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_λ(x)u+( M(x)Du,Du)+h(x),\qquad u\in H_0^1(Ω)\cap L^\infty(Ω), \end{align*} where $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, is a bounded domain with a low-regularity boundary $\partial \Omega$. The coefficients $c,h\in L^p(\Omega )$ for some $p>n$, with $c^{\pm }\ge 0$ 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 ($\lambda \le 0$) 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.

25 pages, 5 figures