Lions Operator Research Articles

Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent

Abstract We study a nonlinear elliptic problem with Robin type boundary condition, governed by a general Leray–Lions operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of a weak solution.

Lewy–Stampacchia’s inequality for a Leray–Lions operator with natural growth in the gradient

We prove that at least one solution of the obstacle problem with a Leray–Lions operator and a perturbation with natural growth with respect to the gradient satisfies Lewy–Stampacchia’s inequality. The proof is based on the natural penalization of the obstacle problem.

Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient

The existence of positive solutions for nonlinear elliptic problems under Dirichlet boundary condition is studied as well as the compactness and directness of the solution set. The main novelties consist in the presence of a Leray–Lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution. Our approach relies on the method of sub-supersolution, nonlinear regularity theory and strong maximum principle.

Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces

In this paper we consider the problem{u∈W01,p(Ω),−diva(x,u,Du)+b(x,u,Du)=h(x,u,Du)in D′(Ω), where −diva(x,u,Du) is a Leray–Lions operator which is defined on W01,p(Ω) with coercivity α, where the growth with respect to Du of h(x,u,Du) is controlled by αγ|Du|p, and where b(x,u,Du) satisfies a similar growth condition but “has the good sign”. The main feature of the problem is that the source terms belong to the Lorentz space LNp,∞(Ω). When two smallness conditions are satisfied (the second one depends on the behavior of b(x,u,Du) when |u| tends to infinity), we prove the existence of a solution which further satisfies eδp−1|u|−1∈W01,p(Ω) for every δ with γ⩽δ<δ0, for some threshold δ0. The key ingredient in the proof of the existence result is an a priori estimate which holds true for every solution to the problem which satisfies the above mentioned exponential regularity condition.

On some nonlinear nonhomogeneous elliptic unilateral problems involving noncontrollable lower order terms with measure right hand side

We prove the existence of entropy solutions to unilateral problems associated to equations of the type $Au-\mathop{\rm div}(\phi (u))=\mu \in L^{1}(\varOmega )+W^{-1,p'(\cdot )}(\varOmega )$, where $A$ is a Leray&#8211;Lions operator acting from $W_{0}^{1

Elliptic problem involving diffuse measure data

In this paper, we study a suitable notion of solution for which a nonlinear elliptic problem governed by a general Leray–Lions operator is well posed for any diffuse measure data. In terms of the paper (Brezis et al., 2007, [10]), we study the notion of solution for which any diffuse measure is “good measure”.

The “strange term” in the periodic homogenization for multivalued Leray–Lions operators in perforated domains

Using the periodic unfolding method of Cioranescu, Damlamian and Griso, we study the homogenization for equations of the form $$-{\rm div}\,\,d_\varepsilon=f,\,\,{\rm with}\,\,\left(\nabla u_{\varepsilon , \delta }(x),d_{\varepsilon , \delta }(x)\right) \in A_\varepsilon(x)$$ in a perforated domain with holes of size $${\varepsilon \delta }$$ periodically distributed in the domain, where $${A_\varepsilon }$$ is a function whose values are maximal monotone graphs (on R N ). Two different unfolding operators are involved in such a geometric situation. Under appropriate growth and coercivity assumptions, if the corresponding two sequences of unfolded maximal monotone graphs converge in the graph sense to the maximal monotone graphs A(x, y) and A 0(x, z) for almost every $${(x,y,z)\in \Omega \times Y \times {\rm {\bf R}}^N}$$ , as $${\varepsilon \to 0}$$ , then every cluster point (u 0, d 0) of the sequence $${(u_{\varepsilon , \delta }, d_{\varepsilon , \delta } )}$$ for the weak topology in the naturally associated Sobolev space is a solution of the homogenized problem which is expressed in terms of u 0 alone. This result applies to the case where $${A_{\varepsilon}(x)}$$ is of the form $${B(x/\varepsilon)}$$ where B(y) is periodic and continuous at y = 0, and, in particular, to the oscillating p-Laplacian.

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.

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.

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) .

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.

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.

Nonlinear unilateral problems in Orlicz spaces

We prove the existence of solutions of the unilateral problem for equations of the type $Au -\textrm{div} \phi (u) = \mu$ in Orlicz spaces, where $A$ is a Leray&#8211;Lions operator defined on ${\cal D}(A)\subset W_0^1L_M({\mit\Omega})$, $\mu\in L^1({\

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.

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.

Existence results for dirichlet problems in L1 via minty's lemma

In some recent papers, existence and qualitative properties of solutions for nonlinear elliptic equations with right hand side measures have been proved (see the references). This paper is concerned with the existence of solutions of elliptic problems as where A(u) = −div(a(x,u, Δu)) is a Leray Lions operator defined from W1,p 0(ω) into its dual, ω is a bounded domain of RN, N≥2,fεL1(ω) and |F|εLp'(ω). We will give an existence result for the so called entropy solutions, using as main tool an L1 version of Minty's Lemma. This approach allows both to extend the known existence results (see [1]) to the case of operators defined by a function a which is not strictly monotone with respect to the gradient, and to avoid the technical result of almost everywhere convergence for gradients of suitable approximating solutions (see [2], [4], [5] and [11]) which was needed in order to prove existence.

Some existence results for quasi-linear elliptic problems via the fixed-point theorem of Tarski

In this paper, we study a family of quasilinear problems, related to Leray–Lions operators. We prove some existence theorems without any continuity condition on the right-hand side of the equation. This is possible by using the fixed-point theorem of Tarski.

A Proof of the Lewy-Stampacchina's Inequality by a Penalization Method

In this paper we prove the Lewy–Stampacchia's inequality for elliptic variational inequalities with obstacle involving fairly general Leray–Lions operators. The main novelty of the paper is the method of proof, which uses the natural penalization. One of the steps of the proof consists in proving, again thanks to the natural penalization, that the nonnegative cone of W 0 1,p (Ω) is dense in the nonnegative cone of W-1,p′(Ω).

Homogenisation of Dirichlet problems for monotone operators in varying domains

We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutionsunof nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient ofunand the corrector for thep-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.