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In this paper, we present global existence results for the following problemwhere ϕ p (x) = |x| p−2 x, p > 1, λ a positive parameter and h a nonnegative measurable function on (0, 1) which may be singular at t = 0 and/or t = 1, and f ∈ C(R + , R + ) with R + = [0, ∞). By applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions, we obtain many different types of global existence results of positive solutions. We also obtain existence results of signchanging solutions for (P λ ) when f is an odd symmetric function.

We prove the existence of ground state positive solutions for a class of semipositone p-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform C1,α a priori estimates and by concentration compactness arguments. Our results are new even in the semilinear case p = 2.

Abstract. We show the various existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators. A suitable condition on degeneracy is discussed and proofs are mainly based on direct methods and critical point theories in Calculus of Variations. In particular, we investigate the various situations of the growth rates between principal operators and nonlinearities.

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