Regularity of the extremal solutions associated to elliptic systems

Abstract

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -Δu =λf(v) \quad \mbox{ in } Ω -Δv =γf(u) \quad \mbox{ in } Ω, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $\lambda ,\gamma$ are positive parameters, $\Omega$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^2$ positive, nondecreasing and convex function in $[0,\infty )$ such that $\frac{f(t)}{t}\to \infty$ as $t\to \infty$. Assuming $$0<{\tau }_{-}:=\underset{t\to \infty }{\mathrm{liminf}}\frac{f(t)f"(t)}{f^{\prime }(t{)}^2}\le {\tau }_{+}:=\underset{t\to \infty }{\mathrm{limsup}}\frac{f(t)f"(t)}{f^{\prime }(t{)}^2}\le 2,$$ we show that the extremal solution $(u^{\ast },v^{\ast })$ associated to the above system is smooth provided\\ $N<\frac{2{\alpha }_{\ast }(2-{\tau }_{+})+2{\tau }_{+}}{{\tau }_{+}}\max \{1,{\tau }_{+}\}$, where ${\alpha }_{\ast }>1$ denotes the largest root of the $2^{nd}$ order polynomial $$P_f(\alpha ,{\tau }_{-},{\tau }_{+}):=(2-{\tau }_{-}{)}^2{\alpha }^2-4(2-{\tau }_{+})\alpha +4(1-{\tau }_{+}).$$ As a consequences, $u^{\ast },v^{\ast }\in L^{\infty }(\Omega )$ for $N<5$. Moreover, if ${\tau }_{-}={\tau }_{+}$, then $u^{\ast },v^{\ast }\in L^{\infty }(\Omega )$ for $N<10$.

References in corpus (1)

• Regularity of minimizers of semilinear elliptic problems up to dimension four