This article deals with the study of the following nonlinear doubly nonlocal equation:where Ω is a bounded domain in R n with smooth boundary, 1 < δ ≤ q ≤ p < r ≤ p * s1 , with p * s1 = np n − ps 1 , 0 < s 2 < s 1 < 1, n > ps 1 and λ, β > 0 are parameters. Here a ∈ L r r−δ (Ω) and b ∈ L ∞ (Ω) are sign changing functions. We prove the L ∞ estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case (r < p * s1 ). Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex-concave problem. In case of δ = q, we show the existence of solution.

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional ( p , q ) {(p,q)} -Laplacian, denoted by ( - Δ ) p s 1 + ( - Δ ) q s 2 {(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}} for s 2 , s 1 ∈ ( 0 , 1 ) {s_{2},s_{1}\in(0,1)} and 1 < p , q < ∞ {1<p,q<\infty} . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional ( p , q ) {(p,q)} -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.

In this article, we study the existence and multiplicity of solutions of the following (p, q)-Laplace equation with singular nonlinearity:where Ω is a bounded domain in R n with smooth boundary, 1 < q < p < r p * , where p * = np n−p , 0 < δ < 1, n > p and λ, β > 0 are parameters. We prove existence, multiplicity and regularity of weak solutions of (P λ ) for suitable range of λ. We also prove the global existence result for problem (P λ ).

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