Electron. J. Differential Equations, Vol. 2016 (2016), No. 261, pp. 1-17.

Solutions to semilinear elliptic PDE's with biharmonic operator and singular potential

Mousomi Bhakta

Abstract:
We study the existence and nonexistence of positive solution to the problem
$$\displaylines{ \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\text{in }\Omega,\cr u>0 \quad\text{in }\Omega,\cr u=0=\Delta u \quad\text{on }\partial\Omega, }$$
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$. We show the existence of a value $\lambda^*>0$ such that when $0<\lambda<\lambda^*$, there is a solution and when $\lambda>\lambda^*$ there is no solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$. Moreover as $\lambda\uparrow\lambda^*$, the minimal positive solution converges to a solution. We also prove that there exists $\tilde{\lambda}^*<\infty$ with $\lambda^*\leq\tilde{\lambda}^*$, and for $\lambda>\tilde{\lambda}^*$, such that the above problem does not have solution even in the distributional sense/very weak sense, and there is a complete blow-up. Under an additional integrability condition on b, we establish the uniqueness of positive solution.

Submitted February 5, 2016. Published September 28, 2016.
Math Subject Classifications: 35B09, 35B25, 35B35, 35G30, 35J91.
Key Words: Semilinear biharmonic equation; singular potential; Navier boundary condition; existence; nonexistence; blow-up phenomenon; stability; uniqueness of extremal solution.

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Mousomi Bhakta
Department of Mathematics
Indian Institute of Science Education and Research
Dr. Homi Bhaba road
Pune-411008, India
email: mousomi@iiserpune.ac.in

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