On nonlinear elliptic problems with discontinuities

Boletim da Sociedade Paranaense de Matemática

In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problems with resonance part and mixed boundary conditions. Our approach is variational and is based on the well known Landesman-Laser type conditions.

Nodea-nonlinear Differential Equations and Applications, 2008

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole $\mathbb {R}^N$ . Via variational arguments, we obtain existence and regularity of nontrivial solutions.

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, 1981

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.

Boundary Value Problems, 2012

We study the Dirichlet problem for linear elliptic second order partial differential equations with discontinuous coefficients in divergence form in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in L p , p > 2. MSC: 35J25; 35B45; 35R05

Journal of Differential Equations, 1978

We prove the existence of a nontrivial solution for the nonlinear elliptic problem −∆u = λh(x)u + a(x)g(u) in R N , where g is superlinear near zero and near infinity, a(x) changes sign, λ is positive, and h(x) 0 is a weight function. For g odd, we prove the existence of an infinite number of solutions. * Mathematics Subject Classifications: 35J25, 35J20, 58E05.

Demonstratio Mathematica, 2022

In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problem with resonance part and nonlinear boundary conditions. Our approach is variational and is based on the well-known Landesman-Laser-type conditions.

Nonlinear Differential Equations and Applications NoDEA, 2008

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole R N . Via variational arguments, we obtain existence and regularity of nontrivial solutions.

Mathematics and Computers in Simulation, 2003

We study the existence of the weak solution of the nonlinear boundary value problem where p and λ are real numbers, p > 1, h ∈ L p (0, π )(p = p/(p -1)) and the nonlinearity g : R → R is a continuous function of the Landesman-Lazer type. Our results generalize previously published results about the solvability of our problem.