Existence of infinitely many solutions for some superlinear elliptic equations
Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ N\) with smooth boundary \(\partial \Omega\) and let g(x,t) be a continuous function which is odd in the t argument and has superlinear growth, i.e., g(x,t)/t\(\to \infty\) as \(| t| \to \infty\) uniformly in \(x\in {\bar \Omega}\). Under an additional, slightly weaker condition on g than by \textit{P. H. Rabinowitz} [Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)], the authors prove by minimax methods that the semilinear problem \[ \Delta u+g(x,u)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] has an unbounded sequence of weak solutions in \(H^ 1_ 0(\Omega)\). A similar result is obtained for the problem \[ \Delta u+g(x,u)+f(x)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(f\in L^{\mu /(\mu -1)}(\Omega)\), \(\mu >1\), and g satisfies an alternative third requirement.
scientific article; zbMATH DE number 4179920
scientific article