Multiple solutions for a coercive quasilinear elliptic equation via Morse theory

We study the quasilinear elliptic problem which is resonant at zero. By using Morse theory, we obtain five nontrivial solutions for the equation with coercive nonlinearities.

Radial quasilinear elliptic problems with singular or vanishing potentials

<p style='text-indent:20px;'>In this paper we continue the work that we began in [<xref ref-type="bibr" rid="b6">6</xref>]. Given <inline-formula><tex-math id="M1">\begin{document}$ 1&lt;p&lt;N $\end{document}</tex-math></inline-formula>, two measurable functions <inline-formula><tex-math id="M2">\begin{document}$ V\left(r \right)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ K\left(r\right)&gt; 0 $\end{document}</tex-math></inline-formula>, and a continuous function <inline-formula><tex-math id="M4">\begin{document}$ A(r) &gt;0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$ r&gt;0 $\end{document}</tex-math></inline-formula>), we consider the quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where all the potentials <inline-formula><tex-math id="M6">\begin{document}$ A,V,K $\end{document}</tex-math></inline-formula> may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space <inline-formula><tex-math id="M7">\begin{document}$ X $\end{document}</tex-math></inline-formula> into the sum of Lebesgue spaces <inline-formula><tex-math id="M8">\begin{document}$ L_{K}^{q_{1}}+L_{K}^{q_{2}} $\end{document}</tex-math></inline-formula>. The nonlinearity has a double-power super <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>-linear behavior, as <inline-formula><tex-math id="M10">\begin{document}$ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ q_1,q_2&gt;p $\end{document}</tex-math></inline-formula> (recovering the power case if <inline-formula><tex-math id="M12">\begin{document}$ q_1 = q_2 $\end{document}</tex-math></inline-formula>). With respect to [<xref ref-type="bibr" rid="b6">6</xref>], in the present paper we assume some more hypotheses on <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula>, and we are able to enlarge the set of values <inline-formula><tex-math id="M14">\begin{document}$ q_1 , q_2 $\end{document}</tex-math></inline-formula> for which we get existence results.</p>

On singular quasilinear elliptic equations with data measures

The aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

Lions-type theorem of the p-Laplacian and applications

In this article, our aim is to establish a generalized version of Lions-type theorem for the p-Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

<div class="abstract"> <p> <div>&lt;div class="abstract"&gt; &lt;div&gt;&lt;p&gt;This paper deals with second-order optimality conditions for a quasilinear elliptic&lt;br /&gt;control problem with a nonlinear coefficient in the principal part that is countably PC&lt;sup&gt;2&lt;/sup&gt;&lt;br /&gt;(continuous and C&lt;sup&gt;2&lt;/sup&gt; apart from countably many points). We prove that the control-to-state&lt;br /&gt;operator is continuously differentiable even though the nonlinear coefficient is non-smooth.&lt;br /&gt;This enables us to establish &amp;ldquo;no-gap&amp;rdquo; second-order necessary and sufficient optimality&lt;br /&gt;conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition&lt;br /&gt;only differs from the necessary one in the fact that the inequality is strict. A condition that&lt;br /&gt;is equivalent to the second-order sucient optimality condition and could be useful for&lt;br /&gt;error estimates in, e.g., finite element discretizations is also provided.&lt;/p&gt; &lt;/div&gt; &lt;/div&gt;</div> </p> </div>