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In this paper, the existence of sign-changing solutions for p-Laplacian equation is obtained via a new linking theorem.

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

In this paper, we study two semilinear singular biharmonic equations: one with sub–critical exponent and critical potential, another with sub–critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile, we have compared the changes of the critical dimensions in singular and non–singular cases, and we get an interesting result.

In this paper, we study the existence of nontrivial solutions for the following Dirichlet problem for the p-Laplacian (p>1): $$ \left\{ {\begin{array}{*{20}l} {{ - \Delta _{p} u \equiv - {\text{div}}{\left( {{\left| {\nabla u} \right|}^{{p - 2}} \nabla u} \right)} = f{\left( {x,u} \right)},} \hfill} & {{x \in \Omega ,} \hfill} \\ {{u = 0,} \hfill} & {{x \in \partial \Omega ,} \hfill} \\ \end{array} } \right. $$ where Ω is a bonded domain in ℝ N (N≥1) and f(x,u) is quasi-asymptotically linear with respect to |u| p −2 u at infinity. Recently it was proved that the above problem has a positive solution under the condition that f(x,s)/s p −1 is nondecreasing with respect to s for all x∈Ω and some others. In this paper, by improving the methods in the literature, we prove that the functional corresponding to the above problem still satisfies a weakened version of (P.S.) condition even if f(x,s)/s p −1 isn't a nondecreasing function with respect to s, and then the above problem has a nontrivial weak solution by Mountain Pass Theorem.