Existence Solutions for a Weighted Biharmonic Equation with Critical Exponential Growth

Abstract

We study a weighted N 2 biharmonic equation involving a positive continuous potential in B. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of R N . It is proved that there is a nontrivial weak solution to this problem by the mountain Pass Theorem. We avoid the loss of compactness by proving a concentration compactness result and by a suitable asymptotic condition.

Ground states of weighted 4D biharmonic equations with exponential growth

“…Consequently, they establish the existence of a ground state weak solution by applying the mountain pass theorem and the Nehari manifold technique. Recently, Dridi et al [20], investigate the existence of least energy nodal solutions for the following nonlocal weighted Schrödinger–Kirchhoff problem: $$$$ \left\{\begin{array}{rcll}{\mathcal{L}}_{\left(\rho, \xi \right)}(u)& =& f\left(x,u\right)& \mathrm{in}\kern3.0235pt \hspace{0.1em}B\\ {}u& =& 0& \mathrm{on}\kern3.0235pt \hspace{0.1em}\partial B,\end{array}\right. $$$$ where $$$ B $$$ is the unit ball of $$$ {\mathrm{\mathbb{R}}}^N,N>2 $$$.…”

“…Consequently, they establish the existence of a ground state weak solution by applying the mountain pass theorem and the Nehari manifold technique. Recently, Dridi et al [20], investigate the existence of least energy nodal solutions for the following nonlocal weighted Schrödinger-Kirchhoff problem:…”

Weigthed elliptic equation of Kirchhoff type with exponential non linear growthWeigthed elliptic equation of Kirchhoff type with exponential non linear growth

“…The existence of solutions to nonlinear weighted elliptic equations involving subcritical and critical growth of the logarithmic weighted Trudinger-Moser type or weighted Adam's inequalities has been studied extensively in recent years, motivated by its applicability in many fields of modern mathematics; see ( [1,8,14,27,28,32]) for a survey on this subject.…”