“…In this connection we observe that actually the conclusion of Theorem 3.1 holds if in the conditions of the theorem we assume that the inequality p 2−1/n is satisfied instead of the inequality p > 2−1/n (see [ This result was established by the first author in [2, Theorem 1.5.6]. The same conclusion as in the given theorem under the conditions p 2−1/n and n/(np−n+1) < m < p * /(p * −1) has already been obtained in [9].…”
Section: Existence and Nonexistence Of Solutions Of Second-order Equasupporting
Abstract. We establish conditions of the nonexistence of weak solutions of the Dirichlet problem for nonlinear elliptic equations of arbitrary even order with some righthand sides from L m where m > 1. The conditions include the requirement of a certain closeness of the parameter m to 1.
AMS Subject Classifications: 35J25, 35J40, 35J60Chinese Library Classifications: O175.8, O175.25, O175.29Key Words: Nonlinear elliptic equations in divergence form; Dirichlet problem; weak solution; existence and nonexistence of weak solutions.
“…In this connection we observe that actually the conclusion of Theorem 3.1 holds if in the conditions of the theorem we assume that the inequality p 2−1/n is satisfied instead of the inequality p > 2−1/n (see [ This result was established by the first author in [2, Theorem 1.5.6]. The same conclusion as in the given theorem under the conditions p 2−1/n and n/(np−n+1) < m < p * /(p * −1) has already been obtained in [9].…”
Section: Existence and Nonexistence Of Solutions Of Second-order Equasupporting
Abstract. We establish conditions of the nonexistence of weak solutions of the Dirichlet problem for nonlinear elliptic equations of arbitrary even order with some righthand sides from L m where m > 1. The conditions include the requirement of a certain closeness of the parameter m to 1.
AMS Subject Classifications: 35J25, 35J40, 35J60Chinese Library Classifications: O175.8, O175.25, O175.29Key Words: Nonlinear elliptic equations in divergence form; Dirichlet problem; weak solution; existence and nonexistence of weak solutions.
“…An equality of the form (4) is used in [1] to determine the gradient of elements of a functional class containing the set • T 1,p (Ω). The direct definition of the functions δ i u for u ∈ • T 1,p (Ω) by equality (3) and the proof of Proposition 1 can be found in [2]. Lemma 1 is, in fact, proved in [3].…”
Section: Remark 1 the Setmentioning
confidence: 92%
“…Lemma 1 is, in fact, proved in [3]. For similar results with more exact estimates used instead of (5), see [4,5]. For θ = 1, inequalities (7) and (8) are established under condition (6) in [1].…”
517.9For sequences of functions from a Sobolev space satisfying special integral estimates, we , in one case, establish a lemma on the choice of pointwise convergent subsequences and, in a different case, prove a theorem on convergence of the corresponding sequences of generalized derivatives in measure. These results are applied to the problem of existence of the entropy solutions of nonlinear equations with degenerate coercivity and L 1 -data.
“…Finally, we remark that L 1,λ ( ) is not contained nor contains L log L( ) as well as L 1 n n−1 ( ) (see Remark 2.3 and Appendix); consequently, our regularity result is independent of that studied in [5,6,11,17].…”
Section: Introductionmentioning
confidence: 91%
“…We remark that a solution u of the problem (1) (very weak or entropy or distributional) belongs only to 1≤q< n n−1 W 1,q 0 ( ); the best regularity for Du, that is Du ∈ L n n−1 ( ), can be achieved under the additional assumption f ∈ L log L( ) or f ∈ L 1 n n−1 ( ) (see [5,6,11,17,21,22]). …”
We prove that if f belongs to the Morrey space L 1,λ ( ), with λ ∈ [0, n − 2], and u is the solution of the problemthen Du belongs to the space L q,λ ( ), for any q ∈ [1, n n−1 [.
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