The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via suband supersolution methods as well as variational techniques for nonsmooth functions. 2000 Mathematics subject classification: 35J20, 35J85, 49J40.

We consider the system ⎧ ⎨ ⎩ −Δ p u = λF (x, u, v), x ∈ Ω, −Δ q v = λH(x, u, v), x ∈ Ω, u = 0 = v, x ∈ ∂Ω, where F (x, u, v) = [g(x)a(u) + f (v)], H(x, u, v) = [g(x)b(v) + h(u)], Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω, λ is a real positive parameter and Δ s z = div (|∇z| s−2 ∇z), s > 1, (s = p, q) is a s-laplacian operator. Here g is a C 1 sign-changing function that may b...

In this paper, we study the multiplicity results of positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Morse theory and the Ljusternik-Schnirelmann category, we investigate how the coefficient h(x) of the critical nonlinearity affects the number of positive solutions of that problem and we get some results as regards the...

Consider the eigenvalue problem generated by a fixed differential operator with sign-changing weight on term. We prove that as part of is rescaled towards negative infinity some subregion, spectrum converges to original restricted complementary region. On interface between regions limiting acquires Dirichlet-type boundary conditions. Our main theorem concerns problems for bilinear forms Hilbert...