Abstract
In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving $$p(\cdot )$$ -Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely $$\begin{aligned} \left\{ \begin{array}{rcll} -{\text {div}}(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla u)&{}=&{}\lambda f(x,u) &{} \text{ in } \Omega ,\\ u&{}=&{}0 &{} \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$ By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter $$\lambda >0$$ small enough, and also that the solution blows up, in the Sobolev norm, as $$\lambda \rightarrow 0^{+}.$$ Finally, by imposing additional hypotheses on the nonlinearity $$f(x,\cdot ),$$ we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have