Non-autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, -\Delta_p^a u-\Delta_q u =\lambda m(x) |u|^{q-2}u \quad \mbox{in} \,\, \R^N, where {$N \geq 2$}, {$1<p, q<N$, $p \neq q$}, ${a \in C^{0, 1}(\R^N, [0, +\infty))}$, $a \not\equiv 0$ and $m: \R^N \to \R$ is {an indefinite sign weight which may admit nontrivial positive and negative parts}. Here $\Delta_q$ is the $q$-Laplacian operator and $\Delta_p^a$ is the weighted $p$-Laplace operator defined by $\Delta_p^a u:=\textnormal{div}(a(x) |\nabla u|^{p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of $a$ in $\R^N$ may be zero. Our main results distinguish between the cases $p<q$ and $q<p$. In the first case, we establish the existence of a {\it continuous} family of eigenvalues, starting from the principal frequency of a suitable single phase eigenvalue problem. In the latter case, we prove the existence of a {\it discrete} family of positive eigenvalues, which diverges to infinity.Comment: 16 page

Existence of infinitely many solutions for degenerate Kirchhoff-type Schrodinger-Choquard equations

In this article we study a class of Kirchhoff-type Schrodinger-Choquard equations involving the fractional p-Laplacian. By means of Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of $\lambda$. The main feature and difficulty of our equations arise in the fact that the Kirchhoff term M could vanish at zero, that is, the problem is degenerate. To our best knowledge, our result is new even in the framework of Schrodinger-Choquard problems

Partial differential equations with variable exponents: variational methods and qualitative analysis

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive met

Nonhomogeneous equations with critical exponential growth and lack of compactness

We study the existence and multiplicity of positive solutions for the following class of quasilinear problems [formula] where e&isin; is a positive parameter. We assume that [formula] is a continuous potential and &fnof; : R &rarr; R is a smooth reaction term with critical exponential growth