On the Neumann problem with combined nonlinearities

Abstract

Abstract. We establish the existence of multiple solutions of an asymptotically linear Neumann problem. These solutions are obtained via the mountain-pass principle and a local minimization.

Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

“…Chabrowski in [15] considered the Neumann boundary problems with singular superlinear nonlinearities by approximation and variational methods. When the superlinear term is subcritical, he obtained two solutions, a mountain-pass solution and a local minimizer solution.…”

Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition

“…To conclude its proof we observe that, if f is odd, then I λ,p is even. Now, the existence of infinite many solutions follows by applying the symmetric version of the Mountain Pass Theorem, see [42,Theorem 9,12]. ✷ s , let (u n ) be a (P S) c sequence in X s p .…”

“…where {λ k } k≥1 denotes the sequence of eigenvalues of (−∆) considered in H 1 0 (Ω). In Chabrowsky and Yang [12] a problem with Neumann boundary condition was considered, while in Motreanu, Motreanu and Papageorgiou [34] the authors study a problem involving a local p-Laplacian. In [37], de Paiva and Massa studied the local problem −∆u = −λ|u| q−2 u + au + g(u) in Ω, u = 0 on ∂Ω,…”

Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

“…The linear case of Poisson's problem and (non-integral) singular Neumann boundary conditions has been discussed in [3]; see [4] for the general linear case and [5] for nonlinear boundary conditions involving a measure. The existence of positive solutions to semilinear singular elliptic problems has been studied in [6] for homogeneous Dirichlet boundary conditions; for (non-integral) homogeneous Neumann boundary conditions, see [7]. For the semilinear problem (1) with (possibly singular) integral Neumann boundary condition (2), existence of very weak solutions u ∈ L p (Ω), 1 < p < ∞, with zero average ∈ L pq p−rq (Ω × ∂Ω) for some 1 ≤ q < (p ) * , 0 ≤ r < max(p/q , 1) and 0 < s < 1, I.e., in the subcritical (w.r.t.…”