Existence of solutions for degenerate quasilinear elliptic equations

Existence of solutions for some doubly degenerate parabolic equations with natural growth terms

“…In the case that b(u) = u and α(0) = 0, existence results are proved in [20,32], existence and uniqueness results are shown in [33] with F = 0, and existence results to the stationary case of (P) can be found in [25,31]. We point out that the distributional solution u obtained in [20] for p = 2, may not belong to L 2 (0, T ; H 1 0 (Ω )), that is, to define ∇u almost everywhere in Q, one may either use the notion of a generalized gradient as in [3] or employ the way as in (2.8) of [6].…”

Existence of weak solutions for some nonlinear elliptic equations with variable exponents

“…Remark 3.1 It is well known that if p 4 1 is a constant function, then one may obtain an L 1 estimate for u provided that q 4 N p and r 4 N pÀ1 (see [26] and the references therein or [25]). It is easy to find that Lemma 3.1 extends the result in the constant exponent case to the variable exponents case.…”

“…Assume that 0 and F 2 (L r ()) N with r 4 N pÀ1 , Rakotoson [24] has proved the existence of bounded solutions to some elliptic variational inequalities related to problem (P). In [25], the existence of weak solutions to problem (P) is obtained for 0, g(x, s) 0, f 2 L q (), F 2 (L r ()) N with q 4 N p and r 4 N pÀ1 . The p(x)-Laplacian is an extension of the p-Laplacian.…”

Existence results for some nonlinear degenerate problems in the anisotropic spaces

“…Published August 25, 2018 where Ω is an open bounded subset of R N , (N ≥ 2), for i = 1, ..., N , a i (x, u, ∇u) is a Carathéodory function and there exists a continuous and bounded function ν: [0, +∞) → [0, +∞) such that ν(0) = 0 and N i=1 a i (x, s, ξ)ξ i ≥ N i=1 ν(|s|)|ξ| pi for every s ∈ R, ξ ∈ R N and a.e x in Ω, and H(x, u, ∇u) is a nonlinear term has a growth condition, and without a sign condition, the source data f and g = (g 1 , ..., g N ) belonging a suitable Lebesgue spaces (see assumptions A 6 )). In problem (1.1), when the norm b L r (Ω) in the growth of H i (x, u, ∇u), is not small enough, the operator becomes non-coercive, moreover, the problem (1.1) is degenerate since its modulus of ellipticity vanishes when either the solution u or its gradient ∇u vanishes [22,25]. Anisotropic operators involve today in various domains of applied Sciences, they provide models for the study of physical and mechanical processus in anisotropic continuous medium ( [11,24]).…”