Mieko Tanaka

Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient

AbstractWe provide the existence of a positive solution for the quasilinear elliptic equation −div(a(x,|∇u|)∇u)=f(x,u,∇u) in Ω under the Dirichlet boundary condition. As a special case (a(x,t)=tp−2), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of f(x,tξ) as t is small. In this paper, we do not impose the sign condition to the nonlinear term f.MSC:35J92, 35P30.

On sign-changing solutions for (p,q)-Laplace equations with two parameters

Abstract We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous ( p , q ) {(p,q)} -Laplace equations - Δ p ⁢ u - Δ q ⁢ u = α ⁢ | u | p - 2 ⁢ u + β ⁢ | u | q - 2 ⁢ u {-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p ≠ q {p\neq q} . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the ( α , β ) {(\alpha,\beta)} -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.

Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

We provide the existence of a solution for quasilinear elliptic equation −div(𝑎∞(𝑥)|∇𝑢|𝑝−2∇𝑢

On sign-changing solutions for resonant (p,q)-Laplace equations

We provide two existence results for sign-changing solutions to the Dirichlet problem for the family of equations −Δpu−Δqu = α |u|p−2u+β |u|q−2u , where 1 < q < p and α , β are parameters. First, we show the existence in the resonant case α ∈ σ(−Δp) for sufficiently large β , thereby generalizing previously known results. The obtained solutions have negative energy. Second, we show the existence for any β λ1(q) and sufficiently large α under an additional nonresonant assumption, where λ1(q) is the first eigenvalue of the q -Laplacian. The obtained solutions have positive energy.

Remarks on minimizers for ( p , q )-Laplace equations with two parameters

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the

Multiple existence results of solutions for the Neumann problems via super- and sub-solutions

(p,q)

Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition

-Laplace equation

Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient

-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u

Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities

in a bounded domain

Generalized eigenvalue problems for (p,q)-Laplacian with indefinite weight

\Omega \subset \mathbb{R}^N