Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension

This paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} $$ { − m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) Δ u + V ( x ) u = f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $N=1,2$ N = 1 , 2 , $m:[0,\infty )\rightarrow (0,\infty )$ m : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, $V:\mathbb{R} ^{N}\rightarrow \mathbb{R} $ V : R N → R is differentiable, and $f\in \mathcal{C}(\mathbb{R} ,\mathbb{R} )$ f ∈ C ( R , R ) . We obtain the existence of a ground state solution of Nehari–Pohozaev type and the least energy solution under some assumptions on V, m, and f. Especially, the existence of nonlocal term $m(\|\nabla u\|^{2}_{L^{2}(\mathbb{R} ^{N})})$ m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) and the lack of Hardy’s inequality and Sobolev’s inequality in low dimension make the problem more complicated. To overcome the above-mentioned difficulties, some new energy inequalities and subtle analyses are introduced.

SOBOLEV’S INEQUALITY FOR MUSIELAK–ORLICZ–MORREY SPACES OVER METRIC MEASURE SPACES

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$ . As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Sobolev’s inequality for double phase functionals with variable exponents

Our aim in this paper is to establish generalizations of Sobolev’s inequality for double phase functionals {\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}}, where {p(\,{\cdot}\,)} and {q(\,{\cdot}\,)} satisfy log-Hölder conditions and {a(\,{\cdot}\,)} is nonnegative, bounded and Hölder continuous of order {\theta\in(0,1]}.