On an anisotropic p-Laplace equation with variable singular exponent

Abstract

We consider Finsler p-Laplace equations under pure and perturbed singular nonlinearities to establish the existence of unique and multiple solutions respectively. Our main emphasis is the case of the variable singular exponent. We employ the method of approximation and variational approach. To the best of our knowledge, the main results are new also for the p = 2 case in the context of the Finsler p-Laplace operator.

Weighted singular problem with variable singular exponent and $p$-admissible weights

“…Remark 2.7. Noting Remark 2.6 and (1.2), Theorem 2.5 extends the existence result in[9, Theorem 1.4] to the weighted anisotropic setting and 1 < p < 2, γ > 1.…”

“…We would like to emphasize that authors in [17] used the algebraic inequality (see Peral [41]) to obtain the approximate solutions associated to the singular p-Laplace equation. Moreover, in the anisotropic case, use of an analogous algebraic inequality resulted the restriction p ≥ 2 in [9]. Here, we show that the such inequality can be replaced by the strict monotonicity hypothesis (H 3 ), which further help us to remove the restriction p ≥ 2 in [9].…”

“…Moreover, in the anisotropic case, use of an analogous algebraic inequality resulted the restriction p ≥ 2 in [9]. Here, we show that the such inequality can be replaced by the strict monotonicity hypothesis (H 3 ), which further help us to remove the restriction p ≥ 2 in [9]. We prove existence of the approximate solutions by a direct implementation of the Minty-Browder theorem, see Lemma 3.2.…”

“…In this article, we provide sufficient conditions on the weight function w, the singular exponent γ and on the nonlinearity f to ensure existence of weak solutions of (1.1). We remark that the weights w and the operator −div(A(x, ∇u)) that we considered in (1.1) is more general than considered in [7,8,17,26,28] and in contrast to [9], we complement on the existence and uniqueness results for 1 < p < 2 and γ > 1 in the weighted setting.…”

“…When γ is a variable, Carmona-Martínez-Aparicio [18] provided sufficient conditions on γ to establish existence of weak solutions for the problem (1.5). Such phenomenon has been extended by many authors to the variants of p-Laplace equations, see Chu-Gao-Gao [19], Zhang [42], Alves-Santos-Siqueira [3], Miri [38], Alves-Moussaoui [2], Byun-Ko [15], Papageorgiou-Scapellato [40], Bal-Garain-Mukherjee [9,29] and the references therein. We would like to point out that for singular problems, when γ > 1, in general solutions u does not belong to W 1,p 0 (Ω).…”