Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

Abstract

Abstract:The existence of multiple solutions to a Dirichlet problem involving the (p, q)-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of −∆ p in W ,p (Ω) and exhibits a concave behavior near zero.

On a Dirichlet problem with (p,q)-Laplacian and parametric concave-convex nonlinearity

“…It should be noted that, unlike our case, λ always multiplies the concave term, which changes the analysis of the problem. Finally, [4,14,23] contain analogous bifurcation theorems for problems of a different kind, whereas [20,21] study (p, q)-Laplace equations having merely concave right-hand side.…”

Nodal solutions for nonlinear nonhomogeneous Robin problems

“…From (10), (12), (16) and (17), we infer that u * is a positive solution of problem (11). From Papageorgiou & Rȃdulescu [20], we have u * ∈ L ∞ (Ω).…”

“…Moreover, reasoning as in the proof of Proposition 5 (with k(z, x) replaced bŷ f (z, x)), we show that Proof. Let u ∈ S + and letk(z, x) be given by (12). We introduce the following truncation ofk(z, ·):…”

“…Next, in (21) we choose h = (û * − u) + ∈ W 1,p (Ω). We have (19), (12), (10) and recall that u ∈ S + )…”

Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations

“…Finally, Section 4.3 deals with asymmetric nonlinearities, meaning that the asymptotic behavior at −∞ and +∞ is different. Due to limited space, we relied only on [10,12,16], [21]- [24], [26]- [29], [31,32], [38]- [40], where bounded domains are considered, and refer to [8,11,15,17] for the case Ω := R N .…”