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

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.