Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient

TL;DR

This paper studies the existence, multiplicity, and bifurcation of solutions for a class of quasilinear elliptic equations with quadratic gradient growth, providing a comprehensive analysis of solution behavior depending on a parameter.

Contribution

It introduces new results on solution existence, uniqueness, and multiplicity for quasilinear elliptic problems with quadratic gradient growth, including bifurcation analysis and solution continuum characterization.

Findings

Existence and uniqueness for coercive case ($\,\lambda \leq 0$)

Multiplicity of solutions for non-coercive case ($\lambda > 0$)

Identification of bifurcation points and solution continuum

Abstract

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_\lambda$} -\mathrm{div}(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x),\qquad u\in H_0^1(\Omega)\cap L^\infty(\Omega), \end{align*} where $\Omega\subset\mathbb{R}^n$, $n\geq 3$, is a bounded domain with a low-regularity boundary $\partial\Omega$. The coefficients $c, h \in L^p(\Omega)$ for some $p > n$, with $c^\pm \geq 0$ and $c_\lambda(x) := \lambda c^+(x) - c^-(x)$ for a real parameter $\lambda$. The matrix $A(x)$ is uniformly positive definite and bounded, while $M(x)$ is positive definite and bounded. Under suitable assumptions, we characterize the solution continuum of $(P_\lambda)$, including its bifurcation points. We establish existence and uniqueness results in the coercive case ($\lambda \leq 0$) and prove multiplicity results in the non-coercive case ($\lambda > 0$). \bigskip \textbf{Keywords}: Quasilinear elliptic equations, quadratic growth on the gradient, sub and super solutions.