We study the existence and non-existence of positive solutions for the (p, q)-Laplace equation −∆pu − ∆qu = α|u| p−2 u + β|u| q−2 u, where p = q, under the zero Dirichlet boundary condition in Ω. The main result of our research is the construction of a continuous curve in (α, β) plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains Ω for which the corresponding first Dirichlet eigenvalue of −∆p is not monotone w.r.t. p > 1.

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the (p, q)-Laplace equation −∆ p u−∆ q u = α|u| p−2 u+ β|u| q−2 u in a bounded domain Ω ⊂ R N under zero Dirichlet boundary condition, where p > q > 1 and α, β ∈ R. A curve on the (α, β)-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the p-and q-Laplacians under zero Dirichlet boundary condition are linearly independent.

The paper is devoted to the study of two-parametric families of Dirichlet problems for systems of equations with p, q-Laplacians and indefinite nonlinearities. Continuous and monotone curves Γ f and Γe on the parametric plane λ × µ, which are the lower and upper bounds for a maximal domain of existence of weak positive solutions are introduced. The curve Γ f is obtained by developing our previous work [4] and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve Γe is derived explicitly via minimax variational principle of the extended functional method.

Let B 1 be a ball in R N centred at the origin and B 0 be a smaller ball compactly contained in B 1 . For p ∈ (1, ∞), using the shape derivative method, we show that the first eigenvalue of the p-Laplacian in annulus B 1 \ B 0 strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 and p → ∞ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p-Laplacian on bounded radial domains. Mathematics Subject Classification (2010): 35J92, 35P30, 35B06, 49R05. Ω |∇u| p dx : u ∈ W 1,p 0 (Ω) \ {0} with u p = 1 .In this article we consider Ω of the formdenotes the open ball of radius r > 0 centred at z ∈ R N . Since the p-Laplacian is invariant under orthogonal transformations, it can be easily seen that λ 1 (B R 1 (x) \ B R 0 (y)) = λ 1 (B R 1 (0) \ B R 0 (se 1 ))

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p, q)-Laplace equations −∆ p u−∆ q u = α|u| p−2 u+ β|u| q−2 u where p = q. By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the (α, β)-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p-and q-Laplacians in one dimension.