Quasilinear degenerated equations withL1datum and without coercivity in perturbation terms

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

“…The estimate (15) follows from (18) and (17). Finally, the inequality (16) follows from (19) and (17).…”

“…Taking into account Hölder inequality, (4), an inclusion H l n O E fju l j < k 1 g, (15), and (93), we established that for every i 2 f1; : : : ; ng …”

“…Moreover, in [14], the existence of weak solutions to the Dirichlet problem for nondegenerate anisotropic equations with right-hand sides from Lebesgues spaces close to L 1 was established. Solvability of the Dirichlet problem for degenerate isotropic equations with L 1 -data and measures as data was studied in [15][16][17][18][19]. Remark that in [15,17], the existence of entropy solutions to the given problem was proved in the case of L 1 -data, and in [16], the existence of a renormalized solution of the problem for the same case was established.…”

“…Solvability of the Dirichlet problem for degenerate isotropic equations with L 1 -data and measures as data was studied in [15][16][17][18][19]. Remark that in [15,17], the existence of entropy solutions to the given problem was proved in the case of L 1 -data, and in [16], the existence of a renormalized solution of the problem for the same case was established. In [16,18,19], the existence of distributional solutions of the problem was obtained in the case of right-hand side measures.…”

“…The exponents q i characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions i characterize degeneration or singularity of the coefficients with respect to the independent variables. This is the most general situation in comparison with the above-mentioned works: the nondegenerate isotropic case means that i Á 1 and q i D q 1 , i D 1; : : : ; n; the nondegenerate anisotropic case means that i Á 1, i D 1; : : : ; n, and q i , i D 1; : : : ; n, are generally different, and the degenerate isotropic case means that i D 1 , i D 1; : : : ; n, as in [16][17][18][19] or i , i D 1; : : : ; n, are generally different as in [15] but q i D q 1 , i D 1; : : : ; n.…”