Enzo Mitidieri

A RELLICH TYPE IDENTITY AND APPLICATIONS

arbitrary given u∈C2(Ω _ ), where Ω is a smooth bounded domain contained in RN. This intrinsic identity has played an important role in many contexts in the theory of partial differential equations. Some earlier applications were considered by Rellich himself, Payne and Weinberger [8] and Nehari [7]. In [9] Pohozaev rediscovered a modified version of RellichOs identity in the study of eigenvalue problems of the form

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L p norms. In the higher-order case further L p norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0.

A maximum principle for an elliptic system and applications to semilinear problems

The Dirichlet problem in a bounded region for elliptic systems of the form (*) \( - \Updelta u = f\left( {x,u} \right) - v,\quad - \Updelta v = \delta u - \gamma v \) is studied. For the question of existence of positive solutions the key ingredient is a maximum principle for a linear elliptic system associated with (*). A priori bounds for the solutions of (*) are proved under various types of growth conditions on f. Variational methods are used to establish the existence of pairs of solutions for (*).

Quasilinear elliptic equations with critical exponents

has no solution if Ω ⊂ R , N ≥ 3, is bounded and starshaped with respect to some point, and 2∗ = 2N/(N − 2). In (P0) the nonlinear term is a power of u with the critical exponent (N + 2)/(N − 2). This terminology comes from the fact that the continuous Sobolev imbeddings H 0 (Ω) ⊂ L(Ω), for p ≤ 2∗ and Ω bounded, are also compact except when p = 2∗. This loss of compactness reflects in that the functional whose Euler–Lagrange equation is (P0) fails to satisfy the Palais–Smale condition. Later Brezis and Nirenberg [BN] observed that the Palais–Smale condition fails at certain levels only. Then they proved that if the nonlinear term is slightly perturbed, the new problem has a solution.

A simple approach to Hardy inequalities

We describe a simple method of proving Hardy-type inequalities of second and higher order with weights for functions defined in ℝn. It is shown that we can obtain such inequalities with sharp constants by applying the divergence theorem to specially chosen vector fields. Another approach to Hardy inequalities based on the application of identities of Rellich-Pokhozhaev type is also proposed.

A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in (P.-L. Lions, SIAM Rev., 24 (1982), pp. 441-467), we partially extend results known for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list those of particular interest in the final section.

Liouville theorems for elliptic inequalities and applications

In this paper we prove nonexistence of positive C2 solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to prove a-priori bounds for a class of semilinear elliptic systems of equations in bounded domains.

Strongly indefinite systems with critical Sobolev exponents

We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case. INTRODUCTION Existence and non-existence of solutions of semilinear elliptic srstems has been a subject of active research recently; see for example [FF], [L3], [PS]. Such systems are called variational if solutions can be viewed as critical points of an associated functional defined on a suitable function space. Restricting our attention to systems with two unknowns, we distinguish two classes of variational elliptic systems: (a) potential systems; (b) Hamiltonian systems. Potential systems are of the form (A) -Llu = ,,> (u, v), ztL2v = ,,> (u, v), where L1 and L2 are second order selfadjoint elliptic operators. Formally these are the Euler-Lagrange equations of the functional f (u, v) = (L1u, u) i (L2v, v)-tH(u, v) . Here (, ) is the L2 inner product for functions defined on the underlying domain, and tt(u, v) = S H(u(x), v(x))dx. The functional f is often called the Lagrangian. Its critical points correspond to weak soltbtions of (A). The appropr:;ate choice of the function spaces for u and v follows by requiring that the quadratic part of f be well defined. This naturally leads to Sobolev spaces with square irltegrable first derivatives. Since also A(, ) must be well defined, by virtue of the embedding theorems, some growth restrictions on this latter function are required. An extra difficulty appears if we consider system (A) with a plus sign in the second equation, for then the corresponding functiorl f has a strongly indefinite quadratic part. For systems of superlinear type and with sllbcritical growth the Mountain Pass Theorem [AR] or, in the strongly indefinite case, the Benci-Rabinowitz Theorem Received by the editors June 5, 1996. 1991 Mathematics Subject Classification. Primary 35J50, 35J55, 35J65.

Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on

No Abstract

{\Bbb R}^n

In this paper we obtain some sufficient (necessary) conditionsfor the validity of the maximum principle for cooperative and non-cooperative elliptic systems.