Abstract
Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the -Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fu ik spectrum of the -Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.
Highlights
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω
Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the p-Laplacian
We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities
Summary
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω. We consider the following nonlinear elliptic boundary value problem. The Fucik spectrum for the p-Laplacian with a nonlinear boundary condition is defined as the set Σp of a, b ∈ R × R such that. One can show that φ1 belongs to L∞ Ω cf., 23, Lemma 5.6 and Theorem 4.3 or 24, Theorem 4.1 , and along with the results of Lieberman in 25, Theorem 2 it holds that φ1 ∈ C1,α Ω This fact combined with φ1 x > 0 in Ω yields φ1 ∈ int C1 Ω , where int C1 Ω denotes the interior of the positive cone C1 Ω {u ∈ C1 Ω : u x ≥ 0, ∀x ∈ Ω} in the Banach space C1 Ω , given by int C1 Ω u ∈ C1 Ω : u x > 0, ∀x ∈ Ω. As pointed out in 1 , there exists a unique solution e ∈ int C1 Ω of problem 1.9 which is required for the construction of sub- and supersolutions of problem 1.1
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