Search results for " Multiplicity of solutions"
showing 3 items of 3 documents
Multiple positive solutions for singularly perturbed elliptic problems in exterior domains
2003
Abstract The equation − e 2 Δ u + a e ( x ) u = u p −1 with boundary Dirichlet zero data is considered in an exterior domain Ω = R N ⧹ ω ( ω bounded and N ⩾2). Under the assumption that a e ⩾ a 0 >0 concentrates round a point of Ω as e →0, that p >2 and p N /( N −2) when N ⩾3, the existence of at least three positive distinct solutions is proved.
Existence and multiplicity of solutions for Dirichlet problems involving nonlinearities with arbitrary growth.
2014
In this article we study the existence and multiplicity of solutions for the Dirichlet problem $$\displaylines{ -\Delta_p u=\lambda f(x,u)+ \mu g(x,u)\quad\hbox{in }\Omega,\cr u=0\quad\hbox{on } \partial \Omega }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are Caratheodory functions, and $\lambda,\mu$ are nonnegative parameters. We impose no growth condition at $\infty$ on the nonlinearities f,g. A corollary to our main result improves an existence result recently obtained by Bonanno via a critical point theorem for $C^1$ functionals which do not satisfy the usual sequential weak lower semicontinuity property.
Three solutions for parametric problems with nonhomogeneous (a,2)-type differential operators and reaction terms sublinear at zero
2019
Abstract We consider parametric Dirichlet problems driven by the sum of a Laplacian and a nonhomogeneous differential operator ( ( a , 2 ) -type equation) and with a reaction term which exhibits arbitrary polynomial growth and a nonlinear dependence on the parameter. We prove the existence of three distinct nontrivial smooth solutions for small values of the parameter, providing sign information for them: one is positive, one is negative and the third one is nodal.