Leray-Lions Operator Research Articles

Eigenvalue problems with fully discontinuous operators and critical exponents

We construct here more examples of ρ ⃗ -multivoque Leray–Lions operators that are strongly monotonic and we introduce a new abstract theorem on critical points for multivalued operators available for unbounded domains.

Anisotropic parabolic problems with measures data

In this work, we prove the existence of a weak solution of an anisotropic parabolic problem with measure data ut +Au+F(u,Du )= μ and u(0 )= μ0 with μ and μ0 two Radon bounded measures. The operator A is a Leray-Lions operator with anisotropic growth condi- tions. Our approach is based on the anisotropic Sobolev inequality, a regularity result, a com- pactness result, and an integration by parts formula.

On a sub–supersolution method for variational inequalities with Leray–Lions operators in variable exponent spaces

In this paper, we consider a sub–supersolution method for variational inequalities with Leray–Lions operators in Sobolev spaces with variable exponents. Existence and qualitative properties of solutions between sub and supersolutions are established.

General Comparison Principle for Variational-Hemivariational Inequalities

We study quasilinear elliptic variational-hemivariational inequalities involving general Leray-Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke's generalized gradient is required. Based on these results, in the second part the theory is extended to discontinuous variational-hemivariational inequalities.

Existence of solutions for degenerated problems in $L^1$ having lower order terms with natural growth

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation Au + g(x,u,∇u) = f, where A is a Leray–Lions operator from the weighted Sobolev space W_0^{1,p}(Ω, w) into its dual W^{ −1,p'}(Ω, w^*) . While g(x,s,ξ) is a nonlinear term having natural growth with respect to ξ and no growth with respect to s , it satisfies a sign condition on s , i.e., g(x,s,ξ) · s ≥ 0 for every s∈ℝ . The right-hand side f belongs to L^1(Ω) .

Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data

We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R N with Dirichlet homogeneous boundary conditions. The initial condition is in L1 and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.

Existence results for some unilateral problems without sign condition with obstacle free in Orlicz spaces

We prove the existence results in the setting of Orlicz spaces for the unilateral problem associated to the following equation, A u + g ( x , u , ∇ u ) = f , where A is a Leray–Lions operator acting from its domain D ( A ) ⊂ W 0 1 L M ( Ω ) into its dual, while g ( x , u , ∇ u ) is a nonlinear term having a growth conditions with respect to ∇ u and no growth with respect to u , but does not satisfy any sign condition. The right-hand side f belongs to L 1 ( Ω ) , and the obstacle is a measurable function.

Comparison and Existence Results for Classes of Nonlinear Elliptic Equations with General Growth in the Gradient

Abstract In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a Leray-Lions operator defined on W0 1,p (Ω), and H(x, u, Du) grows with respect to Du at most like |Du|q, p-1 ≤ q ≤ p. Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.

Existence of Solutions for Unilateral Problems in L 1 Involving Lower Order Terms in Divergence form in Orlicz Spaces

This article is concerned with the existence result of the unilateral problem associated to the equations of the type Au – div φ ( u ) = ƒ ∈ L 1 (Ω), where A is a Leray-Lions operator having a growth not necessarily of polynomial type and φ ∈ C 0 (ℝ, ℝ N ).

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form A u + g ( x , u , D u ) = f , where A is a Leray–Lions operator from a weighted Sobolev space into its dual. We assume that g ( x , s , ξ ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L 1 -setting.

Strongly nonlinear elliptic variational unilateral problems in Orlicz space

We will be concerned with the existence result of unilateral problem associated to the equations of the form Au + g(x, u, ∇u) = f, where A is a Leray‐Lions operator from its domain into . On the nonlinear lower order term g(x, u, ∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, and satisfies the sign condition. The right‐hand side f belongs to .

Symmetrization results for classes of nonlinear elliptic equations with q-growth in the gradient

In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A ( u ) = H ( x , u , Du ) + g ( x , u ) , where the principal term is a Leray–Lions operator defined on W 0 1 , p ( Ω ) . Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.

Branches of positive solutions of quasilinear elliptic equations on non-smooth domains

We prove existence of an unbounded global branch (i.e. connected set) of weak solutions of a second order quasilinear equation depending on a real parameter λ on an arbitrary (possibly non-smooth) bounded domain in R N , with a Leray–Lions operator as the leading part. Here, we can allow lower order nonlinearities which depend on first derivatives, satisfying appropriate growth conditions including the critical case. Furthermore, we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive λ . Our approach also yields a new existence result in the case of critical growth in derivatives of lower order.

Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces

We prove approximation and compactness results in inhomogeneous Orlicz-Sobolev spaces and look at, as an application, the Cauchy-Dirichlet equation u′+A(u)+g(x,t,u,∇u)=f∈W-1,xEM, where A is a Leray-Lions operator having a growth not necessarily of polynomial type. We also give a trace result allowing to deduce the continuity of the solutions with respect to time.

Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces

We prove approximation and compactness results in inhomogeneous Orlicz-Sobolev spaces and look at, as an application, the Cauchy-Dirichlet equation u ′ + A ( u ) + g ( x , t , u , ∇ u ) = f ∈ W - 1 , x E M , where A is a Leray-Lions operator having a growth not necessarily of polynomial type. We also give a trace result allowing to deduce the continuity of the solutions with respect to time.

Existence of solutions of degenerated unilateral problems with L^1 data

In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type Au+g(x,u,∇u)=f- div F, where A is a Leray-Lions operator and g is a Carathéodory function having natural growth with respect to |∇u| and satisfying the sign condition. The second term is such that, f∈L 1 (Ω) and F∈Π i=1 N L p ′ (Ω,w i 1-p ′ ).

Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations

In this paper we study the existence of solutions for quasilinear degenerated elliptic operators A(u) + g(x,u,ru) = f , where A is a Leray-Lions operator from W 1,p 0 ( ,w) into its dual, while g(x,s, ) is a nonlinear term which has a growth condition with respect to and no growth with respect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong either to W 1,p 0 ( ,w ) or to L 1 ().

A Symmetrization Result for Nonlinear Elliptic Equations

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x, u) + f, where the principal term is a Leray-Lions operator defined onW1,p.

Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of p-Laplacian type. If p < γ < N and the right-hand side is a Radon measure with singularity of order γ at x0A ∈ ω, then any supersolution in W1,p(ω) has singularity of order at least (γ−p)/(p−1) at x0. In the proof we exploit a pointwise estimate of A-superharmonic solutions, due to Kilpelainen and Malý, which involves Wolff's potential of Radon's measure.

Comparison results for nonlinear elliptic equations with lower–order terms

AbstractWe consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray–Lions operator defined on $ W ^{1, p} _{0} (\Omega) $ and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.