We consider a nonlinear elliptic equation driven by the sum of a -Laplacian and a -Laplacian, where <!-- MATH $1<q\leq 2\leq p<\infty$ --> <img width="172" height="37" align="MIDDLE" border="0" src="images/img4.gif" alt="$ 1<q\leq 2\leq p<\infty $"> with a -superlinear Carathéodory reaction term which doesn't satisfy the usual Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at least three nontrivial solutions; among them one is positive and one is negative.

We consider a nonlinear eigenvalue problem driven by the sum of and -Laplacians. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ()-differential operator as the parameter goes to .

We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is ( p - 1 ) ${{(p-1)}}$ -superlinear near ± ∞ ${\pm\infty}$ without satisfying the Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.