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

TL;DR

This work addresses existence, nonexistence, and multiplicity for a class of quasilinear elliptic problems with quadratic gradient growth in divergence form: $-\nabla\cdot(A(x)\nabla u)=c_\lambda(x)u+(M(x)\nabla u,\nabla u)+h(x)$ in a bounded domain, with $c_\lambda(x)=\lambda c^+(x)-c^-(x)$. The authors develop a continuum/degree-theory framework built on the Boundary Weak Harnack Inequality, an exponential change of variables, and fixed-point arguments to construct a closed, connected set of solutions \(\Sigma\) and to analyze bifurcation from the axis $\lambda=0$. They prove uniqueness in the coercive regime $\lambda\le 0$ and multiplicity for certain noncoercive regimes $\lambda>0$, describing the global bifurcation diagram: unbounded continua of solutions emanating from infinity to the right of the axis and two-solution branches for small positive $\lambda$ under suitable sign conditions on $u_0$ solving $(P_0)$. Specializing to $h\equiv 0$ yields a sharp two-solution structure and corroborating corollaries. The results extend previous scalar cases to general divergence-form operators with sign-changing lower-order coefficients and rougher domain boundaries, advancing understanding of quasilinear PDEs with quadratic gradient growth and their bifurcation behavior.

Abstract

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_λ$} -\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 $Ω\subset\mathbb{R}^n$, $n\geq 3$, is a bounded domain with a low-regularity boundary $\partialΩ$. The coefficients $c, h \in L^p(Ω)$ for some $p > n$, with $c^\pm \geq 0$ and $c_λ(x) := λc^+(x) - c^-(x)$ for a real parameter $λ$. 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 $(P_λ)$, including its bifurcation points. We establish existence and uniqueness results in the coercive case ($λ\leq 0$) and prove multiplicity results in the non-coercive case ($λ> 0$). \bigskip \textbf{Keywords}: Quasilinear elliptic equations, quadratic growth on the gradient, sub and super solutions.

Figures (3)

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• Figure : Illustration of Theorem 1.3
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Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Strong Maximum Principle - SMP
• Theorem 2.2: Boundary Weak Harnack Inequality, BWHI
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5