Semilinear elliptic problems involving a parameter
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Date
2020-06-29
Authors
Yasmin, Lubana
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Abstract
We consider
\begin{equation}\label{hum}
\left\{
\begin{array}{rrl}
-Δ u &=& λ f(u) \quad \mbox{ in } Ω \\
u&=& 0 \qquad \mbox{ on } ∂Ω,
\end{array}
\right.
\end{equation} in a bounded domain $Ω⊂ \IR^N$. The nonlinear term $f$ is smooth, positive, increasing, convex, superlinear at ∞, and $λ>0$ a parameter. We also consider (\ref{hum}) in case of nonlinearity $\frac{1}{(1-u)^2}$ (MEMS nonlinearity), and with a divergence free advection term, a(x) ($-Δ u+a(x)⋅ ∇ u=λ f(u)$ in $Ω$ and $u=0$ on $∂Ω$). In this thesis, we are interested in talking about the existence of stable minimal solutions to these partial differential equations (pde's). We show, when $λ<λ^*$ (a critical parameter), there is a minimal stable solution and when $λ>λ^*$, there exists no solution. Here, stability of solution means nonnegativeness of the first eigenvalue of the linearized operator associated with the pde. This nonnegative inequality can also be viewed as the second variation of energy functional associated with the pde at $u$. At $λ^*$, we obtain a unique weak solution which is the limit of minimal solutions ($\lim_{λ↗ λ^*} u_λ↗ u^*$), we call it extremal solution. Properties of extremal solution depend strongly on $Ω, f, N$.
For (\ref{hum}), the extremal solution is smooth in $N≤ 9$ with $f(u)=e^u$ while it is singular for $N≥10$, $Ω=B_1$. The best result is by Nedev, which says $u^*$ is bounded for any $f$ and $Ω$ when $N≤ 3$. We discuss the radial case which shows the optimal regularity result for $u^*$ in $N≤ 9$. For the MEMS model, all stable solutions are smooth iff the dimension is $N≤ 7$. For the pde with advection, there is no suitable variational characterization for the stability assumption. To overcome this difficulty, we use a general version of Hardy's inequality to show smoothness of extremal solution in dimension $N≤ 9$ with exponential nonlinearity.
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Keywords
Semilinear elliptic partial differential equation, Sobolev space, Regularity, Extremal solution, Gelfand problem, MEMS model, Radial solution, Linearized operator