Anisotropic singular double phase Dirichlet problems

Abstract

<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

Singular equations with variable exponents and concave-convex nonlinearities

“…Therefore by Proposition A4 of Papageorgiou-Rȃdulescu-Zhang [19], we conclude that for δ ∈ (0, 1) small m δ 0 < u(z) ∀z ∈ Ω 0 , a contradiction to the definition of m 0 . Hence λ ∈ L and we conclude that λ * λ 0 < +∞.…”

“…(Ω) is a positive solution of (P λ ) (λ ∈ (0, 1) small). Proposition A1 of Papageorgiou-Rȃdulescu-Zhang [19] implies that u λ ∈ L ∞ (Ω) and then the singular anisotropic regularity theory (Saoudi-Ghanmi [22]) and the singular anisotropic maximum principle (see Papageorgiou-Rȃdulescu-Zhang [19, Proposition A3]), imply that u λ ∈ intC + .…”

“…We mention that problems with competing nonlinearities (concave-convex problems) and with no singular terms, were first examined by Ambrosetti-Brézis-Cerami [1] (semilinear equations driven by the Laplacian) and García Azorero-Manfredi-Peral Alonso [7] (nonlinear equations driven by the p-Laplacian). Anisotropic singular equations with no concave terms were studied only recently and we refer to the work of Byun-Ko [2], Saoudi-Ghanmi [22] (equations driven by the p(z)-Laplacian) and by Papageorgiou-Rȃdulescu-Zhang [19], Papageorgiou-Winkert [21] (equations driven by the (p(z), q(z))-Laplacian).…”

Anisotropic $(p,q)$-equations with gradient dependent reaction

“…(Ω) is bounded. Then the anisotropic regularity theory (see [19], [38]) implies that (32) y n ∈ L ∞ (Ω), y n ∞ C 9 for some C 9 > 0, all n ∈ N.…”

“…We set xk = w k n ρ p (Dw k n ) 1/p(z) ∈ W 1,p(z) 0(Ω), k ∈ N 0 . On(34) we act with w k n ∈ W 1,p(z) 0(Ω) and obtainρ p (Dw k n ) + ρ q (Dw k n ) = Ω g vn (z, w k−1 n )w k n dz ⇒ ρ p (Dx k ) − Ĉk + vn (z, w k−1 n ) (w k−1 n ) p(z)−1 xp(z)−1 k−1 xk dz, ⇒ ρ(D xk ) Ω g vn (z, w k−1 n ) (w k−1 n ) p(z)−1 xp(z)−1 k−1 xk dz + Ĉk for all k ∈ N.(37)From the proof of Lemma 4, we know that(38) 0 < C 11 xk C 12 for all k k 0 , some 0 < C 11 C 12 .…”