Multiplicity of nontrivial solutions for quasilinear elliptic equation
The purpose of this paper is to study multiplicity of nontrivial solutiond of the following quasilinear equation \[ \begin{cases} -\Delta _pu = f(x,u)\,\, & \mathrm{in}\,\, \Omega ,\\ u =0 \,\, & \mathrm{on}\,\, \partial \Omega , \end{cases} \tag{1} \] where \(\Omega \subset {\mathbb R}^N \,\, (N\geq 3)\) is a bounded smooth domain, \(1<p<N\) and \(\Delta _p= \mathrm{div }\, (| \nabla u | ^{p-2} \nabla u )\) is the \(p\)-Laplacian operator. The nonlinearity \(f\in C(\overline{\Omega } \times {\mathbb R} ,{\mathbb R})\) satisfies the subcritical growth condition: there are \(c_1>0\) and \(q \in (p,p^*)\) such that \[ | f(x,t) | \leq c_1 (1+| t | ^{q-1}),\quad \forall (x,t) \in \Omega \times {\mathbb R}, \tag{2} \] where \(p^+= Np/(N-p)\) is the critical Sobolev exponent of \(p\). First consider the weighted eigenvalue problem \[ \begin{cases} -\Delta _pu = \lambda m(x) | u | ^{p-2} & \mathrm{in}\,\, \Omega ,\\ u=0 & \mathrm{on}\,\, \partial \Omega , \end{cases} \tag{3} \] where \(m(x) \in L^{N/p}(\Omega )\) is a weight function which is positive on a subset of positive measure. Then if one defines the manifold \[ \Sigma = \biggl\{ u \in W^{1,2}_0(\Omega ) ; \int _{\Omega } m(x) | u| ^p dx =1\biggr\}, \] then \(\Sigma \) is a smooth, symmetric non-empty submanifold of \(W^{1,2}_0(\Omega )\). Denote by \({\mathcal A}\) the class of compact symmetric subsets of \(\Sigma \), let \(\Sigma _k = \{ A \in {\mathcal A}; i(A)\geq k\}\) where \(i\) is the cohomological index of Fadell and Rabinowitz. The eigenvalues of (3) can be variationally characterized as \[ \lambda _k (m) = \inf _{A \in \Sigma _k} \sup _{u \in A} \int _{\Omega } | \nabla u | ^p dx. \] They refer to \(\{\lambda _k(m)\}_{k\in {\mathbb N}}\) as the variational eigenvalues of (3). Let \(F(x,t) = \int _0^t f(x,s)ds\), and assume the following \[ \lim _{| t | \to \infty } \frac{pF(x,t)}{| t | ^p} = m_{\infty }(x) \quad \mathrm{uniformly}\,\, \mathrm{ in } \,\,x \in \Omega , \tag{4} \] \[ \lim _{| t | \to 0 } \frac{pF(x,t)}{| t | ^p} = m_{0 }(x) \quad \mathrm{uniformly}\,\, \mathrm{ in }\,\, x \in \Omega , \tag{5} \] \[ \lim _{| t | \to \infty }(f(x,t)t -pF(x,t) ) = \infty \quad \mathrm{uniformly} \mathrm{ in } \,\, x \in \Omega . \tag{6} \] The authors then obtain the following theorem \textbf{Theorem 1.} Assume that \(F\) satisfies (2), (4) and (5). If \(\lambda _k (m_0)<1<\lambda _{k+1}(m_0)\) and one of the following conditions holds: (i) \(\lambda _1(m_{\infty })>1\) or (ii) \(\lambda _1(m_{\infty })=1\) and (6) hold. Then problem (1) has at least two nontrivial solutions. In order to prove the theorem, they use a three-critical point theorem (Theorem 2) related to local linking.
scientific article; zbMATH DE number 6005683
scientific article