Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations (Q1113369)

In connection with the maximizing problem for the functional \(R(u)=\| u\|_{L^ q}/\| \nabla u\|_{L^ p}\) in \(W_ 0^{1,p}(\Omega)\setminus \{0\}\), we consider the equation \[ (E)\quad - div(| \nabla u|^{p-2} \nabla u(x))=| u|^{q-2} u(x),\quad x\in \Omega,\quad 1<p,\quad q<\infty,\quad p\neq q,\quad u(x)=0,\quad x\in \partial \Omega. \] It is shown that for the case \(q<p^*\) \((p^*=\infty\) if \(p\geq N\), and \(p^*=Np/(N-p)\) if \(p<N)\), (E) has always a nonnegative nontrivial solution belonging to \(W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega),\) and for the case \(p<N\) and \(q<p^*\) (resp. \(q=p^*)\), (E) has no nontrivial (resp. nonnegative nontrivial) solution belonging to the class \[ P=\{u\in W_ 0^{1,p}(\Omega)\cap L^ q(\Omega);\quad x_ i| u|^{q-2} u\in L^{p/(p-1)}(\Omega),\quad i=1,2,...,N\}\subset W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega), \] provided that \(\Omega\) is star shaped. The crucial point of the proof of our result is to obtain an \(L^{\infty}\)-estimate of weak solutions and to verify a certain ``Pohozaev-type inequality'' for weak solutions belonging to P.