Existence and Multiplicity Results for a Class of Coupled Quasilinear Elliptic Systems of Gradient Type

Abstract

The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \textup{(P)} { - div ⁡ ( A ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 - 2 ⁢ ∇ ⁡ u ) + 1 p 1 ⁢ A u ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 = G u ⁢ ( x , u , v ) in Ω , - div ⁡ ( B ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 - 2 ⁢ ∇ ⁡ v ) + 1 p 2 ⁢ B v ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 = G v ⁢ ( x , u , v ) in Ω , u = v = 0 on ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1% }-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&amp;\displaystyle=G_{u}(% x,u,v)&amp;&amp;\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{% p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&amp;\displaystyle=G_{v}(x,u,v)&amp;&amp;\displaystyle% \phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle u=v&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is an open bounded domain, p 1 {p_{1}} , p 2 &gt; 1 {p_{2}&gt;1} and A ⁢ ( x , u ) {A(x,u)} , B ⁢ ( x , v ) {B(x,v)} are 𝒞 1 {\mathcal{C}^{1}} -Carathéodory functions on Ω × ℝ {\Omega\times\mathbb{R}} with partial derivatives A u ⁢ ( x , u ) {A_{u}(x,u)} , respectively B v ⁢ ( x , v ) {B_{v}(x,v)} , while G u ⁢ ( x , u , v ) {G_{u}(x,u,v)} , G v ⁢ ( x , u , v ) {G_{v}(x,u,v)} are given Carathéodory maps defined on Ω × ℝ × ℝ {\Omega\times\mathbb{R}\times\mathbb{R}} which are partial derivatives of a function G ⁢ ( x , u , v ) {G(x,u,v)} . We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional 𝒥 {{\mathcal{J}}} , related to problem (P), admits at least one critical point in the “right” Banach space <jats:ital

Existence of minimizers for a quasilinear elliptic system of gradient type

“…Step 3. The proof can be obtained by arguing as in the proof of [11, Proposition 4.9]) (see also [10,Theorem 4.9]) but when the system has only two equations and replacing assumptions (3.5) and (3.6) in [11] with (2.3) and (2.4) and using conditions (4.19) and the fact that we can choose s 5 such that    1 < s 5 < p 2 (1 + σ 2 ) 0 ≤ s 6 < p2p * 1 N (1 + σ 1 ), instead of (3.24) and (3.25) in [11]. Steps 4. and 5.…”

“…To this aim, following the approach introduced in [4,5] for quasilinear equations and extended in [10] to quasilinear systems, we look for solution of (1.1) as critical points of the functional:…”

Nontrivial solutions for a class of gradient-type quasilinear elliptic systems

“…. , m}, u = 0 on ∂Ω, with Āi,t (x, t) = ∂ Āi ∂t (x, t), which has been studied in [12] if m = 2, and generalizes the classical gradient-type (p 1 , . .…”

“…On the contrary, here we use the variational approach introduced in [7,8], and already applied to systems in [12], so that, under suitable hypotheses, finding solutions of problem (1.1) turns into searching critical points of the functional…”

“…Differently from the problem in [12], here we deal with a more general system. In spite of it, a regularity result on J in X can be proved under basic assumptions on G(x, u) and suitable growth conditions on the C 1 -Carathéodory function A i (x, t, ξ) and its partial derivatives, i ∈ {1, .…”