Discontinuous quasilinear elliptic problems at resonance

In this paper we examine nonlinear elliptic equations driven by the p-Laplacian and with a discontinuous forcing term. To develop an existence theory we pass to an elliptic inclusion by filling in the gaps at the discontinuity points of the forcing term. We prove three existence theorems. The first is a multiplicity result and proves the existence of two bounded solutions one strictly positive and the other strictly negative. The other two theorems deal with problems at resonance and prove the existence of solutions using Landesman-Lazer type conditions.

Journal of Mathematical Analysis and Applications, 1999

We study a quasilinear elliptic equation at resonance with discontinuous right hand side. To have an existence theory, we pass to a multivalued version of the problem by filling in the gaps at the discontinuity points. Using the nonsmooth critical point theory of Chang for locally Lipschitz functionals and the Ekeland variational principle, we show that the resulting elliptic inclusion has three distinct nontrivial solutions.

Nonlinear Analysis: Theory, Methods & Applications, 1994

Nonlinear Analysis: Theory, Methods & Applications, 2010

In this paper, using nonsmooth critical point theory, we study the existence of multiple solutions of the following class of problems: −div (|x| −ap |∇u| p−2 ∇u) ∈ λ|x| −(a+1)p+c [f − (u), f + (u)], x ∈ Ω, u = 0, x ∈ ∂Ω, where a, c, λ and f − , f + satisfy certain conditions.

Journal of Mathematical Analysis and Applications, 1989

The main purpose of this paper is to study elliptic boundary value problems of the type i -du=f(u)+p(x), XC!2 u = 0, .-tEaR, (*I where Q is a bounded domain in RN and f has, possibly, "upward" discontinuities. The idea is to find solutions of (*) by using Clarke's Dual Action Principle [S]. This approach has a remarkable smoothing effect, in the sense that it allows one to look for solutions of (*) as critical points of a functional which, in spite of the discontinuity of A is C '. The Variational Principle is discussed in Section 1 and is applied in Section 2 to (*), leading one to proofs of existence and multiplicity results for various kinds of nonlinearities. Among other things, we can find the results of both [4, 51 and [ 111 in a quite direct way. Notations. Q is a bounded domain in RN, N >, 2, with smooth boundary af2;2; ).ly denotes the norm in Ly(Q), q 2 1; (., .) denotes the scalar product in L*(Q); * Supported by the Minister0 P.I. (40%), Gruppo Naz. "Calcolo delle Variazioni.

Glasgow Mathematical Journal, 2000

In this paper we study quasilinear second order boundary value problems with multivalued right hand side and Dirichlet boundary conditions. We prove three existence theorems. The ®rst two deal with the``convex'' and``nonconvex'' problems respectively, while the third establishes the existence of extremal solutions. For the ®rst two the proof is based on the theory of nonlinear operators of monotone type, while the proof of the third uses a ®xed point argument.

2001

The aim of this paper is to study the existence of solutions for a quasilinear elliptic system where the nonlinear term is a Caratheodory function on a bounded domain of R N , by proving the well known unique continuation property for elliptic system in all dimensions: 1, 2, 3, . . . and the strict monotonocity of eigensurfaces. These properties let us to consider the above problem as a nonresonance problem. * Mathematics Subject Classifications: 35J05, 35J45, 35J65.

Journal of Mathematical Analysis and Applications, 1978

Nonlinear Analysis: Theory, Methods & Applications, 1984

Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan

We study a strong solvability of the Dirichlet problem n i,j=1 a ij (x)u xixj + g(x, u) = f (x), x ∈ Ω, u| ∂Ω = 0 for a class of semilinear elliptic equations with discontinuous coefficients satisfying the Cordes condition. For this problem we get the existence results inẆ 2 2 (Ω) Sobolev space whenever the norm f L2(Ω) is sufficiently small. We also have proved the strong solvability of the Dirichlet problem in spaceẆ 2 p (Ω) with 1 < p < ∞ for the equation considered above with continuous leading coefficients and a small f Lp(Ω) norm.