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We consider a nonlinear Dirichlet boundary value problem involving the $p(x)$-Laplacian and a concave term. Our main result shows the existence of at least three nontrivial solutions. We use truncation techniques and the method of sub- and supersolutions.
We study the uniqueness of weak solutions for Dirichlet problems with variable exponent and non-standard growth conditions. First, we provide two uniqueness results under ellipticity type hypotheses. Next, we provide a uniqueness result when the operator driving the problem is in the form of the divergence of a monotone map. Finally, we derive a fourth uniqueness result under homogeneity type hypotheses, by means of a comparison result and approximation.
A nonautonomous second order system with a nonsmooth potential is studied. It is assumed that the system is asymptotically linear at infinity and resonant (both at infinity and at the origin), with respect to the zero eigenvalue. Also, it is assumed that the linearization of the system is indefinite. Using a nonsmooth variant of the reduction method and the local linking theorem, we show that the system has at least two nontrivial solutions.
We consider second order periodic systems with a nonsmooth potential and an indefinite linear part. We impose conditions under which the nonsmooth Euler functional is unbounded. Then using a nonsmooth variant of the reduction method and the nonsmooth local linking theorem, we establish the existence of at least two nontrivial solutions.
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