinear elliptic equations having a gradient term with natural growth

2006

Abstract In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: (1) { − Δ p u = g ( u ) | ∇ u | p + f , in Ω , u = 0 , on ∂ Ω . Here Ω is a bounded open set in R N ( N ⩾ 2 ), Δ p denotes the so-called p-Laplace operator ( p > 1 ) and g is a continuous real function. Given f ∈ L m ( Ω ) ( m > 1 ), we study under which growth conditions on g problem (1) admits a solution. If m ⩾ N / p , we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N p ; while if 1 m N / p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condit…

Dirichlet problemMathematics(all)Pure mathematicsApplied MathematicsGeneral MathematicsWeak solutionNonlinear elliptic operatorsMathematical analysisGradient term; Nonlinear elliptic operators; Unbounded solutionsType (model theory)Elliptic curveElliptic operatorCompact spaceUnbounded solutionsSettore MAT/05 - Analisi MatematicaBounded functionp-LaplacianGradient termMathematicsJournal de Mathématiques Pures et Appliquées
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Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term

2006

We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following (1){ut−div(α(u)∇u)=β(u)|∇u|2+f(x,t),in Ω×]0,T[;u(x,t)=0,on ∂Ω×]0,T[;u(x,0)=u0(x),in Ω. Here Ω is a bounded open set in RN, T>0. The unknown function u=u(x,t) depends on x∈Ω and t∈]0,T[. The symbol ∇u denotes the gradient of u with respect to x. The real functions α, β are continuous; moreover α is positive, bounded and may vanish at ±∞. As far as the data are concerned, we require the following assumptions: ∫ΩΦ(u0(x))dx<∞ where Φ is a convenient function which …

Quadratic growthNonlinear parabolic problems; gradient term with quadratic growth; existence and regularity; bounded and unbounded solutions; lack of coercivenesstermine quadratico nel gradienteApplied MathematicsOperator (physics)existence and regularityMathematical analysisDegenerate energy levelsFunction (mathematics)equazioni parabolichebounded and unbounded solutionsParabolic partial differential equationBounded functioncoercività degenerePrincipal partOrder (group theory)gradient term with quadratic growthNonlinear parabolic problemsMathematical PhysicsAnalysislack of coercivenessMathematics
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