Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities. (English) Zbl 1210.35309
Summary: The main goal of this paper is to present multiple solution results for elliptic inclusions of Clarke’s gradient type under nonlinear Neumann boundary conditions involving the \(p\)-Laplacian and set-valued nonlinearities. To be more precise, we study the inclusion
\[ -\Delta_pu\in\partial F(x,u)-|u|^{p-2}u \quad\text{in }\Omega \]
with the boundary condition
\[ |\nabla u|^{p-2} \frac{\partial u}{\partial\nu}\in a(u^+)^{p-1}- b(u^-)^{p-1}+\partial G(x,u) \quad\text{on }\partial\Omega. \]
We prove the existence of two constant-sign solutions and one sign-changing solution depending on the parameters \(a\) and \(b\). Our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth Mountain-Pass Theorem, the Second Deformation Lemma for locally Lipschitz functionals as well as comparison results of local \(C^1(\overline{\Omega})\)-minimizers and local \(W^{1,p}(\Omega)\)-minimizers of nonsmooth functionals.
\[ -\Delta_pu\in\partial F(x,u)-|u|^{p-2}u \quad\text{in }\Omega \]
with the boundary condition
\[ |\nabla u|^{p-2} \frac{\partial u}{\partial\nu}\in a(u^+)^{p-1}- b(u^-)^{p-1}+\partial G(x,u) \quad\text{on }\partial\Omega. \]
We prove the existence of two constant-sign solutions and one sign-changing solution depending on the parameters \(a\) and \(b\). Our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth Mountain-Pass Theorem, the Second Deformation Lemma for locally Lipschitz functionals as well as comparison results of local \(C^1(\overline{\Omega})\)-minimizers and local \(W^{1,p}(\Omega)\)-minimizers of nonsmooth functionals.
MSC:
| 35R70 | PDEs with multivalued right-hand sides |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B51 | Comparison principles in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
Keywords:
Clarke’s generalized gradient; mountain-pass theorem; multiple solutions; sub-supersolution method; \(p\)-LaplacianReferences:
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